The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^5).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^5).


The TRS R consists of the following rules:

f(s(x1), x2, x3, x4, x5) → f(x1, x2, x3, x4, x5)
f(0, s(x2), x3, x4, x5) → f(x2, x2, x3, x4, x5)
f(0, 0, s(x3), x4, x5) → f(x3, x3, x3, x4, x5)
f(0, 0, 0, s(x4), x5) → f(x4, x4, x4, x4, x5)
f(0, 0, 0, 0, s(x5)) → f(x5, x5, x5, x5, x5)
f(0, 0, 0, 0, 0) → 0

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^5).


The TRS R consists of the following rules:

f(s(x1), x2, x3, x4, x5) → f(x1, x2, x3, x4, x5) [1]
f(0, s(x2), x3, x4, x5) → f(x2, x2, x3, x4, x5) [1]
f(0, 0, s(x3), x4, x5) → f(x3, x3, x3, x4, x5) [1]
f(0, 0, 0, s(x4), x5) → f(x4, x4, x4, x4, x5) [1]
f(0, 0, 0, 0, s(x5)) → f(x5, x5, x5, x5, x5) [1]
f(0, 0, 0, 0, 0) → 0 [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:

f(s(x1), x2, x3, x4, x5) → f(x1, x2, x3, x4, x5) [1]
f(0, s(x2), x3, x4, x5) → f(x2, x2, x3, x4, x5) [1]
f(0, 0, s(x3), x4, x5) → f(x3, x3, x3, x4, x5) [1]
f(0, 0, 0, s(x4), x5) → f(x4, x4, x4, x4, x5) [1]
f(0, 0, 0, 0, s(x5)) → f(x5, x5, x5, x5, x5) [1]
f(0, 0, 0, 0, 0) → 0 [1]

The TRS has the following type information:

f :: s:0 → s:0 → s:0 → s:0 → s:0 → s:0 s :: s:0 → s:0 0 :: s:0

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:

f

(c) The following functions are completely defined:
none

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: f(s(x1), x2, x3, x4, x5) → f(x1, x2, x3, x4, x5) [1] f(0, s(x2), x3, x4, x5) → f(x2, x2, x3, x4, x5) [1] f(0, 0, s(x3), x4, x5) → f(x3, x3, x3, x4, x5) [1] f(0, 0, 0, s(x4), x5) → f(x4, x4, x4, x4, x5) [1] f(0, 0, 0, 0, s(x5)) → f(x5, x5, x5, x5, x5) [1] f(0, 0, 0, 0, 0) → 0 [1] The TRS has the following type information: f :: s:0 → s:0 → s:0 → s:0 → s:0 → s:0 s :: s:0 → s:0 0 :: s:0 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: f(s(x1), x2, x3, x4, x5) → f(x1, x2, x3, x4, x5) [1] f(0, s(x2), x3, x4, x5) → f(x2, x2, x3, x4, x5) [1] f(0, 0, s(x3), x4, x5) → f(x3, x3, x3, x4, x5) [1] f(0, 0, 0, s(x4), x5) → f(x4, x4, x4, x4, x5) [1] f(0, 0, 0, 0, s(x5)) → f(x5, x5, x5, x5, x5) [1] f(0, 0, 0, 0, 0) → 0 [1] The TRS has the following type information: f :: s:0 → s:0 → s:0 → s:0 → s:0 → s:0 s :: s:0 → s:0 0 :: s:0 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

(10) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z', z'', z1, z2) -{ 1 }→ f(x1, x2, x3, x4, x5) :|: z' = x2, z1 = x4, x1 >= 0, x4 >= 0, x5 >= 0, z = 1 + x1, z'' = x3, z2 = x5, x2 >= 0, x3 >= 0
f(z, z', z'', z1, z2) -{ 1 }→ f(x2, x2, x3, x4, x5) :|: z1 = x4, x4 >= 0, x5 >= 0, z' = 1 + x2, z'' = x3, z = 0, z2 = x5, x2 >= 0, x3 >= 0
f(z, z', z'', z1, z2) -{ 1 }→ f(x3, x3, x3, x4, x5) :|: z1 = x4, x4 >= 0, x5 >= 0, z'' = 1 + x3, z = 0, z2 = x5, x3 >= 0, z' = 0
f(z, z', z'', z1, z2) -{ 1 }→ f(x4, x4, x4, x4, x5) :|: z'' = 0, x4 >= 0, x5 >= 0, z1 = 1 + x4, z = 0, z2 = x5, z' = 0
f(z, z', z'', z1, z2) -{ 1 }→ f(x5, x5, x5, x5, x5) :|: z'' = 0, z1 = 0, x5 >= 0, z = 0, z2 = 1 + x5, z' = 0
f(z, z', z'', z1, z2) -{ 1 }→ 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 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:

f(z, z', z'', z1, z2) -{ 1 }→ f(z - 1, z', z'', z1, z2) :|: z - 1 >= 0, z1 >= 0, z2 >= 0, z' >= 0, z'' >= 0
f(z, z', z'', z1, z2) -{ 1 }→ f(z' - 1, z' - 1, z'', z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z' - 1 >= 0, z'' >= 0
f(z, z', z'', z1, z2) -{ 1 }→ f(z'' - 1, z'' - 1, z'' - 1, z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z'' - 1 >= 0, z' = 0
f(z, z', z'', z1, z2) -{ 1 }→ f(z1 - 1, z1 - 1, z1 - 1, z1 - 1, z2) :|: z'' = 0, z1 - 1 >= 0, z2 >= 0, z = 0, z' = 0
f(z, z', z'', z1, z2) -{ 1 }→ f(z2 - 1, z2 - 1, z2 - 1, z2 - 1, z2 - 1) :|: z'' = 0, z1 = 0, z2 - 1 >= 0, z = 0, z' = 0
f(z, z', z'', z1, z2) -{ 1 }→ 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 0, z' = 0

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

Found the following analysis order by SCC decomposition:

{ f }

(14) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z', z'', z1, z2) -{ 1 }→ f(z - 1, z', z'', z1, z2) :|: z - 1 >= 0, z1 >= 0, z2 >= 0, z' >= 0, z'' >= 0
f(z, z', z'', z1, z2) -{ 1 }→ f(z' - 1, z' - 1, z'', z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z' - 1 >= 0, z'' >= 0
f(z, z', z'', z1, z2) -{ 1 }→ f(z'' - 1, z'' - 1, z'' - 1, z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z'' - 1 >= 0, z' = 0
f(z, z', z'', z1, z2) -{ 1 }→ f(z1 - 1, z1 - 1, z1 - 1, z1 - 1, z2) :|: z'' = 0, z1 - 1 >= 0, z2 >= 0, z = 0, z' = 0
f(z, z', z'', z1, z2) -{ 1 }→ f(z2 - 1, z2 - 1, z2 - 1, z2 - 1, z2 - 1) :|: z'' = 0, z1 = 0, z2 - 1 >= 0, z = 0, z' = 0
f(z, z', z'', z1, z2) -{ 1 }→ 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 0, z' = 0

Function symbols to be analyzed: {f}

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

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z', z'', z1, z2) -{ 1 }→ f(z - 1, z', z'', z1, z2) :|: z - 1 >= 0, z1 >= 0, z2 >= 0, z' >= 0, z'' >= 0
f(z, z', z'', z1, z2) -{ 1 }→ f(z' - 1, z' - 1, z'', z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z' - 1 >= 0, z'' >= 0
f(z, z', z'', z1, z2) -{ 1 }→ f(z'' - 1, z'' - 1, z'' - 1, z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z'' - 1 >= 0, z' = 0
f(z, z', z'', z1, z2) -{ 1 }→ f(z1 - 1, z1 - 1, z1 - 1, z1 - 1, z2) :|: z'' = 0, z1 - 1 >= 0, z2 >= 0, z = 0, z' = 0
f(z, z', z'', z1, z2) -{ 1 }→ f(z2 - 1, z2 - 1, z2 - 1, z2 - 1, z2 - 1) :|: z'' = 0, z1 = 0, z2 - 1 >= 0, z = 0, z' = 0
f(z, z', z'', z1, z2) -{ 1 }→ 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 0, z' = 0

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

f: runtime: ?, size: O(1) [0]

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

Computed RUNTIME bound using KoAT for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n⁵) with polynomial bound: 1 + z + z' + z'·z'' + z'·z''·z1 + z'·z''·z1·z2 + z'·z''·z1·z2² + z'·z''·z1² + z'·z''·z2 + z'·z''·z2² + z'·z''·z2³ + z'·z''² + z'·z1 + z'·z1·z2 + 3·z'·z1·z2² + 2·z'·z1·z2³ + z'·z1² + 2·z'·z1²·z2 + z'·z1²·z2² + z'·z1³ + z'·z2 + z'·z2² + 2·z'·z2³ + z'·z2⁴ + z'² + z'' + 2·z''·z1 + 5·z''·z1·z2 + 7·z''·z1·z2² + 4·z''·z1·z2³ + 4·z''·z1² + 4·z''·z1²·z2 + 2·z''·z1²·z2² + 2·z''·z1³ + 2·z''·z2 + 4·z''·z2² + 5·z''·z2³ + 2·z''·z2⁴ + 2·z''² + 2·z''²·z1 + z''²·z1·z2 + z''²·z1·z2² + z''²·z1² + 2·z''²·z2 + z''²·z2² + z''²·z2³ + z''³ + z1 + 3·z1·z2 + 9·z1·z2² + 9·z1·z2³ + 3·z1·z2⁴ + 3·z1² + 6·z1²·z2 + 7·z1²·z2² + 3·z1²·z2³ + 3·z1³ + 3·z1³·z2 + z1³·z2² + z1⁴ + z2 + 4·z2² + 6·z2³ + 4·z2⁴ + z2⁵

(18) Obligation:

Complexity RNTS consisting of the following rules:

f(z, z', z'', z1, z2) -{ 1 }→ f(z - 1, z', z'', z1, z2) :|: z - 1 >= 0, z1 >= 0, z2 >= 0, z' >= 0, z'' >= 0
f(z, z', z'', z1, z2) -{ 1 }→ f(z' - 1, z' - 1, z'', z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z' - 1 >= 0, z'' >= 0
f(z, z', z'', z1, z2) -{ 1 }→ f(z'' - 1, z'' - 1, z'' - 1, z1, z2) :|: z1 >= 0, z2 >= 0, z = 0, z'' - 1 >= 0, z' = 0
f(z, z', z'', z1, z2) -{ 1 }→ f(z1 - 1, z1 - 1, z1 - 1, z1 - 1, z2) :|: z'' = 0, z1 - 1 >= 0, z2 >= 0, z = 0, z' = 0
f(z, z', z'', z1, z2) -{ 1 }→ f(z2 - 1, z2 - 1, z2 - 1, z2 - 1, z2 - 1) :|: z'' = 0, z1 = 0, z2 - 1 >= 0, z = 0, z' = 0
f(z, z', z'', z1, z2) -{ 1 }→ 0 :|: z'' = 0, z1 = 0, z2 = 0, z = 0, z' = 0

Function symbols to be analyzed:
Previous analysis results are:

f: runtime: O(n5) [1 + z + z' + z'·z'' + z'·z''·z1 + z'·z''·z1·z2 + z'·z''·z1·z22 + z'·z''·z12 + z'·z''·z2 + z'·z''·z22 + z'·z''·z23 + z'·z''2 + z'·z1 + z'·z1·z2 + 3·z'·z1·z22 + 2·z'·z1·z23 + z'·z12 + 2·z'·z12·z2 + z'·z12·z22 + z'·z13 + z'·z2 + z'·z22 + 2·z'·z23 + z'·z24 + z'2 + z'' + 2·z''·z1 + 5·z''·z1·z2 + 7·z''·z1·z22 + 4·z''·z1·z23 + 4·z''·z12 + 4·z''·z12·z2 + 2·z''·z12·z22 + 2·z''·z13 + 2·z''·z2 + 4·z''·z22 + 5·z''·z23 + 2·z''·z24 + 2·z''2 + 2·z''2·z1 + z''2·z1·z2 + z''2·z1·z22 + z''2·z12 + 2·z''2·z2 + z''2·z22 + z''2·z23 + z''3 + z1 + 3·z1·z2 + 9·z1·z22 + 9·z1·z23 + 3·z1·z24 + 3·z12 + 6·z12·z2 + 7·z12·z22 + 3·z12·z23 + 3·z13 + 3·z13·z2 + z13·z22 + z14 + z2 + 4·z22 + 6·z23 + 4·z24 + z25], size: O(1) [0]

(19) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(20) BOUNDS(1, n^5)