WORST_CASE(?,O(n^1))
proof of input_lmGH4pRUVq.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 110 ms]
(2) CpxRelTRS
(3) RelTrsToWeightedTrsProof [UPPER BOUND(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) InliningProof [UPPER BOUND(ID), 1017 ms]
(14) CpxRNTS
(15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxRNTS
(17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) CpxRNTS
(19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(20) CpxRNTS
(21) IntTrsBoundProof [UPPER BOUND(ID), 264 ms]
(22) CpxRNTS
(23) IntTrsBoundProof [UPPER BOUND(ID), 59 ms]
(24) CpxRNTS
(25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 130 ms]
(28) CpxRNTS
(29) IntTrsBoundProof [UPPER BOUND(ID), 43 ms]
(30) CpxRNTS
(31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 128 ms]
(34) CpxRNTS
(35) IntTrsBoundProof [UPPER BOUND(ID), 43 ms]
(36) CpxRNTS
(37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(38) CpxRNTS
(39) IntTrsBoundProof [UPPER BOUND(ID), 254 ms]
(40) CpxRNTS
(41) IntTrsBoundProof [UPPER BOUND(ID), 116 ms]
(42) CpxRNTS
(43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(44) CpxRNTS
(45) IntTrsBoundProof [UPPER BOUND(ID), 1048 ms]
(46) CpxRNTS
(47) IntTrsBoundProof [UPPER BOUND(ID), 1475 ms]
(48) CpxRNTS
(49) FinalProof [FINISHED, 0 ms]
(50) BOUNDS(1, n^1)


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   a(C(x1, x2), y) -> C(a(x1, y), a(x2, C(x1, x2)))
   a(Z, y) -> Z
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
   eqZList(C(x1, x2), Z) -> False
   eqZList(Z, C(y1, y2)) -> False
   eqZList(Z, Z) -> True
   second(C(x1, x2)) -> x2
   first(C(x1, x2)) -> x1

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

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

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


The TRS R consists of the following rules:

   a(C(x1, x2), y) -> C(a(x1, y), a(x2, C(x1, x2)))
   a(Z, y) -> Z
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
   eqZList(C(x1, x2), Z) -> False
   eqZList(Z, C(y1, y2)) -> False
   eqZList(Z, Z) -> True
   second(C(x1, x2)) -> x2
   first(C(x1, x2)) -> x1

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(3) RelTrsToWeightedTrsProof (UPPER BOUND(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^1).


The TRS R consists of the following rules:

   a(C(x1, x2), y) -> C(a(x1, y), a(x2, C(x1, x2))) [1]
   a(Z, y) -> Z [1]
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) [1]
   eqZList(C(x1, x2), Z) -> False [1]
   eqZList(Z, C(y1, y2)) -> False [1]
   eqZList(Z, Z) -> True [1]
   second(C(x1, x2)) -> x2 [1]
   first(C(x1, x2)) -> x1 [1]
   and(False, False) -> False [0]
   and(True, False) -> False [0]
   and(False, True) -> False [0]
   and(True, True) -> True [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:

   a(C(x1, x2), y) -> C(a(x1, y), a(x2, C(x1, x2))) [1]
   a(Z, y) -> Z [1]
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) [1]
   eqZList(C(x1, x2), Z) -> False [1]
   eqZList(Z, C(y1, y2)) -> False [1]
   eqZList(Z, Z) -> True [1]
   second(C(x1, x2)) -> x2 [1]
   first(C(x1, x2)) -> x1 [1]
   and(False, False) -> False [0]
   and(True, False) -> False [0]
   and(False, True) -> False [0]
   and(True, True) -> True [0]

The TRS has the following type information:
a :: C:Z -> C:Z -> C:Z
C :: C:Z -> C:Z -> C:Z
Z :: C:Z
eqZList :: C:Z -> C:Z -> False:True
and :: False:True -> False:True -> False:True
False :: False:True
True :: False:True
second :: C:Z -> C:Z
first :: C:Z -> C:Z

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:

   a_2
   second_1
   first_1

(c) The following functions are completely defined:

   eqZList_2
   and_2

Due to the following rules being added:

   and(v0, v1) -> null_and [0]

And the following fresh constants:    null_and

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

(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:

   a(C(x1, x2), y) -> C(a(x1, y), a(x2, C(x1, x2))) [1]
   a(Z, y) -> Z [1]
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2)) [1]
   eqZList(C(x1, x2), Z) -> False [1]
   eqZList(Z, C(y1, y2)) -> False [1]
   eqZList(Z, Z) -> True [1]
   second(C(x1, x2)) -> x2 [1]
   first(C(x1, x2)) -> x1 [1]
   and(False, False) -> False [0]
   and(True, False) -> False [0]
   and(False, True) -> False [0]
   and(True, True) -> True [0]
   and(v0, v1) -> null_and [0]

The TRS has the following type information:
a :: C:Z -> C:Z -> C:Z
C :: C:Z -> C:Z -> C:Z
Z :: C:Z
eqZList :: C:Z -> C:Z -> False:True:null_and
and :: False:True:null_and -> False:True:null_and -> False:True:null_and
False :: False:True:null_and
True :: False:True:null_and
second :: C:Z -> C:Z
first :: C:Z -> C:Z
null_and :: False:True:null_and

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:

   a(C(x1, x2), y) -> C(a(x1, y), a(x2, C(x1, x2))) [1]
   a(Z, y) -> Z [1]
   eqZList(C(C(x1', x2'), C(x11, x21)), C(C(y1', y2'), C(y11, y21))) -> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) [3]
   eqZList(C(C(x1', x2'), C(x12, x22)), C(C(y1', y2'), Z)) -> and(and(eqZList(x1', y1'), eqZList(x2', y2')), False) [3]
   eqZList(C(C(x1', x2'), Z), C(C(y1', y2'), C(y12, y22))) -> and(and(eqZList(x1', y1'), eqZList(x2', y2')), False) [3]
   eqZList(C(C(x1', x2'), Z), C(C(y1', y2'), Z)) -> and(and(eqZList(x1', y1'), eqZList(x2', y2')), True) [3]
   eqZList(C(C(x1'', x2''), C(x13, x23)), C(Z, C(y13, y23))) -> and(False, and(eqZList(x13, y13), eqZList(x23, y23))) [3]
   eqZList(C(C(x1'', x2''), C(x14, x24)), C(Z, Z)) -> and(False, False) [3]
   eqZList(C(C(x1'', x2''), Z), C(Z, C(y14, y24))) -> and(False, False) [3]
   eqZList(C(C(x1'', x2''), Z), C(Z, Z)) -> and(False, True) [3]
   eqZList(C(Z, C(x15, x25)), C(C(y1'', y2''), C(y15, y25))) -> and(False, and(eqZList(x15, y15), eqZList(x25, y25))) [3]
   eqZList(C(Z, C(x16, x26)), C(C(y1'', y2''), Z)) -> and(False, False) [3]
   eqZList(C(Z, Z), C(C(y1'', y2''), C(y16, y26))) -> and(False, False) [3]
   eqZList(C(Z, Z), C(C(y1'', y2''), Z)) -> and(False, True) [3]
   eqZList(C(Z, C(x17, x27)), C(Z, C(y17, y27))) -> and(True, and(eqZList(x17, y17), eqZList(x27, y27))) [3]
   eqZList(C(Z, C(x18, x28)), C(Z, Z)) -> and(True, False) [3]
   eqZList(C(Z, Z), C(Z, C(y18, y28))) -> and(True, False) [3]
   eqZList(C(Z, Z), C(Z, Z)) -> and(True, True) [3]
   eqZList(C(x1, x2), Z) -> False [1]
   eqZList(Z, C(y1, y2)) -> False [1]
   eqZList(Z, Z) -> True [1]
   second(C(x1, x2)) -> x2 [1]
   first(C(x1, x2)) -> x1 [1]
   and(False, False) -> False [0]
   and(True, False) -> False [0]
   and(False, True) -> False [0]
   and(True, True) -> True [0]
   and(v0, v1) -> null_and [0]

The TRS has the following type information:
a :: C:Z -> C:Z -> C:Z
C :: C:Z -> C:Z -> C:Z
Z :: C:Z
eqZList :: C:Z -> C:Z -> False:True:null_and
and :: False:True:null_and -> False:True:null_and -> False:True:null_and
False :: False:True:null_and
True :: False:True:null_and
second :: C:Z -> C:Z
first :: C:Z -> C:Z
null_and :: False:True:null_and

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:

   Z => 0
   False => 1
   True => 2
   null_and => 0

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

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

   a(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   a(z, z') -{ 1 }-> 1 + a(x1, y) + a(x2, 1 + x1 + x2) :|: x1 >= 0, y >= 0, z = 1 + x1 + x2, z' = y, x2 >= 0
   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22)
   eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27)
   eqZList(z, z') -{ 3 }-> and(2, 2) :|: z = 1 + 0 + 0, z' = 1 + 0 + 0
   eqZList(z, z') -{ 3 }-> and(2, 1) :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0
   eqZList(z, z') -{ 3 }-> and(2, 1) :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0
   eqZList(z, z') -{ 3 }-> and(1, 2) :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0
   eqZList(z, z') -{ 3 }-> and(1, 2) :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0
   eqZList(z, z') -{ 3 }-> and(1, 1) :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0
   eqZList(z, z') -{ 3 }-> and(1, 1) :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0
   eqZList(z, z') -{ 3 }-> and(1, 1) :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0
   eqZList(z, z') -{ 3 }-> and(1, 1) :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0
   eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0
   eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0
   eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0
   first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0
   second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0


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

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

   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1

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

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

   a(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   a(z, z') -{ 1 }-> 1 + a(x1, y) + a(x2, 1 + x1 + x2) :|: x1 >= 0, y >= 0, z = 1 + x1 + x2, z' = y, x2 >= 0
   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22)
   eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27)
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0
   eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0
   eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2
   eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0
   eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0
   second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0


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

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

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

   a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0
   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22)
   eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27)
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0
   eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0
   eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2
   eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0
   eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0
   second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0


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

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

   { and }
   { first }
   { second }
   { a }
   { eqZList }

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

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

   a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0
   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22)
   eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27)
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0
   eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0
   eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2
   eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0
   eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0
   second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0

Function symbols to be analyzed: {and}, {first}, {second}, {a}, {eqZList}

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

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

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

   a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0
   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22)
   eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27)
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0
   eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0
   eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2
   eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0
   eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0
   second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0

Function symbols to be analyzed: {and}, {first}, {second}, {a}, {eqZList}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0
   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22)
   eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27)
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0
   eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0
   eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2
   eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0
   eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0
   second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0

Function symbols to be analyzed: {and}, {first}, {second}, {a}, {eqZList}
Previous analysis results are:
  and: runtime: ?, size: O(1) [2]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0
   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22)
   eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27)
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0
   eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0
   eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2
   eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0
   eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0
   second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0

Function symbols to be analyzed: {first}, {second}, {a}, {eqZList}
Previous analysis results are:
  and: runtime: O(1) [0], size: O(1) [2]

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

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

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

   a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0
   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22)
   eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27)
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0
   eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0
   eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2
   eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0
   eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0
   second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0

Function symbols to be analyzed: {first}, {second}, {a}, {eqZList}
Previous analysis results are:
  and: runtime: O(1) [0], size: O(1) [2]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0
   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22)
   eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27)
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0
   eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0
   eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2
   eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0
   eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0
   second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0

Function symbols to be analyzed: {first}, {second}, {a}, {eqZList}
Previous analysis results are:
  and: runtime: O(1) [0], size: O(1) [2]
  first: runtime: ?, size: O(n^1) [z]

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

(29) 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: 1

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

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

   a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0
   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22)
   eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27)
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0
   eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0
   eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2
   eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0
   eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0
   second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0

Function symbols to be analyzed: {second}, {a}, {eqZList}
Previous analysis results are:
  and: runtime: O(1) [0], size: O(1) [2]
  first: runtime: O(1) [1], size: O(n^1) [z]

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

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

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

   a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0
   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22)
   eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27)
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0
   eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0
   eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2
   eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0
   eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0
   second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0

Function symbols to be analyzed: {second}, {a}, {eqZList}
Previous analysis results are:
  and: runtime: O(1) [0], size: O(1) [2]
  first: runtime: O(1) [1], size: O(n^1) [z]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0
   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22)
   eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27)
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0
   eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0
   eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2
   eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0
   eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0
   second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0

Function symbols to be analyzed: {second}, {a}, {eqZList}
Previous analysis results are:
  and: runtime: O(1) [0], size: O(1) [2]
  first: runtime: O(1) [1], size: O(n^1) [z]
  second: runtime: ?, size: O(n^1) [z]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0
   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22)
   eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27)
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0
   eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0
   eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2
   eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0
   eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0
   second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0

Function symbols to be analyzed: {a}, {eqZList}
Previous analysis results are:
  and: runtime: O(1) [0], size: O(1) [2]
  first: runtime: O(1) [1], size: O(n^1) [z]
  second: runtime: O(1) [1], size: O(n^1) [z]

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

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

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

   a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0
   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22)
   eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27)
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0
   eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0
   eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2
   eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0
   eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0
   second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0

Function symbols to be analyzed: {a}, {eqZList}
Previous analysis results are:
  and: runtime: O(1) [0], size: O(1) [2]
  first: runtime: O(1) [1], size: O(n^1) [z]
  second: runtime: O(1) [1], size: O(n^1) [z]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0
   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22)
   eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27)
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0
   eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0
   eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2
   eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0
   eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0
   second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0

Function symbols to be analyzed: {a}, {eqZList}
Previous analysis results are:
  and: runtime: O(1) [0], size: O(1) [2]
  first: runtime: O(1) [1], size: O(n^1) [z]
  second: runtime: O(1) [1], size: O(n^1) [z]
  a: runtime: ?, size: O(n^1) [z]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   a(z, z') -{ 1 }-> 1 + a(x1, z') + a(x2, 1 + x1 + x2) :|: x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0
   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22)
   eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27)
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0
   eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0
   eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2
   eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0
   eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0
   second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0

Function symbols to be analyzed: {eqZList}
Previous analysis results are:
  and: runtime: O(1) [0], size: O(1) [2]
  first: runtime: O(1) [1], size: O(n^1) [z]
  second: runtime: O(1) [1], size: O(n^1) [z]
  a: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z]

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

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

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

   a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   a(z, z') -{ 3 + 2*x1 + 2*x2 }-> 1 + s + s' :|: s >= 0, s <= x1, s' >= 0, s' <= x2, x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0
   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22)
   eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27)
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0
   eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0
   eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2
   eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0
   eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0
   second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0

Function symbols to be analyzed: {eqZList}
Previous analysis results are:
  and: runtime: O(1) [0], size: O(1) [2]
  first: runtime: O(1) [1], size: O(n^1) [z]
  second: runtime: O(1) [1], size: O(n^1) [z]
  a: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   a(z, z') -{ 3 + 2*x1 + 2*x2 }-> 1 + s + s' :|: s >= 0, s <= x1, s' >= 0, s' <= x2, x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0
   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22)
   eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27)
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0
   eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0
   eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2
   eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0
   eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0
   second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0

Function symbols to be analyzed: {eqZList}
Previous analysis results are:
  and: runtime: O(1) [0], size: O(1) [2]
  first: runtime: O(1) [1], size: O(n^1) [z]
  second: runtime: O(1) [1], size: O(n^1) [z]
  a: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z]
  eqZList: runtime: ?, size: O(1) [2]

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: eqZList
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 78 + 156*z'

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

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

   a(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   a(z, z') -{ 3 + 2*x1 + 2*x2 }-> 1 + s + s' :|: s >= 0, s <= x1, s' >= 0, s' <= x2, x1 >= 0, z' >= 0, z = 1 + x1 + x2, x2 >= 0
   and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
   and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
   and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
   and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), and(eqZList(x11, y11), eqZList(x21, y21))) :|: x1' >= 0, x2' >= 0, y2' >= 0, y21 >= 0, y11 >= 0, x11 >= 0, x21 >= 0, z = 1 + (1 + x1' + x2') + (1 + x11 + x21), y1' >= 0, z' = 1 + (1 + y1' + y2') + (1 + y11 + y21)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 2) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: x1' >= 0, x2' >= 0, y2' >= 0, z' = 1 + (1 + y1' + y2') + 0, x12 >= 0, x22 >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + (1 + x12 + x22)
   eqZList(z, z') -{ 3 }-> and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, y2' >= 0, y1' >= 0, z = 1 + (1 + x1' + x2') + 0, z' = 1 + (1 + y1' + y2') + (1 + y12 + y22)
   eqZList(z, z') -{ 3 }-> and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, x17 >= 0, y27 >= 0, z = 1 + 0 + (1 + x17 + x27), y17 >= 0, z' = 1 + 0 + (1 + y17 + y27)
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, z' = 1 + 0 + (1 + y13 + y23), x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), y13 >= 0, y23 >= 0, x2'' >= 0
   eqZList(z, z') -{ 3 }-> and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z = 1 + 0 + (1 + x15 + x25), z' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0
   eqZList(z, z') -{ 1 }-> 2 :|: z = 0, z' = 0
   eqZList(z, z') -{ 3 }-> 2 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, 2 = 2
   eqZList(z, z') -{ 1 }-> 1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0, z' = 0
   eqZList(z, z') -{ 1 }-> 1 :|: y1 >= 0, z' = 1 + y1 + y2, y2 >= 0, z = 0
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 1 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, 2 = 2, 1 = 1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, z' = 1 + 0 + (1 + y14 + y24), x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z' = 1 + 0 + 0, x1'' >= 0, z = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x16 + x26), y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: y26 >= 0, z = 1 + 0 + 0, y16 >= 0, z' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, y2'' >= 0, y1'' >= 0, z' = 1 + (1 + y1'' + y2'') + 0, v0 >= 0, v1 >= 0, 1 = v0, 2 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + (1 + x18 + x28), z' = 1 + 0 + 0, x28 >= 0, x18 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + (1 + y18 + y28), y18 >= 0, y28 >= 0, v0 >= 0, v1 >= 0, 2 = v0, 1 = v1
   eqZList(z, z') -{ 3 }-> 0 :|: z = 1 + 0 + 0, z' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v0, 2 = v1
   first(z) -{ 1 }-> x1 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0
   second(z) -{ 1 }-> x2 :|: x1 >= 0, z = 1 + x1 + x2, x2 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  and: runtime: O(1) [0], size: O(1) [2]
  first: runtime: O(1) [1], size: O(n^1) [z]
  second: runtime: O(1) [1], size: O(n^1) [z]
  a: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z]
  eqZList: runtime: O(n^1) [78 + 156*z'], size: O(1) [2]

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(49) FinalProof (FINISHED)
Computed overall runtime complexity
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(50)
BOUNDS(1, n^1)