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

(0) Obligation:

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


The TRS R consists of the following rules:

p(m, n, s(r)) → p(m, r, n)
p(m, s(n), 0) → p(0, n, m)
p(m, 0, 0) → m

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^1).


The TRS R consists of the following rules:

p(m, n, s(r)) → p(m, r, n) [1]
p(m, s(n), 0) → p(0, n, m) [1]
p(m, 0, 0) → m [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:

p(m, n, s(r)) → p(m, r, n) [1]
p(m, s(n), 0) → p(0, n, m) [1]
p(m, 0, 0) → m [1]

The TRS has the following type information:

p :: 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:

p

(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: p(m, n, s(r)) → p(m, r, n) [1] p(m, s(n), 0) → p(0, n, m) [1] p(m, 0, 0) → m [1] The TRS has the following type information: p :: 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: p(m, n, s(r)) → p(m, r, n) [1] p(m, s(n), 0) → p(0, n, m) [1] p(m, 0, 0) → m [1] The TRS has the following type information: p :: 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:

p(z, z', z'') -{ 1 }→ m :|: z = m, z'' = 0, z' = 0, m >= 0
p(z, z', z'') -{ 1 }→ p(m, r, n) :|: z = m, z'' = 1 + r, n >= 0, r >= 0, z' = n, m >= 0
p(z, z', z'') -{ 1 }→ p(0, n, m) :|: z = m, z'' = 0, n >= 0, z' = 1 + n, m >= 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:

p(z, z', z'') -{ 1 }→ z :|: z'' = 0, z' = 0, z >= 0
p(z, z', z'') -{ 1 }→ p(z, z'' - 1, z') :|: z' >= 0, z'' - 1 >= 0, z >= 0
p(z, z', z'') -{ 1 }→ p(0, z' - 1, z) :|: z'' = 0, z' - 1 >= 0, z >= 0

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

Found the following analysis order by SCC decomposition:

{ p }

(14) Obligation:

Complexity RNTS consisting of the following rules:

p(z, z', z'') -{ 1 }→ z :|: z'' = 0, z' = 0, z >= 0
p(z, z', z'') -{ 1 }→ p(z, z'' - 1, z') :|: z' >= 0, z'' - 1 >= 0, z >= 0
p(z, z', z'') -{ 1 }→ p(0, z' - 1, z) :|: z'' = 0, z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {p}

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

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

p(z, z', z'') -{ 1 }→ z :|: z'' = 0, z' = 0, z >= 0
p(z, z', z'') -{ 1 }→ p(z, z'' - 1, z') :|: z' >= 0, z'' - 1 >= 0, z >= 0
p(z, z', z'') -{ 1 }→ p(0, z' - 1, z) :|: z'' = 0, z' - 1 >= 0, z >= 0

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

p: runtime: ?, size: O(n1) [z]

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

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

p(z, z', z'') -{ 1 }→ z :|: z'' = 0, z' = 0, z >= 0
p(z, z', z'') -{ 1 }→ p(z, z'' - 1, z') :|: z' >= 0, z'' - 1 >= 0, z >= 0
p(z, z', z'') -{ 1 }→ p(0, z' - 1, z) :|: z'' = 0, z' - 1 >= 0, z >= 0

Function symbols to be analyzed:
Previous analysis results are:

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

(19) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(20) BOUNDS(1, n^1)