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

(0) Obligation:

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


The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(0, y) → 0
minus(s(x), y) → if_minus(le(s(x), y), s(x), y)
if_minus(true, s(x), y) → 0
if_minus(false, s(x), y) → s(minus(x, y))
gcd(0, y) → y
gcd(s(x), 0) → s(x)
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y))
if_gcd(true, s(x), s(y)) → gcd(minus(x, y), s(y))
if_gcd(false, s(x), s(y)) → gcd(minus(y, x), s(x))

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^3).


The TRS R consists of the following rules:

le(0, y) → true [1]
le(s(x), 0) → false [1]
le(s(x), s(y)) → le(x, y) [1]
minus(0, y) → 0 [1]
minus(s(x), y) → if_minus(le(s(x), y), s(x), y) [1]
if_minus(true, s(x), y) → 0 [1]
if_minus(false, s(x), y) → s(minus(x, y)) [1]
gcd(0, y) → y [1]
gcd(s(x), 0) → s(x) [1]
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y)) [1]
if_gcd(true, s(x), s(y)) → gcd(minus(x, y), s(y)) [1]
if_gcd(false, s(x), s(y)) → gcd(minus(y, x), s(x)) [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:

le(0, y) → true [1]
le(s(x), 0) → false [1]
le(s(x), s(y)) → le(x, y) [1]
minus(0, y) → 0 [1]
minus(s(x), y) → if_minus(le(s(x), y), s(x), y) [1]
if_minus(true, s(x), y) → 0 [1]
if_minus(false, s(x), y) → s(minus(x, y)) [1]
gcd(0, y) → y [1]
gcd(s(x), 0) → s(x) [1]
gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y)) [1]
if_gcd(true, s(x), s(y)) → gcd(minus(x, y), s(y)) [1]
if_gcd(false, s(x), s(y)) → gcd(minus(y, x), s(x)) [1]

The TRS has the following type information:

le :: 0:s → 0:s → true:false 0 :: 0:s true :: true:false s :: 0:s → 0:s false :: true:false minus :: 0:s → 0:s → 0:s if_minus :: true:false → 0:s → 0:s → 0:s gcd :: 0:s → 0:s → 0:s if_gcd :: 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:

gcd
if_gcd

(c) The following functions are completely defined:

le
minus
if_minus

Due to the following rules being added:

if_minus(v0, v1, v2) → 0 [0]

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: le(0, y) → true [1] le(s(x), 0) → false [1] le(s(x), s(y)) → le(x, y) [1] minus(0, y) → 0 [1] minus(s(x), y) → if_minus(le(s(x), y), s(x), y) [1] if_minus(true, s(x), y) → 0 [1] if_minus(false, s(x), y) → s(minus(x, y)) [1] gcd(0, y) → y [1] gcd(s(x), 0) → s(x) [1] gcd(s(x), s(y)) → if_gcd(le(y, x), s(x), s(y)) [1] if_gcd(true, s(x), s(y)) → gcd(minus(x, y), s(y)) [1] if_gcd(false, s(x), s(y)) → gcd(minus(y, x), s(x)) [1] if_minus(v0, v1, v2) → 0 [0] The TRS has the following type information: le :: 0:s → 0:s → true:false 0 :: 0:s true :: true:false s :: 0:s → 0:s false :: true:false minus :: 0:s → 0:s → 0:s if_minus :: true:false → 0:s → 0:s → 0:s gcd :: 0:s → 0:s → 0:s if_gcd :: 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: le(0, y) → true [1] le(s(x), 0) → false [1] le(s(x), s(y)) → le(x, y) [1] minus(0, y) → 0 [1] minus(s(x), 0) → if_minus(false, s(x), 0) [2] minus(s(x), s(y')) → if_minus(le(x, y'), s(x), s(y')) [2] if_minus(true, s(x), y) → 0 [1] if_minus(false, s(x), y) → s(minus(x, y)) [1] gcd(0, y) → y [1] gcd(s(x), 0) → s(x) [1] gcd(s(x), s(0)) → if_gcd(true, s(x), s(0)) [2] gcd(s(0), s(s(x'))) → if_gcd(false, s(0), s(s(x'))) [2] gcd(s(s(y'')), s(s(x''))) → if_gcd(le(x'', y''), s(s(y'')), s(s(x''))) [2] if_gcd(true, s(0), s(y)) → gcd(0, s(y)) [2] if_gcd(true, s(s(x1)), s(y)) → gcd(if_minus(le(s(x1), y), s(x1), y), s(y)) [2] if_gcd(false, s(x), s(0)) → gcd(0, s(x)) [2] if_gcd(false, s(x), s(s(x2))) → gcd(if_minus(le(s(x2), x), s(x2), x), s(x)) [2] if_minus(v0, v1, v2) → 0 [0] The TRS has the following type information: le :: 0:s → 0:s → true:false 0 :: 0:s true :: true:false s :: 0:s → 0:s false :: true:false minus :: 0:s → 0:s → 0:s if_minus :: true:false → 0:s → 0:s → 0:s gcd :: 0:s → 0:s → 0:s if_gcd :: 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:

gcd(z, z') -{ 1 }→ y :|: y >= 0, z = 0, z' = y
gcd(z, z') -{ 2 }→ if_gcd(le(x'', y''), 1 + (1 + y''), 1 + (1 + x'')) :|: z' = 1 + (1 + x''), z = 1 + (1 + y''), y'' >= 0, x'' >= 0
gcd(z, z') -{ 2 }→ if_gcd(1, 1 + x, 1 + 0) :|: x >= 0, z' = 1 + 0, z = 1 + x
gcd(z, z') -{ 2 }→ if_gcd(0, 1 + 0, 1 + (1 + x')) :|: z' = 1 + (1 + x'), z = 1 + 0, x' >= 0
gcd(z, z') -{ 1 }→ 1 + x :|: x >= 0, z = 1 + x, z' = 0
if_gcd(z, z', z'') -{ 2 }→ gcd(if_minus(le(1 + x1, y), 1 + x1, y), 1 + y) :|: x1 >= 0, z' = 1 + (1 + x1), z = 1, y >= 0, z'' = 1 + y
if_gcd(z, z', z'') -{ 2 }→ gcd(if_minus(le(1 + x2, x), 1 + x2, x), 1 + x) :|: z' = 1 + x, x >= 0, z'' = 1 + (1 + x2), z = 0, x2 >= 0
if_gcd(z, z', z'') -{ 2 }→ gcd(0, 1 + x) :|: z' = 1 + x, x >= 0, z = 0, z'' = 1 + 0
if_gcd(z, z', z'') -{ 2 }→ gcd(0, 1 + y) :|: z = 1, y >= 0, z'' = 1 + y, z' = 1 + 0
if_minus(z, z', z'') -{ 1 }→ 0 :|: z' = 1 + x, z'' = y, z = 1, x >= 0, y >= 0
if_minus(z, z', z'') -{ 0 }→ 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
if_minus(z, z', z'') -{ 1 }→ 1 + minus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0, z = 0
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
minus(z, z') -{ 2 }→ if_minus(le(x, y'), 1 + x, 1 + y') :|: x >= 0, y' >= 0, z = 1 + x, z' = 1 + y'
minus(z, z') -{ 2 }→ if_minus(0, 1 + x, 0) :|: x >= 0, z = 1 + x, z' = 0
minus(z, z') -{ 1 }→ 0 :|: y >= 0, z = 0, z' = y

(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:

gcd(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
gcd(z, z') -{ 2 }→ if_gcd(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
gcd(z, z') -{ 2 }→ if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
gcd(z, z') -{ 2 }→ if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
gcd(z, z') -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0, z' = 0
if_gcd(z, z', z'') -{ 2 }→ gcd(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z' - 2 >= 0, z = 1, z'' - 1 >= 0
if_gcd(z, z', z'') -{ 2 }→ gcd(if_minus(le(1 + (z'' - 2), z' - 1), 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' - 2 >= 0
if_gcd(z, z', z'') -{ 2 }→ gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
if_gcd(z, z', z'') -{ 2 }→ gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
if_minus(z, z', z'') -{ 1 }→ 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
if_minus(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
if_minus(z, z', z'') -{ 1 }→ 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 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
minus(z, z') -{ 2 }→ if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 2 }→ if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

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

Found the following analysis order by SCC decomposition:

{ le } { minus, if_minus } { gcd, if_gcd }

(14) Obligation:

Complexity RNTS consisting of the following rules:

gcd(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
gcd(z, z') -{ 2 }→ if_gcd(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
gcd(z, z') -{ 2 }→ if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
gcd(z, z') -{ 2 }→ if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
gcd(z, z') -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0, z' = 0
if_gcd(z, z', z'') -{ 2 }→ gcd(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z' - 2 >= 0, z = 1, z'' - 1 >= 0
if_gcd(z, z', z'') -{ 2 }→ gcd(if_minus(le(1 + (z'' - 2), z' - 1), 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' - 2 >= 0
if_gcd(z, z', z'') -{ 2 }→ gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
if_gcd(z, z', z'') -{ 2 }→ gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
if_minus(z, z', z'') -{ 1 }→ 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
if_minus(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
if_minus(z, z', z'') -{ 1 }→ 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 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
minus(z, z') -{ 2 }→ if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 2 }→ if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {le}, {minus,if_minus}, {gcd,if_gcd}

(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:

gcd(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
gcd(z, z') -{ 2 }→ if_gcd(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
gcd(z, z') -{ 2 }→ if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
gcd(z, z') -{ 2 }→ if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
gcd(z, z') -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0, z' = 0
if_gcd(z, z', z'') -{ 2 }→ gcd(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z' - 2 >= 0, z = 1, z'' - 1 >= 0
if_gcd(z, z', z'') -{ 2 }→ gcd(if_minus(le(1 + (z'' - 2), z' - 1), 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' - 2 >= 0
if_gcd(z, z', z'') -{ 2 }→ gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
if_gcd(z, z', z'') -{ 2 }→ gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
if_minus(z, z', z'') -{ 1 }→ 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
if_minus(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
if_minus(z, z', z'') -{ 1 }→ 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 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
minus(z, z') -{ 2 }→ if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 2 }→ if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {le}, {minus,if_minus}, {gcd,if_gcd}
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:

gcd(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
gcd(z, z') -{ 2 }→ if_gcd(le(z' - 2, z - 2), 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
gcd(z, z') -{ 2 }→ if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
gcd(z, z') -{ 2 }→ if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
gcd(z, z') -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0, z' = 0
if_gcd(z, z', z'') -{ 2 }→ gcd(if_minus(le(1 + (z' - 2), z'' - 1), 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: z' - 2 >= 0, z = 1, z'' - 1 >= 0
if_gcd(z, z', z'') -{ 2 }→ gcd(if_minus(le(1 + (z'' - 2), z' - 1), 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' - 2 >= 0
if_gcd(z, z', z'') -{ 2 }→ gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
if_gcd(z, z', z'') -{ 2 }→ gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
if_minus(z, z', z'') -{ 1 }→ 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
if_minus(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
if_minus(z, z', z'') -{ 1 }→ 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 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
minus(z, z') -{ 2 }→ if_minus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 2 }→ if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {minus,if_minus}, {gcd,if_gcd}
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:

gcd(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
gcd(z, z') -{ 1 + z }→ if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
gcd(z, z') -{ 2 }→ if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
gcd(z, z') -{ 2 }→ if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
gcd(z, z') -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0, z' = 0
if_gcd(z, z', z'') -{ 2 + z'' }→ gcd(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
if_gcd(z, z', z'') -{ 2 + z' }→ gcd(if_minus(s2, 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z = 0, z'' - 2 >= 0
if_gcd(z, z', z'') -{ 2 }→ gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
if_gcd(z, z', z'') -{ 2 }→ gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
if_minus(z, z', z'') -{ 1 }→ 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
if_minus(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
if_minus(z, z', z'') -{ 1 }→ 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 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
minus(z, z') -{ 2 + z' }→ if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 2 }→ if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {minus,if_minus}, {gcd,if_gcd}
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: minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z

Computed SIZE bound using CoFloCo for: if_minus
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:

gcd(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
gcd(z, z') -{ 1 + z }→ if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
gcd(z, z') -{ 2 }→ if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
gcd(z, z') -{ 2 }→ if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
gcd(z, z') -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0, z' = 0
if_gcd(z, z', z'') -{ 2 + z'' }→ gcd(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
if_gcd(z, z', z'') -{ 2 + z' }→ gcd(if_minus(s2, 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z = 0, z'' - 2 >= 0
if_gcd(z, z', z'') -{ 2 }→ gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
if_gcd(z, z', z'') -{ 2 }→ gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
if_minus(z, z', z'') -{ 1 }→ 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
if_minus(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
if_minus(z, z', z'') -{ 1 }→ 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 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
minus(z, z') -{ 2 + z' }→ if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 2 }→ if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {minus,if_minus}, {gcd,if_gcd}
Previous analysis results are:

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

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

Computed RUNTIME bound using PUBS for: minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 3 + 3·z + z·z' + z'

Computed RUNTIME bound using PUBS for: if_minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 1 + 3·z' + z'·z''

(24) Obligation:

Complexity RNTS consisting of the following rules:

gcd(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
gcd(z, z') -{ 1 + z }→ if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
gcd(z, z') -{ 2 }→ if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
gcd(z, z') -{ 2 }→ if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
gcd(z, z') -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0, z' = 0
if_gcd(z, z', z'') -{ 2 + z'' }→ gcd(if_minus(s1, 1 + (z' - 2), z'' - 1), 1 + (z'' - 1)) :|: s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
if_gcd(z, z', z'') -{ 2 + z' }→ gcd(if_minus(s2, 1 + (z'' - 2), z' - 1), 1 + (z' - 1)) :|: s2 >= 0, s2 <= 1, z' - 1 >= 0, z = 0, z'' - 2 >= 0
if_gcd(z, z', z'') -{ 2 }→ gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
if_gcd(z, z', z'') -{ 2 }→ gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
if_minus(z, z', z'') -{ 1 }→ 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
if_minus(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
if_minus(z, z', z'') -{ 1 }→ 1 + minus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0, z = 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
minus(z, z') -{ 2 + z' }→ if_minus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 2 }→ if_minus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {gcd,if_gcd}
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1] minus: runtime: O(n2) [3 + 3·z + z·z' + z'], size: O(n1) [z] if_minus: runtime: O(n2) [1 + 3·z' + z'·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:

gcd(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
gcd(z, z') -{ 1 + z }→ if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
gcd(z, z') -{ 2 }→ if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
gcd(z, z') -{ 2 }→ if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
gcd(z, z') -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0, z' = 0
if_gcd(z, z', z'') -{ 1 + 2·z' + z'·z'' }→ gcd(s6, 1 + (z'' - 1)) :|: s6 >= 0, s6 <= 1 * (1 + (z' - 2)), s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
if_gcd(z, z', z'') -{ 1 + z'·z'' + 2·z'' }→ gcd(s7, 1 + (z' - 1)) :|: s7 >= 0, s7 <= 1 * (1 + (z'' - 2)), s2 >= 0, s2 <= 1, z' - 1 >= 0, z = 0, z'' - 2 >= 0
if_gcd(z, z', z'') -{ 2 }→ gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
if_gcd(z, z', z'') -{ 2 }→ gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
if_minus(z, z', z'') -{ 1 }→ 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
if_minus(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
if_minus(z, z', z'') -{ 1 + 3·z' + z'·z'' }→ 1 + s5 :|: s5 >= 0, s5 <= 1 * (z' - 1), z' - 1 >= 0, z'' >= 0, z = 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
minus(z, z') -{ 3 + 3·z }→ s3 :|: s3 >= 0, s3 <= 1 * (1 + (z - 1)), z - 1 >= 0, z' = 0
minus(z, z') -{ 3 + 3·z + z·z' + z' }→ s4 :|: s4 >= 0, s4 <= 1 * (1 + (z - 1)), s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {gcd,if_gcd}
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1] minus: runtime: O(n2) [3 + 3·z + z·z' + z'], size: O(n1) [z] if_minus: runtime: O(n2) [1 + 3·z' + z'·z''], size: O(n1) [z']

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

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

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

(28) Obligation:

Complexity RNTS consisting of the following rules:

gcd(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
gcd(z, z') -{ 1 + z }→ if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
gcd(z, z') -{ 2 }→ if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
gcd(z, z') -{ 2 }→ if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
gcd(z, z') -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0, z' = 0
if_gcd(z, z', z'') -{ 1 + 2·z' + z'·z'' }→ gcd(s6, 1 + (z'' - 1)) :|: s6 >= 0, s6 <= 1 * (1 + (z' - 2)), s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
if_gcd(z, z', z'') -{ 1 + z'·z'' + 2·z'' }→ gcd(s7, 1 + (z' - 1)) :|: s7 >= 0, s7 <= 1 * (1 + (z'' - 2)), s2 >= 0, s2 <= 1, z' - 1 >= 0, z = 0, z'' - 2 >= 0
if_gcd(z, z', z'') -{ 2 }→ gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
if_gcd(z, z', z'') -{ 2 }→ gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
if_minus(z, z', z'') -{ 1 }→ 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
if_minus(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
if_minus(z, z', z'') -{ 1 + 3·z' + z'·z'' }→ 1 + s5 :|: s5 >= 0, s5 <= 1 * (z' - 1), z' - 1 >= 0, z'' >= 0, z = 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
minus(z, z') -{ 3 + 3·z }→ s3 :|: s3 >= 0, s3 <= 1 * (1 + (z - 1)), z - 1 >= 0, z' = 0
minus(z, z') -{ 3 + 3·z + z·z' + z' }→ s4 :|: s4 >= 0, s4 <= 1 * (1 + (z - 1)), s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {gcd,if_gcd}
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1] minus: runtime: O(n2) [3 + 3·z + z·z' + z'], size: O(n1) [z] if_minus: runtime: O(n2) [1 + 3·z' + z'·z''], size: O(n1) [z'] gcd: runtime: ?, size: O(n1) [z + z'] if_gcd: runtime: ?, size: O(n1) [z' + z'']

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

Computed RUNTIME bound using PUBS for: gcd
after applying outer abstraction to obtain an ITS,
resulting in: O(n³) with polynomial bound: 13 + 19·z + 16·z·z' + 3·z·z'² + 8·z² + 3·z²·z' + z³ + 19·z' + 8·z'² + z'³

Computed RUNTIME bound using PUBS for: if_gcd
after applying outer abstraction to obtain an ITS,
resulting in: O(n³) with polynomial bound: 15 + 19·z' + 11·z'·z'' + 3·z'·z''² + 8·z'² + 3·z'²·z'' + z'³ + 19·z'' + 8·z''² + z''³

(30) Obligation:

Complexity RNTS consisting of the following rules:

gcd(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
gcd(z, z') -{ 1 + z }→ if_gcd(s'', 1 + (1 + (z - 2)), 1 + (1 + (z' - 2))) :|: s'' >= 0, s'' <= 1, z - 2 >= 0, z' - 2 >= 0
gcd(z, z') -{ 2 }→ if_gcd(1, 1 + (z - 1), 1 + 0) :|: z - 1 >= 0, z' = 1 + 0
gcd(z, z') -{ 2 }→ if_gcd(0, 1 + 0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
gcd(z, z') -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0, z' = 0
if_gcd(z, z', z'') -{ 1 + 2·z' + z'·z'' }→ gcd(s6, 1 + (z'' - 1)) :|: s6 >= 0, s6 <= 1 * (1 + (z' - 2)), s1 >= 0, s1 <= 1, z' - 2 >= 0, z = 1, z'' - 1 >= 0
if_gcd(z, z', z'') -{ 1 + z'·z'' + 2·z'' }→ gcd(s7, 1 + (z' - 1)) :|: s7 >= 0, s7 <= 1 * (1 + (z'' - 2)), s2 >= 0, s2 <= 1, z' - 1 >= 0, z = 0, z'' - 2 >= 0
if_gcd(z, z', z'') -{ 2 }→ gcd(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 0, z'' = 1 + 0
if_gcd(z, z', z'') -{ 2 }→ gcd(0, 1 + (z'' - 1)) :|: z = 1, z'' - 1 >= 0, z' = 1 + 0
if_minus(z, z', z'') -{ 1 }→ 0 :|: z = 1, z' - 1 >= 0, z'' >= 0
if_minus(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
if_minus(z, z', z'') -{ 1 + 3·z' + z'·z'' }→ 1 + s5 :|: s5 >= 0, s5 <= 1 * (z' - 1), z' - 1 >= 0, z'' >= 0, z = 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
minus(z, z') -{ 3 + 3·z }→ s3 :|: s3 >= 0, s3 <= 1 * (1 + (z - 1)), z - 1 >= 0, z' = 0
minus(z, z') -{ 3 + 3·z + z·z' + z' }→ s4 :|: s4 >= 0, s4 <= 1 * (1 + (z - 1)), s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed:
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1] minus: runtime: O(n2) [3 + 3·z + z·z' + z'], size: O(n1) [z] if_minus: runtime: O(n2) [1 + 3·z' + z'·z''], size: O(n1) [z'] gcd: runtime: O(n3) [13 + 19·z + 16·z·z' + 3·z·z'2 + 8·z2 + 3·z2·z' + z3 + 19·z' + 8·z'2 + z'3], size: O(n1) [z + z'] if_gcd: runtime: O(n3) [15 + 19·z' + 11·z'·z'' + 3·z'·z''2 + 8·z'2 + 3·z'2·z'' + z'3 + 19·z'' + 8·z''2 + z''3], size: O(n1) [z' + z'']

(31) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(32) BOUNDS(1, n^3)