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

0 CpxRelTRS
↳1 RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CpxWeightedTrs
↳3 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CpxTypedWeightedTrs
↳5 CompletionProof (UPPER BOUND(ID), 0 ms)
↳6 CpxTypedWeightedCompleteTrs
↳7 NarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)
↳8 CpxTypedWeightedCompleteTrs
↳9 CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID), 4 ms)
↳10 CpxRNTS
↳11 InliningProof (UPPER BOUND(ID), 889 ms)
↳12 CpxRNTS
↳13 SimplificationProof (BOTH BOUNDS(ID, ID), 0 ms)
↳14 CpxRNTS
↳15 CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID), 0 ms)
↳16 CpxRNTS
↳17 IntTrsBoundProof (UPPER BOUND(ID), 291 ms)
↳18 CpxRNTS
↳19 IntTrsBoundProof (UPPER BOUND(ID), 48 ms)
↳20 CpxRNTS
↳21 ResultPropagationProof (UPPER BOUND(ID), 1 ms)
↳22 CpxRNTS
↳23 IntTrsBoundProof (UPPER BOUND(ID), 157 ms)
↳24 CpxRNTS
↳25 IntTrsBoundProof (UPPER BOUND(ID), 43 ms)
↳26 CpxRNTS
↳27 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳28 CpxRNTS
↳29 IntTrsBoundProof (UPPER BOUND(ID), 186 ms)
↳30 CpxRNTS
↳31 IntTrsBoundProof (UPPER BOUND(ID), 7 ms)
↳32 CpxRNTS
↳33 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳34 CpxRNTS
↳35 IntTrsBoundProof (UPPER BOUND(ID), 540 ms)
↳36 CpxRNTS
↳37 IntTrsBoundProof (UPPER BOUND(ID), 109 ms)
↳38 CpxRNTS
↳39 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳40 CpxRNTS
↳41 IntTrsBoundProof (UPPER BOUND(ID), 1417 ms)
↳42 CpxRNTS
↳43 IntTrsBoundProof (UPPER BOUND(ID), 2635 ms)
↳44 CpxRNTS
↳45 FinalProof (⇔, 0 ms)
↳46 BOUNDS(1, n^1)

(0) Obligation:

The Runtime Complexity (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, z) → C(a(x1, y, z), a(x2, y, y))
a(Z, y, z) → 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: INNERMOST

(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed relative TRS to weighted TRS

(2) 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, z) → C(a(x1, y, z), a(x2, y, y)) [1]
a(Z, y, z) → 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

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(4) Obligation:

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

a(C(x1, x2), y, z) → C(a(x1, y, z), a(x2, y, y)) [1]
a(Z, y, z) → 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 → a → a → 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

(5) CompletionProof (UPPER BOUND(ID) transformation)

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
second
first

(c) The following functions are completely defined:

eqZList
and

Due to the following rules being added:

and(v0, v1) → null_and [0]

And the following fresh constants:

null_and, const

(6) 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, z) → C(a(x1, y, z), a(x2, y, y)) [1]
a(Z, y, z) → 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 → a → a → 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
const :: a

Rewrite Strategy: INNERMOST

(7) NarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Narrowed the inner basic terms of all right-hand sides by a single narrowing step.

(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, z) → C(a(x1, y, z), a(x2, y, y)) [1]
a(Z, y, z) → 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 → a → a → 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
const :: a

Rewrite Strategy: INNERMOST

(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

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
const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

a(z', z'', z1) -{ 1 }→ 0 :|: z1 = z, z >= 0, z'' = y, y >= 0, z' = 0
a(z', z'', z1) -{ 1 }→ 1 + a(x1, y, z) + a(x2, y, y) :|: z' = 1 + x1 + x2, z1 = z, x1 >= 0, z >= 0, z'' = y, y >= 0, 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' = 1, z'' = 2
and(z', z'') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
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, z'' = 1 + (1 + y1' + y2') + (1 + y11 + y21), y11 >= 0, z' = 1 + (1 + x1' + x2') + (1 + x11 + x21), x11 >= 0, x21 >= 0, y1' >= 0
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, z' = 1 + (1 + x1' + x2') + (1 + x12 + x22), y2' >= 0, x12 >= 0, x22 >= 0, z'' = 1 + (1 + y1' + y2') + 0, y1' >= 0
eqZList(z', z'') -{ 3 }→ and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, z'' = 1 + (1 + y1' + y2') + (1 + y12 + y22), y2' >= 0, y1' >= 0, z' = 1 + (1 + x1' + x2') + 0
eqZList(z', z'') -{ 3 }→ and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, z' = 1 + 0 + (1 + x17 + x27), x17 >= 0, y27 >= 0, 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) :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0
eqZList(z', z'') -{ 3 }→ and(2, 1) :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0
eqZList(z', z'') -{ 3 }→ and(1, and(eqZList(x13, y13), eqZList(x23, y23))) :|: x23 >= 0, x13 >= 0, x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), z'' = 1 + 0 + (1 + y13 + y23), y13 >= 0, y23 >= 0, x2'' >= 0
eqZList(z', z'') -{ 3 }→ and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z'' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), z' = 1 + 0 + (1 + x15 + x25), y2'' >= 0, x25 >= 0, y1'' >= 0, y25 >= 0, x15 >= 0, y15 >= 0
eqZList(z', z'') -{ 3 }→ and(1, 2) :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0
eqZList(z', z'') -{ 3 }→ and(1, 2) :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0
eqZList(z', z'') -{ 3 }→ and(1, 1) :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0
eqZList(z', z'') -{ 3 }→ and(1, 1) :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0
eqZList(z', z'') -{ 3 }→ and(1, 1) :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0
eqZList(z', z'') -{ 3 }→ and(1, 1) :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 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 :|: z'' = 0, z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
eqZList(z', z'') -{ 1 }→ 1 :|: y1 >= 0, z'' = 1 + y1 + y2, y2 >= 0, z' = 0
first(z') -{ 1 }→ x1 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
second(z') -{ 1 }→ x2 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0

(11) InliningProof (UPPER BOUND(ID) transformation)

Inlined the following terminating rules on right-hand sides where appropriate:

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

(12) Obligation:

Complexity RNTS consisting of the following rules:

a(z', z'', z1) -{ 1 }→ 0 :|: z1 = z, z >= 0, z'' = y, y >= 0, z' = 0
a(z', z'', z1) -{ 1 }→ 1 + a(x1, y, z) + a(x2, y, y) :|: z' = 1 + x1 + x2, z1 = z, x1 >= 0, z >= 0, z'' = y, y >= 0, 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' = 1, z'' = 2
and(z', z'') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
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, z'' = 1 + (1 + y1' + y2') + (1 + y11 + y21), y11 >= 0, z' = 1 + (1 + x1' + x2') + (1 + x11 + x21), x11 >= 0, x21 >= 0, y1' >= 0
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, z' = 1 + (1 + x1' + x2') + (1 + x12 + x22), y2' >= 0, x12 >= 0, x22 >= 0, z'' = 1 + (1 + y1' + y2') + 0, y1' >= 0
eqZList(z', z'') -{ 3 }→ and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, z'' = 1 + (1 + y1' + y2') + (1 + y12 + y22), y2' >= 0, y1' >= 0, z' = 1 + (1 + x1' + x2') + 0
eqZList(z', z'') -{ 3 }→ and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, z' = 1 + 0 + (1 + x17 + x27), x17 >= 0, y27 >= 0, 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, x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), z'' = 1 + 0 + (1 + y13 + y23), y13 >= 0, y23 >= 0, x2'' >= 0
eqZList(z', z'') -{ 3 }→ and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z'' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), z' = 1 + 0 + (1 + x15 + x25), 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 :|: z'' = 0, z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
eqZList(z', z'') -{ 1 }→ 1 :|: y1 >= 0, z'' = 1 + y1 + y2, y2 >= 0, z' = 0
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
first(z') -{ 1 }→ x1 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
second(z') -{ 1 }→ x2 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0

(13) SimplificationProof (BOTH BOUNDS(ID, ID) transformation)

Simplified the RNTS by moving equalities from the constraints into the right-hand sides.

(14) Obligation:

Complexity RNTS consisting of the following rules:

a(z', z'', z1) -{ 1 }→ 0 :|: z1 >= 0, z'' >= 0, z' = 0
a(z', z'', z1) -{ 1 }→ 1 + a(x1, z'', z1) + a(x2, z'', z'') :|: z' = 1 + x1 + x2, x1 >= 0, z1 >= 0, z'' >= 0, 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' = 1, z'' = 2
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, z'' = 1 + (1 + y1' + y2') + (1 + y11 + y21), y11 >= 0, z' = 1 + (1 + x1' + x2') + (1 + x11 + x21), x11 >= 0, x21 >= 0, y1' >= 0
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, z' = 1 + (1 + x1' + x2') + (1 + x12 + x22), y2' >= 0, x12 >= 0, x22 >= 0, z'' = 1 + (1 + y1' + y2') + 0, y1' >= 0
eqZList(z', z'') -{ 3 }→ and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, z'' = 1 + (1 + y1' + y2') + (1 + y12 + y22), y2' >= 0, y1' >= 0, z' = 1 + (1 + x1' + x2') + 0
eqZList(z', z'') -{ 3 }→ and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, z' = 1 + 0 + (1 + x17 + x27), x17 >= 0, y27 >= 0, 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, x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), z'' = 1 + 0 + (1 + y13 + y23), y13 >= 0, y23 >= 0, x2'' >= 0
eqZList(z', z'') -{ 3 }→ and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z'' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), z' = 1 + 0 + (1 + x15 + x25), 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 :|: z'' = 0, z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
eqZList(z', z'') -{ 1 }→ 1 :|: y1 >= 0, z'' = 1 + y1 + y2, y2 >= 0, z' = 0
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
first(z') -{ 1 }→ x1 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
second(z') -{ 1 }→ x2 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0

(15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID) transformation)

Found the following analysis order by SCC decomposition:

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

a(z', z'', z1) -{ 1 }→ 0 :|: z1 >= 0, z'' >= 0, z' = 0
a(z', z'', z1) -{ 1 }→ 1 + a(x1, z'', z1) + a(x2, z'', z'') :|: z' = 1 + x1 + x2, x1 >= 0, z1 >= 0, z'' >= 0, 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' = 1, z'' = 2
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, z'' = 1 + (1 + y1' + y2') + (1 + y11 + y21), y11 >= 0, z' = 1 + (1 + x1' + x2') + (1 + x11 + x21), x11 >= 0, x21 >= 0, y1' >= 0
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, z' = 1 + (1 + x1' + x2') + (1 + x12 + x22), y2' >= 0, x12 >= 0, x22 >= 0, z'' = 1 + (1 + y1' + y2') + 0, y1' >= 0
eqZList(z', z'') -{ 3 }→ and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, z'' = 1 + (1 + y1' + y2') + (1 + y12 + y22), y2' >= 0, y1' >= 0, z' = 1 + (1 + x1' + x2') + 0
eqZList(z', z'') -{ 3 }→ and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, z' = 1 + 0 + (1 + x17 + x27), x17 >= 0, y27 >= 0, 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, x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), z'' = 1 + 0 + (1 + y13 + y23), y13 >= 0, y23 >= 0, x2'' >= 0
eqZList(z', z'') -{ 3 }→ and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z'' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), z' = 1 + 0 + (1 + x15 + x25), 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 :|: z'' = 0, z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
eqZList(z', z'') -{ 1 }→ 1 :|: y1 >= 0, z'' = 1 + y1 + y2, y2 >= 0, z' = 0
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
first(z') -{ 1 }→ x1 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
second(z') -{ 1 }→ x2 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0

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

(17) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(18) Obligation:

Complexity RNTS consisting of the following rules:

a(z', z'', z1) -{ 1 }→ 0 :|: z1 >= 0, z'' >= 0, z' = 0
a(z', z'', z1) -{ 1 }→ 1 + a(x1, z'', z1) + a(x2, z'', z'') :|: z' = 1 + x1 + x2, x1 >= 0, z1 >= 0, z'' >= 0, 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' = 1, z'' = 2
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, z'' = 1 + (1 + y1' + y2') + (1 + y11 + y21), y11 >= 0, z' = 1 + (1 + x1' + x2') + (1 + x11 + x21), x11 >= 0, x21 >= 0, y1' >= 0
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, z' = 1 + (1 + x1' + x2') + (1 + x12 + x22), y2' >= 0, x12 >= 0, x22 >= 0, z'' = 1 + (1 + y1' + y2') + 0, y1' >= 0
eqZList(z', z'') -{ 3 }→ and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, z'' = 1 + (1 + y1' + y2') + (1 + y12 + y22), y2' >= 0, y1' >= 0, z' = 1 + (1 + x1' + x2') + 0
eqZList(z', z'') -{ 3 }→ and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, z' = 1 + 0 + (1 + x17 + x27), x17 >= 0, y27 >= 0, 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, x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), z'' = 1 + 0 + (1 + y13 + y23), y13 >= 0, y23 >= 0, x2'' >= 0
eqZList(z', z'') -{ 3 }→ and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z'' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), z' = 1 + 0 + (1 + x15 + x25), 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 :|: z'' = 0, z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
eqZList(z', z'') -{ 1 }→ 1 :|: y1 >= 0, z'' = 1 + y1 + y2, y2 >= 0, z' = 0
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
first(z') -{ 1 }→ x1 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
second(z') -{ 1 }→ x2 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0

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

(19) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(20) Obligation:

Complexity RNTS consisting of the following rules:

a(z', z'', z1) -{ 1 }→ 0 :|: z1 >= 0, z'' >= 0, z' = 0
a(z', z'', z1) -{ 1 }→ 1 + a(x1, z'', z1) + a(x2, z'', z'') :|: z' = 1 + x1 + x2, x1 >= 0, z1 >= 0, z'' >= 0, 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' = 1, z'' = 2
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, z'' = 1 + (1 + y1' + y2') + (1 + y11 + y21), y11 >= 0, z' = 1 + (1 + x1' + x2') + (1 + x11 + x21), x11 >= 0, x21 >= 0, y1' >= 0
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, z' = 1 + (1 + x1' + x2') + (1 + x12 + x22), y2' >= 0, x12 >= 0, x22 >= 0, z'' = 1 + (1 + y1' + y2') + 0, y1' >= 0
eqZList(z', z'') -{ 3 }→ and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, z'' = 1 + (1 + y1' + y2') + (1 + y12 + y22), y2' >= 0, y1' >= 0, z' = 1 + (1 + x1' + x2') + 0
eqZList(z', z'') -{ 3 }→ and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, z' = 1 + 0 + (1 + x17 + x27), x17 >= 0, y27 >= 0, 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, x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), z'' = 1 + 0 + (1 + y13 + y23), y13 >= 0, y23 >= 0, x2'' >= 0
eqZList(z', z'') -{ 3 }→ and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z'' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), z' = 1 + 0 + (1 + x15 + x25), 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 :|: z'' = 0, z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
eqZList(z', z'') -{ 1 }→ 1 :|: y1 >= 0, z'' = 1 + y1 + y2, y2 >= 0, z' = 0
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
first(z') -{ 1 }→ x1 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
second(z') -{ 1 }→ x2 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0

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

(21) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(22) Obligation:

Complexity RNTS consisting of the following rules:

a(z', z'', z1) -{ 1 }→ 0 :|: z1 >= 0, z'' >= 0, z' = 0
a(z', z'', z1) -{ 1 }→ 1 + a(x1, z'', z1) + a(x2, z'', z'') :|: z' = 1 + x1 + x2, x1 >= 0, z1 >= 0, z'' >= 0, 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' = 1, z'' = 2
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, z'' = 1 + (1 + y1' + y2') + (1 + y11 + y21), y11 >= 0, z' = 1 + (1 + x1' + x2') + (1 + x11 + x21), x11 >= 0, x21 >= 0, y1' >= 0
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, z' = 1 + (1 + x1' + x2') + (1 + x12 + x22), y2' >= 0, x12 >= 0, x22 >= 0, z'' = 1 + (1 + y1' + y2') + 0, y1' >= 0
eqZList(z', z'') -{ 3 }→ and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, z'' = 1 + (1 + y1' + y2') + (1 + y12 + y22), y2' >= 0, y1' >= 0, z' = 1 + (1 + x1' + x2') + 0
eqZList(z', z'') -{ 3 }→ and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, z' = 1 + 0 + (1 + x17 + x27), x17 >= 0, y27 >= 0, 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, x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), z'' = 1 + 0 + (1 + y13 + y23), y13 >= 0, y23 >= 0, x2'' >= 0
eqZList(z', z'') -{ 3 }→ and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z'' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), z' = 1 + 0 + (1 + x15 + x25), 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 :|: z'' = 0, z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
eqZList(z', z'') -{ 1 }→ 1 :|: y1 >= 0, z'' = 1 + y1 + y2, y2 >= 0, z' = 0
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
first(z') -{ 1 }→ x1 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
second(z') -{ 1 }→ x2 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0

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

(23) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using KoAT for: first
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: z'

(24) Obligation:

Complexity RNTS consisting of the following rules:

a(z', z'', z1) -{ 1 }→ 0 :|: z1 >= 0, z'' >= 0, z' = 0
a(z', z'', z1) -{ 1 }→ 1 + a(x1, z'', z1) + a(x2, z'', z'') :|: z' = 1 + x1 + x2, x1 >= 0, z1 >= 0, z'' >= 0, 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' = 1, z'' = 2
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, z'' = 1 + (1 + y1' + y2') + (1 + y11 + y21), y11 >= 0, z' = 1 + (1 + x1' + x2') + (1 + x11 + x21), x11 >= 0, x21 >= 0, y1' >= 0
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, z' = 1 + (1 + x1' + x2') + (1 + x12 + x22), y2' >= 0, x12 >= 0, x22 >= 0, z'' = 1 + (1 + y1' + y2') + 0, y1' >= 0
eqZList(z', z'') -{ 3 }→ and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, z'' = 1 + (1 + y1' + y2') + (1 + y12 + y22), y2' >= 0, y1' >= 0, z' = 1 + (1 + x1' + x2') + 0
eqZList(z', z'') -{ 3 }→ and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, z' = 1 + 0 + (1 + x17 + x27), x17 >= 0, y27 >= 0, 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, x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), z'' = 1 + 0 + (1 + y13 + y23), y13 >= 0, y23 >= 0, x2'' >= 0
eqZList(z', z'') -{ 3 }→ and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z'' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), z' = 1 + 0 + (1 + x15 + x25), 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 :|: z'' = 0, z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
eqZList(z', z'') -{ 1 }→ 1 :|: y1 >= 0, z'' = 1 + y1 + y2, y2 >= 0, z' = 0
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
first(z') -{ 1 }→ x1 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
second(z') -{ 1 }→ x2 :|: z' = 1 + x1 + x2, x1 >= 0, 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(n1) [z']

(25) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(26) Obligation:

Complexity RNTS consisting of the following rules:

a(z', z'', z1) -{ 1 }→ 0 :|: z1 >= 0, z'' >= 0, z' = 0
a(z', z'', z1) -{ 1 }→ 1 + a(x1, z'', z1) + a(x2, z'', z'') :|: z' = 1 + x1 + x2, x1 >= 0, z1 >= 0, z'' >= 0, 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' = 1, z'' = 2
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, z'' = 1 + (1 + y1' + y2') + (1 + y11 + y21), y11 >= 0, z' = 1 + (1 + x1' + x2') + (1 + x11 + x21), x11 >= 0, x21 >= 0, y1' >= 0
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, z' = 1 + (1 + x1' + x2') + (1 + x12 + x22), y2' >= 0, x12 >= 0, x22 >= 0, z'' = 1 + (1 + y1' + y2') + 0, y1' >= 0
eqZList(z', z'') -{ 3 }→ and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, z'' = 1 + (1 + y1' + y2') + (1 + y12 + y22), y2' >= 0, y1' >= 0, z' = 1 + (1 + x1' + x2') + 0
eqZList(z', z'') -{ 3 }→ and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, z' = 1 + 0 + (1 + x17 + x27), x17 >= 0, y27 >= 0, 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, x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), z'' = 1 + 0 + (1 + y13 + y23), y13 >= 0, y23 >= 0, x2'' >= 0
eqZList(z', z'') -{ 3 }→ and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z'' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), z' = 1 + 0 + (1 + x15 + x25), 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 :|: z'' = 0, z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
eqZList(z', z'') -{ 1 }→ 1 :|: y1 >= 0, z'' = 1 + y1 + y2, y2 >= 0, z' = 0
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
first(z') -{ 1 }→ x1 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
second(z') -{ 1 }→ x2 :|: z' = 1 + x1 + x2, x1 >= 0, 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(n1) [z']

(27) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(28) Obligation:

Complexity RNTS consisting of the following rules:

a(z', z'', z1) -{ 1 }→ 0 :|: z1 >= 0, z'' >= 0, z' = 0
a(z', z'', z1) -{ 1 }→ 1 + a(x1, z'', z1) + a(x2, z'', z'') :|: z' = 1 + x1 + x2, x1 >= 0, z1 >= 0, z'' >= 0, 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' = 1, z'' = 2
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, z'' = 1 + (1 + y1' + y2') + (1 + y11 + y21), y11 >= 0, z' = 1 + (1 + x1' + x2') + (1 + x11 + x21), x11 >= 0, x21 >= 0, y1' >= 0
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, z' = 1 + (1 + x1' + x2') + (1 + x12 + x22), y2' >= 0, x12 >= 0, x22 >= 0, z'' = 1 + (1 + y1' + y2') + 0, y1' >= 0
eqZList(z', z'') -{ 3 }→ and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, z'' = 1 + (1 + y1' + y2') + (1 + y12 + y22), y2' >= 0, y1' >= 0, z' = 1 + (1 + x1' + x2') + 0
eqZList(z', z'') -{ 3 }→ and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, z' = 1 + 0 + (1 + x17 + x27), x17 >= 0, y27 >= 0, 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, x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), z'' = 1 + 0 + (1 + y13 + y23), y13 >= 0, y23 >= 0, x2'' >= 0
eqZList(z', z'') -{ 3 }→ and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z'' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), z' = 1 + 0 + (1 + x15 + x25), 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 :|: z'' = 0, z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
eqZList(z', z'') -{ 1 }→ 1 :|: y1 >= 0, z'' = 1 + y1 + y2, y2 >= 0, z' = 0
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
first(z') -{ 1 }→ x1 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
second(z') -{ 1 }→ x2 :|: z' = 1 + x1 + x2, x1 >= 0, 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(n1) [z']

(29) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using KoAT for: second
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: z'

(30) Obligation:

Complexity RNTS consisting of the following rules:

a(z', z'', z1) -{ 1 }→ 0 :|: z1 >= 0, z'' >= 0, z' = 0
a(z', z'', z1) -{ 1 }→ 1 + a(x1, z'', z1) + a(x2, z'', z'') :|: z' = 1 + x1 + x2, x1 >= 0, z1 >= 0, z'' >= 0, 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' = 1, z'' = 2
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, z'' = 1 + (1 + y1' + y2') + (1 + y11 + y21), y11 >= 0, z' = 1 + (1 + x1' + x2') + (1 + x11 + x21), x11 >= 0, x21 >= 0, y1' >= 0
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, z' = 1 + (1 + x1' + x2') + (1 + x12 + x22), y2' >= 0, x12 >= 0, x22 >= 0, z'' = 1 + (1 + y1' + y2') + 0, y1' >= 0
eqZList(z', z'') -{ 3 }→ and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, z'' = 1 + (1 + y1' + y2') + (1 + y12 + y22), y2' >= 0, y1' >= 0, z' = 1 + (1 + x1' + x2') + 0
eqZList(z', z'') -{ 3 }→ and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, z' = 1 + 0 + (1 + x17 + x27), x17 >= 0, y27 >= 0, 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, x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), z'' = 1 + 0 + (1 + y13 + y23), y13 >= 0, y23 >= 0, x2'' >= 0
eqZList(z', z'') -{ 3 }→ and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z'' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), z' = 1 + 0 + (1 + x15 + x25), 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 :|: z'' = 0, z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
eqZList(z', z'') -{ 1 }→ 1 :|: y1 >= 0, z'' = 1 + y1 + y2, y2 >= 0, z' = 0
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
first(z') -{ 1 }→ x1 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
second(z') -{ 1 }→ x2 :|: z' = 1 + x1 + x2, x1 >= 0, 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(n1) [z']
second: runtime: ?, size: O(n1) [z']

(31) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(32) Obligation:

Complexity RNTS consisting of the following rules:

a(z', z'', z1) -{ 1 }→ 0 :|: z1 >= 0, z'' >= 0, z' = 0
a(z', z'', z1) -{ 1 }→ 1 + a(x1, z'', z1) + a(x2, z'', z'') :|: z' = 1 + x1 + x2, x1 >= 0, z1 >= 0, z'' >= 0, 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' = 1, z'' = 2
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, z'' = 1 + (1 + y1' + y2') + (1 + y11 + y21), y11 >= 0, z' = 1 + (1 + x1' + x2') + (1 + x11 + x21), x11 >= 0, x21 >= 0, y1' >= 0
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, z' = 1 + (1 + x1' + x2') + (1 + x12 + x22), y2' >= 0, x12 >= 0, x22 >= 0, z'' = 1 + (1 + y1' + y2') + 0, y1' >= 0
eqZList(z', z'') -{ 3 }→ and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, z'' = 1 + (1 + y1' + y2') + (1 + y12 + y22), y2' >= 0, y1' >= 0, z' = 1 + (1 + x1' + x2') + 0
eqZList(z', z'') -{ 3 }→ and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, z' = 1 + 0 + (1 + x17 + x27), x17 >= 0, y27 >= 0, 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, x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), z'' = 1 + 0 + (1 + y13 + y23), y13 >= 0, y23 >= 0, x2'' >= 0
eqZList(z', z'') -{ 3 }→ and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z'' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), z' = 1 + 0 + (1 + x15 + x25), 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 :|: z'' = 0, z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
eqZList(z', z'') -{ 1 }→ 1 :|: y1 >= 0, z'' = 1 + y1 + y2, y2 >= 0, z' = 0
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
first(z') -{ 1 }→ x1 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
second(z') -{ 1 }→ x2 :|: z' = 1 + x1 + x2, x1 >= 0, 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(n1) [z']
second: runtime: O(1) [1], size: O(n1) [z']

(33) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(34) Obligation:

Complexity RNTS consisting of the following rules:

a(z', z'', z1) -{ 1 }→ 0 :|: z1 >= 0, z'' >= 0, z' = 0
a(z', z'', z1) -{ 1 }→ 1 + a(x1, z'', z1) + a(x2, z'', z'') :|: z' = 1 + x1 + x2, x1 >= 0, z1 >= 0, z'' >= 0, 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' = 1, z'' = 2
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, z'' = 1 + (1 + y1' + y2') + (1 + y11 + y21), y11 >= 0, z' = 1 + (1 + x1' + x2') + (1 + x11 + x21), x11 >= 0, x21 >= 0, y1' >= 0
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, z' = 1 + (1 + x1' + x2') + (1 + x12 + x22), y2' >= 0, x12 >= 0, x22 >= 0, z'' = 1 + (1 + y1' + y2') + 0, y1' >= 0
eqZList(z', z'') -{ 3 }→ and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, z'' = 1 + (1 + y1' + y2') + (1 + y12 + y22), y2' >= 0, y1' >= 0, z' = 1 + (1 + x1' + x2') + 0
eqZList(z', z'') -{ 3 }→ and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, z' = 1 + 0 + (1 + x17 + x27), x17 >= 0, y27 >= 0, 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, x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), z'' = 1 + 0 + (1 + y13 + y23), y13 >= 0, y23 >= 0, x2'' >= 0
eqZList(z', z'') -{ 3 }→ and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z'' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), z' = 1 + 0 + (1 + x15 + x25), 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 :|: z'' = 0, z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
eqZList(z', z'') -{ 1 }→ 1 :|: y1 >= 0, z'' = 1 + y1 + y2, y2 >= 0, z' = 0
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
first(z') -{ 1 }→ x1 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
second(z') -{ 1 }→ x2 :|: z' = 1 + x1 + x2, x1 >= 0, 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(n1) [z']
second: runtime: O(1) [1], size: O(n1) [z']

(35) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using CoFloCo for: a
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: z'

(36) Obligation:

Complexity RNTS consisting of the following rules:

a(z', z'', z1) -{ 1 }→ 0 :|: z1 >= 0, z'' >= 0, z' = 0
a(z', z'', z1) -{ 1 }→ 1 + a(x1, z'', z1) + a(x2, z'', z'') :|: z' = 1 + x1 + x2, x1 >= 0, z1 >= 0, z'' >= 0, 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' = 1, z'' = 2
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, z'' = 1 + (1 + y1' + y2') + (1 + y11 + y21), y11 >= 0, z' = 1 + (1 + x1' + x2') + (1 + x11 + x21), x11 >= 0, x21 >= 0, y1' >= 0
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, z' = 1 + (1 + x1' + x2') + (1 + x12 + x22), y2' >= 0, x12 >= 0, x22 >= 0, z'' = 1 + (1 + y1' + y2') + 0, y1' >= 0
eqZList(z', z'') -{ 3 }→ and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, z'' = 1 + (1 + y1' + y2') + (1 + y12 + y22), y2' >= 0, y1' >= 0, z' = 1 + (1 + x1' + x2') + 0
eqZList(z', z'') -{ 3 }→ and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, z' = 1 + 0 + (1 + x17 + x27), x17 >= 0, y27 >= 0, 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, x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), z'' = 1 + 0 + (1 + y13 + y23), y13 >= 0, y23 >= 0, x2'' >= 0
eqZList(z', z'') -{ 3 }→ and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z'' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), z' = 1 + 0 + (1 + x15 + x25), 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 :|: z'' = 0, z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
eqZList(z', z'') -{ 1 }→ 1 :|: y1 >= 0, z'' = 1 + y1 + y2, y2 >= 0, z' = 0
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
first(z') -{ 1 }→ x1 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
second(z') -{ 1 }→ x2 :|: z' = 1 + x1 + x2, x1 >= 0, 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(n1) [z']
second: runtime: O(1) [1], size: O(n1) [z']
a: runtime: ?, size: O(n1) [z']

(37) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using CoFloCo for: a
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: 1 + 2·z'

(38) Obligation:

Complexity RNTS consisting of the following rules:

a(z', z'', z1) -{ 1 }→ 0 :|: z1 >= 0, z'' >= 0, z' = 0
a(z', z'', z1) -{ 1 }→ 1 + a(x1, z'', z1) + a(x2, z'', z'') :|: z' = 1 + x1 + x2, x1 >= 0, z1 >= 0, z'' >= 0, 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' = 1, z'' = 2
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, z'' = 1 + (1 + y1' + y2') + (1 + y11 + y21), y11 >= 0, z' = 1 + (1 + x1' + x2') + (1 + x11 + x21), x11 >= 0, x21 >= 0, y1' >= 0
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, z' = 1 + (1 + x1' + x2') + (1 + x12 + x22), y2' >= 0, x12 >= 0, x22 >= 0, z'' = 1 + (1 + y1' + y2') + 0, y1' >= 0
eqZList(z', z'') -{ 3 }→ and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, z'' = 1 + (1 + y1' + y2') + (1 + y12 + y22), y2' >= 0, y1' >= 0, z' = 1 + (1 + x1' + x2') + 0
eqZList(z', z'') -{ 3 }→ and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, z' = 1 + 0 + (1 + x17 + x27), x17 >= 0, y27 >= 0, 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, x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), z'' = 1 + 0 + (1 + y13 + y23), y13 >= 0, y23 >= 0, x2'' >= 0
eqZList(z', z'') -{ 3 }→ and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z'' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), z' = 1 + 0 + (1 + x15 + x25), 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 :|: z'' = 0, z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
eqZList(z', z'') -{ 1 }→ 1 :|: y1 >= 0, z'' = 1 + y1 + y2, y2 >= 0, z' = 0
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
first(z') -{ 1 }→ x1 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
second(z') -{ 1 }→ x2 :|: z' = 1 + x1 + x2, x1 >= 0, 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(n1) [z']
second: runtime: O(1) [1], size: O(n1) [z']
a: runtime: O(n1) [1 + 2·z'], size: O(n1) [z']

(39) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(40) Obligation:

Complexity RNTS consisting of the following rules:

a(z', z'', z1) -{ 1 }→ 0 :|: z1 >= 0, z'' >= 0, z' = 0
a(z', z'', z1) -{ 3 + 2·x1 + 2·x2 }→ 1 + s + s' :|: s >= 0, s <= 1 * x1, s' >= 0, s' <= 1 * x2, z' = 1 + x1 + x2, x1 >= 0, z1 >= 0, z'' >= 0, 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' = 1, z'' = 2
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, z'' = 1 + (1 + y1' + y2') + (1 + y11 + y21), y11 >= 0, z' = 1 + (1 + x1' + x2') + (1 + x11 + x21), x11 >= 0, x21 >= 0, y1' >= 0
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, z' = 1 + (1 + x1' + x2') + (1 + x12 + x22), y2' >= 0, x12 >= 0, x22 >= 0, z'' = 1 + (1 + y1' + y2') + 0, y1' >= 0
eqZList(z', z'') -{ 3 }→ and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, z'' = 1 + (1 + y1' + y2') + (1 + y12 + y22), y2' >= 0, y1' >= 0, z' = 1 + (1 + x1' + x2') + 0
eqZList(z', z'') -{ 3 }→ and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, z' = 1 + 0 + (1 + x17 + x27), x17 >= 0, y27 >= 0, 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, x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), z'' = 1 + 0 + (1 + y13 + y23), y13 >= 0, y23 >= 0, x2'' >= 0
eqZList(z', z'') -{ 3 }→ and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z'' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), z' = 1 + 0 + (1 + x15 + x25), 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 :|: z'' = 0, z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
eqZList(z', z'') -{ 1 }→ 1 :|: y1 >= 0, z'' = 1 + y1 + y2, y2 >= 0, z' = 0
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
first(z') -{ 1 }→ x1 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
second(z') -{ 1 }→ x2 :|: z' = 1 + x1 + x2, x1 >= 0, 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(n1) [z']
second: runtime: O(1) [1], size: O(n1) [z']
a: runtime: O(n1) [1 + 2·z'], size: O(n1) [z']

(41) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(42) Obligation:

Complexity RNTS consisting of the following rules:

a(z', z'', z1) -{ 1 }→ 0 :|: z1 >= 0, z'' >= 0, z' = 0
a(z', z'', z1) -{ 3 + 2·x1 + 2·x2 }→ 1 + s + s' :|: s >= 0, s <= 1 * x1, s' >= 0, s' <= 1 * x2, z' = 1 + x1 + x2, x1 >= 0, z1 >= 0, z'' >= 0, 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' = 1, z'' = 2
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, z'' = 1 + (1 + y1' + y2') + (1 + y11 + y21), y11 >= 0, z' = 1 + (1 + x1' + x2') + (1 + x11 + x21), x11 >= 0, x21 >= 0, y1' >= 0
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, z' = 1 + (1 + x1' + x2') + (1 + x12 + x22), y2' >= 0, x12 >= 0, x22 >= 0, z'' = 1 + (1 + y1' + y2') + 0, y1' >= 0
eqZList(z', z'') -{ 3 }→ and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, z'' = 1 + (1 + y1' + y2') + (1 + y12 + y22), y2' >= 0, y1' >= 0, z' = 1 + (1 + x1' + x2') + 0
eqZList(z', z'') -{ 3 }→ and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, z' = 1 + 0 + (1 + x17 + x27), x17 >= 0, y27 >= 0, 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, x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), z'' = 1 + 0 + (1 + y13 + y23), y13 >= 0, y23 >= 0, x2'' >= 0
eqZList(z', z'') -{ 3 }→ and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z'' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), z' = 1 + 0 + (1 + x15 + x25), 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 :|: z'' = 0, z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
eqZList(z', z'') -{ 1 }→ 1 :|: y1 >= 0, z'' = 1 + y1 + y2, y2 >= 0, z' = 0
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
first(z') -{ 1 }→ x1 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
second(z') -{ 1 }→ x2 :|: z' = 1 + x1 + x2, x1 >= 0, 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(n1) [z']
second: runtime: O(1) [1], size: O(n1) [z']
a: runtime: O(n1) [1 + 2·z'], size: O(n1) [z']
eqZList: runtime: ?, size: O(1) [2]

(43) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using KoAT for: eqZList
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: 78 + 156·z''

(44) Obligation:

Complexity RNTS consisting of the following rules:

a(z', z'', z1) -{ 1 }→ 0 :|: z1 >= 0, z'' >= 0, z' = 0
a(z', z'', z1) -{ 3 + 2·x1 + 2·x2 }→ 1 + s + s' :|: s >= 0, s <= 1 * x1, s' >= 0, s' <= 1 * x2, z' = 1 + x1 + x2, x1 >= 0, z1 >= 0, z'' >= 0, 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' = 1, z'' = 2
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, z'' = 1 + (1 + y1' + y2') + (1 + y11 + y21), y11 >= 0, z' = 1 + (1 + x1' + x2') + (1 + x11 + x21), x11 >= 0, x21 >= 0, y1' >= 0
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, z' = 1 + (1 + x1' + x2') + (1 + x12 + x22), y2' >= 0, x12 >= 0, x22 >= 0, z'' = 1 + (1 + y1' + y2') + 0, y1' >= 0
eqZList(z', z'') -{ 3 }→ and(and(eqZList(x1', y1'), eqZList(x2', y2')), 1) :|: y22 >= 0, x1' >= 0, x2' >= 0, y12 >= 0, z'' = 1 + (1 + y1' + y2') + (1 + y12 + y22), y2' >= 0, y1' >= 0, z' = 1 + (1 + x1' + x2') + 0
eqZList(z', z'') -{ 3 }→ and(2, and(eqZList(x17, y17), eqZList(x27, y27))) :|: x27 >= 0, z' = 1 + 0 + (1 + x17 + x27), x17 >= 0, y27 >= 0, 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, x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x13 + x23), z'' = 1 + 0 + (1 + y13 + y23), y13 >= 0, y23 >= 0, x2'' >= 0
eqZList(z', z'') -{ 3 }→ and(1, and(eqZList(x15, y15), eqZList(x25, y25))) :|: z'' = 1 + (1 + y1'' + y2'') + (1 + y15 + y25), z' = 1 + 0 + (1 + x15 + x25), 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 :|: z'' = 0, z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
eqZList(z', z'') -{ 1 }→ 1 :|: y1 >= 0, z'' = 1 + y1 + y2, y2 >= 0, z' = 0
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, 1 = 1, 2 = 2
eqZList(z', z'') -{ 3 }→ 1 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 1 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, 2 = 2, 1 = 1
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + (1 + x14 + x24), x14 >= 0, x24 >= 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + 0 + (1 + y14 + y24), x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, y24 >= 0, x2'' >= 0, y14 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x1'' >= 0, z' = 1 + (1 + x1'' + x2'') + 0, x2'' >= 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, z' = 1 + 0 + (1 + x16 + x26), y1'' >= 0, x26 >= 0, x16 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: y26 >= 0, y16 >= 0, z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + (1 + y16 + y26), y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 1 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + (1 + y1'' + y2'') + 0, y2'' >= 0, y1'' >= 0, v0 >= 0, v1 >= 0, 2 = v1, 1 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: x28 >= 0, x18 >= 0, z' = 1 + 0 + (1 + x18 + x28), z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, y18 >= 0, z'' = 1 + 0 + (1 + y18 + y28), y28 >= 0, v0 >= 0, v1 >= 0, 1 = v1, 2 = v0
eqZList(z', z'') -{ 3 }→ 0 :|: z' = 1 + 0 + 0, z'' = 1 + 0 + 0, v0 >= 0, v1 >= 0, 2 = v1, 2 = v0
first(z') -{ 1 }→ x1 :|: z' = 1 + x1 + x2, x1 >= 0, x2 >= 0
second(z') -{ 1 }→ x2 :|: z' = 1 + x1 + x2, x1 >= 0, 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(n1) [z']
second: runtime: O(1) [1], size: O(n1) [z']
a: runtime: O(n1) [1 + 2·z'], size: O(n1) [z']
eqZList: runtime: O(n1) [78 + 156·z''], size: O(1) [2]

(45) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(46) BOUNDS(1, n^1)