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

(0) Obligation:

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


The TRS R consists of the following rules:

half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
inc(0) → 0
inc(s(x)) → s(inc(x))
log(x) → log2(x, 0)
log2(x, y) → if(le(x, s(0)), x, inc(y))
if(true, x, s(y)) → y
if(false, x, y) → log2(half(x), y)

Rewrite Strategy: INNERMOST

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

Transformed TRS to weighted TRS

(2) Obligation:

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


The TRS R consists of the following rules:

half(0) → 0 [1]
half(s(0)) → 0 [1]
half(s(s(x))) → s(half(x)) [1]
le(0, y) → true [1]
le(s(x), 0) → false [1]
le(s(x), s(y)) → le(x, y) [1]
inc(0) → 0 [1]
inc(s(x)) → s(inc(x)) [1]
log(x) → log2(x, 0) [1]
log2(x, y) → if(le(x, s(0)), x, inc(y)) [1]
if(true, x, s(y)) → y [1]
if(false, x, y) → log2(half(x), y) [1]

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:

half(0) → 0 [1]
half(s(0)) → 0 [1]
half(s(s(x))) → s(half(x)) [1]
le(0, y) → true [1]
le(s(x), 0) → false [1]
le(s(x), s(y)) → le(x, y) [1]
inc(0) → 0 [1]
inc(s(x)) → s(inc(x)) [1]
log(x) → log2(x, 0) [1]
log2(x, y) → if(le(x, s(0)), x, inc(y)) [1]
if(true, x, s(y)) → y [1]
if(false, x, y) → log2(half(x), y) [1]

The TRS has the following type information:

half :: 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s le :: 0:s → 0:s → true:false true :: true:false false :: true:false inc :: 0:s → 0:s log :: 0:s → 0:s log2 :: 0:s → 0:s → 0:s if :: true:false → 0:s → 0:s → 0:s

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:

log
log2
if

(c) The following functions are completely defined:

half
le
inc

Due to the following rules being added:
none

And the following fresh constants: none

(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: half(0) → 0 [1] half(s(0)) → 0 [1] half(s(s(x))) → s(half(x)) [1] le(0, y) → true [1] le(s(x), 0) → false [1] le(s(x), s(y)) → le(x, y) [1] inc(0) → 0 [1] inc(s(x)) → s(inc(x)) [1] log(x) → log2(x, 0) [1] log2(x, y) → if(le(x, s(0)), x, inc(y)) [1] if(true, x, s(y)) → y [1] if(false, x, y) → log2(half(x), y) [1] The TRS has the following type information: half :: 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s le :: 0:s → 0:s → true:false true :: true:false false :: true:false inc :: 0:s → 0:s log :: 0:s → 0:s log2 :: 0:s → 0:s → 0:s if :: true:false → 0:s → 0:s → 0:s 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: half(0) → 0 [1] half(s(0)) → 0 [1] half(s(s(x))) → s(half(x)) [1] le(0, y) → true [1] le(s(x), 0) → false [1] le(s(x), s(y)) → le(x, y) [1] inc(0) → 0 [1] inc(s(x)) → s(inc(x)) [1] log(x) → log2(x, 0) [1] log2(0, 0) → if(true, 0, 0) [3] log2(0, s(x'')) → if(true, 0, s(inc(x''))) [3] log2(s(x'), 0) → if(le(x', 0), s(x'), 0) [3] log2(s(x'), s(x1)) → if(le(x', 0), s(x'), s(inc(x1))) [3] if(true, x, s(y)) → y [1] if(false, 0, y) → log2(0, y) [2] if(false, s(0), y) → log2(0, y) [2] if(false, s(s(x2)), y) → log2(s(half(x2)), y) [2] The TRS has the following type information: half :: 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s le :: 0:s → 0:s → true:false true :: true:false false :: true:false inc :: 0:s → 0:s log :: 0:s → 0:s log2 :: 0:s → 0:s → 0:s if :: true:false → 0:s → 0:s → 0:s 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:

0 => 0
true => 1
false => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 1 + half(x) :|: x >= 0, z = 1 + (1 + x)
if(z, z', z'') -{ 1 }→ y :|: z' = x, z = 1, x >= 0, y >= 0, z'' = 1 + y
if(z, z', z'') -{ 2 }→ log2(0, y) :|: z'' = y, y >= 0, z = 0, z' = 0
if(z, z', z'') -{ 2 }→ log2(0, y) :|: z'' = y, y >= 0, z' = 1 + 0, z = 0
if(z, z', z'') -{ 2 }→ log2(1 + half(x2), y) :|: z'' = y, z' = 1 + (1 + x2), y >= 0, z = 0, x2 >= 0
inc(z) -{ 1 }→ 0 :|: z = 0
inc(z) -{ 1 }→ 1 + inc(x) :|: x >= 0, z = 1 + x
le(z, z') -{ 1 }→ le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
le(z, z') -{ 1 }→ 1 :|: y >= 0, z = 0, z' = y
le(z, z') -{ 1 }→ 0 :|: x >= 0, z = 1 + x, z' = 0
log(z) -{ 1 }→ log2(x, 0) :|: x >= 0, z = x
log2(z, z') -{ 3 }→ if(le(x', 0), 1 + x', 0) :|: z = 1 + x', x' >= 0, z' = 0
log2(z, z') -{ 3 }→ if(le(x', 0), 1 + x', 1 + inc(x1)) :|: z = 1 + x', x1 >= 0, x' >= 0, z' = 1 + x1
log2(z, z') -{ 3 }→ if(1, 0, 0) :|: z = 0, z' = 0
log2(z, z') -{ 3 }→ if(1, 0, 1 + inc(x'')) :|: z' = 1 + x'', x'' >= 0, z = 0

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

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

(12) Obligation:

Complexity RNTS consisting of the following rules:

half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 1 + half(z - 2) :|: z - 2 >= 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
if(z, z', z'') -{ 2 }→ log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 0, z' - 2 >= 0
if(z, z', z'') -{ 1 }→ z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
inc(z) -{ 1 }→ 0 :|: z = 0
inc(z) -{ 1 }→ 1 + inc(z - 1) :|: z - 1 >= 0
le(z, z') -{ 1 }→ le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
log(z) -{ 1 }→ log2(z, 0) :|: z >= 0
log2(z, z') -{ 3 }→ if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
log2(z, z') -{ 3 }→ if(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
log2(z, z') -{ 3 }→ if(1, 0, 0) :|: z = 0, z' = 0
log2(z, z') -{ 3 }→ if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0

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

Found the following analysis order by SCC decomposition:

{ le } { half } { inc } { log2, if } { log }

(14) Obligation:

Complexity RNTS consisting of the following rules:

half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 1 + half(z - 2) :|: z - 2 >= 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
if(z, z', z'') -{ 2 }→ log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 0, z' - 2 >= 0
if(z, z', z'') -{ 1 }→ z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
inc(z) -{ 1 }→ 0 :|: z = 0
inc(z) -{ 1 }→ 1 + inc(z - 1) :|: z - 1 >= 0
le(z, z') -{ 1 }→ le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
log(z) -{ 1 }→ log2(z, 0) :|: z >= 0
log2(z, z') -{ 3 }→ if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
log2(z, z') -{ 3 }→ if(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
log2(z, z') -{ 3 }→ if(1, 0, 0) :|: z = 0, z' = 0
log2(z, z') -{ 3 }→ if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0

Function symbols to be analyzed: {le}, {half}, {inc}, {log2,if}, {log}

(15) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 1 + half(z - 2) :|: z - 2 >= 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
if(z, z', z'') -{ 2 }→ log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 0, z' - 2 >= 0
if(z, z', z'') -{ 1 }→ z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
inc(z) -{ 1 }→ 0 :|: z = 0
inc(z) -{ 1 }→ 1 + inc(z - 1) :|: z - 1 >= 0
le(z, z') -{ 1 }→ le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
log(z) -{ 1 }→ log2(z, 0) :|: z >= 0
log2(z, z') -{ 3 }→ if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
log2(z, z') -{ 3 }→ if(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
log2(z, z') -{ 3 }→ if(1, 0, 0) :|: z = 0, z' = 0
log2(z, z') -{ 3 }→ if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0

Function symbols to be analyzed: {le}, {half}, {inc}, {log2,if}, {log}
Previous analysis results are:

le: runtime: ?, size: O(1) [1]

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

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 1 + half(z - 2) :|: z - 2 >= 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
if(z, z', z'') -{ 2 }→ log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 0, z' - 2 >= 0
if(z, z', z'') -{ 1 }→ z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
inc(z) -{ 1 }→ 0 :|: z = 0
inc(z) -{ 1 }→ 1 + inc(z - 1) :|: z - 1 >= 0
le(z, z') -{ 1 }→ le(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
log(z) -{ 1 }→ log2(z, 0) :|: z >= 0
log2(z, z') -{ 3 }→ if(le(z - 1, 0), 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
log2(z, z') -{ 3 }→ if(le(z - 1, 0), 1 + (z - 1), 1 + inc(z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
log2(z, z') -{ 3 }→ if(1, 0, 0) :|: z = 0, z' = 0
log2(z, z') -{ 3 }→ if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0

Function symbols to be analyzed: {half}, {inc}, {log2,if}, {log}
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1]

(19) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(20) Obligation:

Complexity RNTS consisting of the following rules:

half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 1 + half(z - 2) :|: z - 2 >= 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
if(z, z', z'') -{ 2 }→ log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 0, z' - 2 >= 0
if(z, z', z'') -{ 1 }→ z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
inc(z) -{ 1 }→ 0 :|: z = 0
inc(z) -{ 1 }→ 1 + inc(z - 1) :|: z - 1 >= 0
le(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
log(z) -{ 1 }→ log2(z, 0) :|: z >= 0
log2(z, z') -{ 4 }→ if(s', 1 + (z - 1), 0) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' = 0
log2(z, z') -{ 4 }→ if(s'', 1 + (z - 1), 1 + inc(z' - 1)) :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
log2(z, z') -{ 3 }→ if(1, 0, 0) :|: z = 0, z' = 0
log2(z, z') -{ 3 }→ if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0

Function symbols to be analyzed: {half}, {inc}, {log2,if}, {log}
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1]

(21) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 1 + half(z - 2) :|: z - 2 >= 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
if(z, z', z'') -{ 2 }→ log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 0, z' - 2 >= 0
if(z, z', z'') -{ 1 }→ z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
inc(z) -{ 1 }→ 0 :|: z = 0
inc(z) -{ 1 }→ 1 + inc(z - 1) :|: z - 1 >= 0
le(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
log(z) -{ 1 }→ log2(z, 0) :|: z >= 0
log2(z, z') -{ 4 }→ if(s', 1 + (z - 1), 0) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' = 0
log2(z, z') -{ 4 }→ if(s'', 1 + (z - 1), 1 + inc(z' - 1)) :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
log2(z, z') -{ 3 }→ if(1, 0, 0) :|: z = 0, z' = 0
log2(z, z') -{ 3 }→ if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0

Function symbols to be analyzed: {half}, {inc}, {log2,if}, {log}
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1] half: runtime: ?, size: O(n1) [z]

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

Computed RUNTIME bound using PUBS for: half
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z

(24) Obligation:

Complexity RNTS consisting of the following rules:

half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ 1 }→ 1 + half(z - 2) :|: z - 2 >= 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
if(z, z', z'') -{ 2 }→ log2(1 + half(z' - 2), z'') :|: z'' >= 0, z = 0, z' - 2 >= 0
if(z, z', z'') -{ 1 }→ z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
inc(z) -{ 1 }→ 0 :|: z = 0
inc(z) -{ 1 }→ 1 + inc(z - 1) :|: z - 1 >= 0
le(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
log(z) -{ 1 }→ log2(z, 0) :|: z >= 0
log2(z, z') -{ 4 }→ if(s', 1 + (z - 1), 0) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' = 0
log2(z, z') -{ 4 }→ if(s'', 1 + (z - 1), 1 + inc(z' - 1)) :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
log2(z, z') -{ 3 }→ if(1, 0, 0) :|: z = 0, z' = 0
log2(z, z') -{ 3 }→ if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0

Function symbols to be analyzed: {inc}, {log2,if}, {log}
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1] half: runtime: O(n1) [1 + z], size: O(n1) [z]

(25) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(26) Obligation:

Complexity RNTS consisting of the following rules:

half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 2), z - 2 >= 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
if(z, z', z'') -{ 1 + z' }→ log2(1 + s2, z'') :|: s2 >= 0, s2 <= 1 * (z' - 2), z'' >= 0, z = 0, z' - 2 >= 0
if(z, z', z'') -{ 1 }→ z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
inc(z) -{ 1 }→ 0 :|: z = 0
inc(z) -{ 1 }→ 1 + inc(z - 1) :|: z - 1 >= 0
le(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
log(z) -{ 1 }→ log2(z, 0) :|: z >= 0
log2(z, z') -{ 4 }→ if(s', 1 + (z - 1), 0) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' = 0
log2(z, z') -{ 4 }→ if(s'', 1 + (z - 1), 1 + inc(z' - 1)) :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
log2(z, z') -{ 3 }→ if(1, 0, 0) :|: z = 0, z' = 0
log2(z, z') -{ 3 }→ if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0

Function symbols to be analyzed: {inc}, {log2,if}, {log}
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1] half: runtime: O(n1) [1 + z], size: O(n1) [z]

(27) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(28) Obligation:

Complexity RNTS consisting of the following rules:

half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 2), z - 2 >= 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
if(z, z', z'') -{ 1 + z' }→ log2(1 + s2, z'') :|: s2 >= 0, s2 <= 1 * (z' - 2), z'' >= 0, z = 0, z' - 2 >= 0
if(z, z', z'') -{ 1 }→ z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
inc(z) -{ 1 }→ 0 :|: z = 0
inc(z) -{ 1 }→ 1 + inc(z - 1) :|: z - 1 >= 0
le(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
log(z) -{ 1 }→ log2(z, 0) :|: z >= 0
log2(z, z') -{ 4 }→ if(s', 1 + (z - 1), 0) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' = 0
log2(z, z') -{ 4 }→ if(s'', 1 + (z - 1), 1 + inc(z' - 1)) :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
log2(z, z') -{ 3 }→ if(1, 0, 0) :|: z = 0, z' = 0
log2(z, z') -{ 3 }→ if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0

Function symbols to be analyzed: {inc}, {log2,if}, {log}
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1] half: runtime: O(n1) [1 + z], size: O(n1) [z] inc: runtime: ?, size: O(n1) [z]

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

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

(30) Obligation:

Complexity RNTS consisting of the following rules:

half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 2), z - 2 >= 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
if(z, z', z'') -{ 1 + z' }→ log2(1 + s2, z'') :|: s2 >= 0, s2 <= 1 * (z' - 2), z'' >= 0, z = 0, z' - 2 >= 0
if(z, z', z'') -{ 1 }→ z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
inc(z) -{ 1 }→ 0 :|: z = 0
inc(z) -{ 1 }→ 1 + inc(z - 1) :|: z - 1 >= 0
le(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
log(z) -{ 1 }→ log2(z, 0) :|: z >= 0
log2(z, z') -{ 4 }→ if(s', 1 + (z - 1), 0) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' = 0
log2(z, z') -{ 4 }→ if(s'', 1 + (z - 1), 1 + inc(z' - 1)) :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
log2(z, z') -{ 3 }→ if(1, 0, 0) :|: z = 0, z' = 0
log2(z, z') -{ 3 }→ if(1, 0, 1 + inc(z' - 1)) :|: z' - 1 >= 0, z = 0

Function symbols to be analyzed: {log2,if}, {log}
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1] half: runtime: O(n1) [1 + z], size: O(n1) [z] inc: runtime: O(n1) [1 + z], size: O(n1) [z]

(31) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(32) Obligation:

Complexity RNTS consisting of the following rules:

half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 2), z - 2 >= 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
if(z, z', z'') -{ 1 + z' }→ log2(1 + s2, z'') :|: s2 >= 0, s2 <= 1 * (z' - 2), z'' >= 0, z = 0, z' - 2 >= 0
if(z, z', z'') -{ 1 }→ z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
inc(z) -{ 1 }→ 0 :|: z = 0
inc(z) -{ 1 + z }→ 1 + s3 :|: s3 >= 0, s3 <= 1 * (z - 1), z - 1 >= 0
le(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
log(z) -{ 1 }→ log2(z, 0) :|: z >= 0
log2(z, z') -{ 4 }→ if(s', 1 + (z - 1), 0) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' = 0
log2(z, z') -{ 4 + z' }→ if(s'', 1 + (z - 1), 1 + s5) :|: s5 >= 0, s5 <= 1 * (z' - 1), s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
log2(z, z') -{ 3 }→ if(1, 0, 0) :|: z = 0, z' = 0
log2(z, z') -{ 3 + z' }→ if(1, 0, 1 + s4) :|: s4 >= 0, s4 <= 1 * (z' - 1), z' - 1 >= 0, z = 0

Function symbols to be analyzed: {log2,if}, {log}
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1] half: runtime: O(n1) [1 + z], size: O(n1) [z] inc: runtime: O(n1) [1 + z], size: O(n1) [z]

(33) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using KoAT for: log2
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z'

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

(34) Obligation:

Complexity RNTS consisting of the following rules:

half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 2), z - 2 >= 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
if(z, z', z'') -{ 1 + z' }→ log2(1 + s2, z'') :|: s2 >= 0, s2 <= 1 * (z' - 2), z'' >= 0, z = 0, z' - 2 >= 0
if(z, z', z'') -{ 1 }→ z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
inc(z) -{ 1 }→ 0 :|: z = 0
inc(z) -{ 1 + z }→ 1 + s3 :|: s3 >= 0, s3 <= 1 * (z - 1), z - 1 >= 0
le(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
log(z) -{ 1 }→ log2(z, 0) :|: z >= 0
log2(z, z') -{ 4 }→ if(s', 1 + (z - 1), 0) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' = 0
log2(z, z') -{ 4 + z' }→ if(s'', 1 + (z - 1), 1 + s5) :|: s5 >= 0, s5 <= 1 * (z' - 1), s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
log2(z, z') -{ 3 }→ if(1, 0, 0) :|: z = 0, z' = 0
log2(z, z') -{ 3 + z' }→ if(1, 0, 1 + s4) :|: s4 >= 0, s4 <= 1 * (z' - 1), z' - 1 >= 0, z = 0

Function symbols to be analyzed: {log2,if}, {log}
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1] half: runtime: O(n1) [1 + z], size: O(n1) [z] inc: runtime: O(n1) [1 + z], size: O(n1) [z] log2: runtime: ?, size: O(n1) [z'] if: runtime: ?, size: O(n1) [z'']

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

Computed RUNTIME bound using PUBS for: log2
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 5 + 6·z + z·z' + z² + z'

Computed RUNTIME bound using PUBS for: if
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 7 + 5·z' + z'·z'' + z'² + z''

(36) Obligation:

Complexity RNTS consisting of the following rules:

half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 2), z - 2 >= 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z = 0, z' = 0
if(z, z', z'') -{ 2 }→ log2(0, z'') :|: z'' >= 0, z' = 1 + 0, z = 0
if(z, z', z'') -{ 1 + z' }→ log2(1 + s2, z'') :|: s2 >= 0, s2 <= 1 * (z' - 2), z'' >= 0, z = 0, z' - 2 >= 0
if(z, z', z'') -{ 1 }→ z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
inc(z) -{ 1 }→ 0 :|: z = 0
inc(z) -{ 1 + z }→ 1 + s3 :|: s3 >= 0, s3 <= 1 * (z - 1), z - 1 >= 0
le(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
log(z) -{ 1 }→ log2(z, 0) :|: z >= 0
log2(z, z') -{ 4 }→ if(s', 1 + (z - 1), 0) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' = 0
log2(z, z') -{ 4 + z' }→ if(s'', 1 + (z - 1), 1 + s5) :|: s5 >= 0, s5 <= 1 * (z' - 1), s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
log2(z, z') -{ 3 }→ if(1, 0, 0) :|: z = 0, z' = 0
log2(z, z') -{ 3 + z' }→ if(1, 0, 1 + s4) :|: s4 >= 0, s4 <= 1 * (z' - 1), z' - 1 >= 0, z = 0

Function symbols to be analyzed: {log}
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1] half: runtime: O(n1) [1 + z], size: O(n1) [z] inc: runtime: O(n1) [1 + z], size: O(n1) [z] log2: runtime: O(n2) [5 + 6·z + z·z' + z2 + z'], size: O(n1) [z'] if: runtime: O(n2) [7 + 5·z' + z'·z'' + z'2 + z''], size: O(n1) [z'']

(37) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(38) Obligation:

Complexity RNTS consisting of the following rules:

half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 2), z - 2 >= 0
if(z, z', z'') -{ 7 + z'' }→ s11 :|: s11 >= 0, s11 <= 1 * z'', z'' >= 0, z = 0, z' = 0
if(z, z', z'') -{ 7 + z'' }→ s12 :|: s12 >= 0, s12 <= 1 * z'', z'' >= 0, z' = 1 + 0, z = 0
if(z, z', z'') -{ 13 + 8·s2 + s2·z'' + s2² + z' + 2·z'' }→ s13 :|: s13 >= 0, s13 <= 1 * z'', s2 >= 0, s2 <= 1 * (z' - 2), z'' >= 0, z = 0, z' - 2 >= 0
if(z, z', z'') -{ 1 }→ z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
inc(z) -{ 1 }→ 0 :|: z = 0
inc(z) -{ 1 + z }→ 1 + s3 :|: s3 >= 0, s3 <= 1 * (z - 1), z - 1 >= 0
le(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
log(z) -{ 6 + 6·z + z² }→ s6 :|: s6 >= 0, s6 <= 1 * 0, z >= 0
log2(z, z') -{ 12 + s5 + s5·z + 6·z + z² + z' }→ s10 :|: s10 >= 0, s10 <= 1 * (1 + s5), s5 >= 0, s5 <= 1 * (z' - 1), s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
log2(z, z') -{ 10 }→ s7 :|: s7 >= 0, s7 <= 1 * 0, z = 0, z' = 0
log2(z, z') -{ 11 + s4 + z' }→ s8 :|: s8 >= 0, s8 <= 1 * (1 + s4), s4 >= 0, s4 <= 1 * (z' - 1), z' - 1 >= 0, z = 0
log2(z, z') -{ 11 + 5·z + z² }→ s9 :|: s9 >= 0, s9 <= 1 * 0, s' >= 0, s' <= 1, z - 1 >= 0, z' = 0

Function symbols to be analyzed: {log}
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1] half: runtime: O(n1) [1 + z], size: O(n1) [z] inc: runtime: O(n1) [1 + z], size: O(n1) [z] log2: runtime: O(n2) [5 + 6·z + z·z' + z2 + z'], size: O(n1) [z'] if: runtime: O(n2) [7 + 5·z' + z'·z'' + z'2 + z''], size: O(n1) [z'']

(39) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(40) Obligation:

Complexity RNTS consisting of the following rules:

half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 2), z - 2 >= 0
if(z, z', z'') -{ 7 + z'' }→ s11 :|: s11 >= 0, s11 <= 1 * z'', z'' >= 0, z = 0, z' = 0
if(z, z', z'') -{ 7 + z'' }→ s12 :|: s12 >= 0, s12 <= 1 * z'', z'' >= 0, z' = 1 + 0, z = 0
if(z, z', z'') -{ 13 + 8·s2 + s2·z'' + s2² + z' + 2·z'' }→ s13 :|: s13 >= 0, s13 <= 1 * z'', s2 >= 0, s2 <= 1 * (z' - 2), z'' >= 0, z = 0, z' - 2 >= 0
if(z, z', z'') -{ 1 }→ z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
inc(z) -{ 1 }→ 0 :|: z = 0
inc(z) -{ 1 + z }→ 1 + s3 :|: s3 >= 0, s3 <= 1 * (z - 1), z - 1 >= 0
le(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
log(z) -{ 6 + 6·z + z² }→ s6 :|: s6 >= 0, s6 <= 1 * 0, z >= 0
log2(z, z') -{ 12 + s5 + s5·z + 6·z + z² + z' }→ s10 :|: s10 >= 0, s10 <= 1 * (1 + s5), s5 >= 0, s5 <= 1 * (z' - 1), s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
log2(z, z') -{ 10 }→ s7 :|: s7 >= 0, s7 <= 1 * 0, z = 0, z' = 0
log2(z, z') -{ 11 + s4 + z' }→ s8 :|: s8 >= 0, s8 <= 1 * (1 + s4), s4 >= 0, s4 <= 1 * (z' - 1), z' - 1 >= 0, z = 0
log2(z, z') -{ 11 + 5·z + z² }→ s9 :|: s9 >= 0, s9 <= 1 * 0, s' >= 0, s' <= 1, z - 1 >= 0, z' = 0

Function symbols to be analyzed: {log}
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1] half: runtime: O(n1) [1 + z], size: O(n1) [z] inc: runtime: O(n1) [1 + z], size: O(n1) [z] log2: runtime: O(n2) [5 + 6·z + z·z' + z2 + z'], size: O(n1) [z'] if: runtime: O(n2) [7 + 5·z' + z'·z'' + z'2 + z''], size: O(n1) [z''] log: runtime: ?, size: O(1) [0]

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

Computed RUNTIME bound using KoAT for: log
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 6 + 6·z + z²

(42) Obligation:

Complexity RNTS consisting of the following rules:

half(z) -{ 1 }→ 0 :|: z = 0
half(z) -{ 1 }→ 0 :|: z = 1 + 0
half(z) -{ z }→ 1 + s1 :|: s1 >= 0, s1 <= 1 * (z - 2), z - 2 >= 0
if(z, z', z'') -{ 7 + z'' }→ s11 :|: s11 >= 0, s11 <= 1 * z'', z'' >= 0, z = 0, z' = 0
if(z, z', z'') -{ 7 + z'' }→ s12 :|: s12 >= 0, s12 <= 1 * z'', z'' >= 0, z' = 1 + 0, z = 0
if(z, z', z'') -{ 13 + 8·s2 + s2·z'' + s2² + z' + 2·z'' }→ s13 :|: s13 >= 0, s13 <= 1 * z'', s2 >= 0, s2 <= 1 * (z' - 2), z'' >= 0, z = 0, z' - 2 >= 0
if(z, z', z'') -{ 1 }→ z'' - 1 :|: z = 1, z' >= 0, z'' - 1 >= 0
inc(z) -{ 1 }→ 0 :|: z = 0
inc(z) -{ 1 + z }→ 1 + s3 :|: s3 >= 0, s3 <= 1 * (z - 1), z - 1 >= 0
le(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
le(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
le(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
log(z) -{ 6 + 6·z + z² }→ s6 :|: s6 >= 0, s6 <= 1 * 0, z >= 0
log2(z, z') -{ 12 + s5 + s5·z + 6·z + z² + z' }→ s10 :|: s10 >= 0, s10 <= 1 * (1 + s5), s5 >= 0, s5 <= 1 * (z' - 1), s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
log2(z, z') -{ 10 }→ s7 :|: s7 >= 0, s7 <= 1 * 0, z = 0, z' = 0
log2(z, z') -{ 11 + s4 + z' }→ s8 :|: s8 >= 0, s8 <= 1 * (1 + s4), s4 >= 0, s4 <= 1 * (z' - 1), z' - 1 >= 0, z = 0
log2(z, z') -{ 11 + 5·z + z² }→ s9 :|: s9 >= 0, s9 <= 1 * 0, s' >= 0, s' <= 1, z - 1 >= 0, z' = 0

Function symbols to be analyzed:
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1] half: runtime: O(n1) [1 + z], size: O(n1) [z] inc: runtime: O(n1) [1 + z], size: O(n1) [z] log2: runtime: O(n2) [5 + 6·z + z·z' + z2 + z'], size: O(n1) [z'] if: runtime: O(n2) [7 + 5·z' + z'·z'' + z'2 + z''], size: O(n1) [z''] log: runtime: O(n2) [6 + 6·z + z2], size: O(1) [0]

(43) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(44) BOUNDS(1, n^2)