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:

D(t) → 1
D(constant) → 0
D(+(x, y)) → +(D(x), D(y))
D(*(x, y)) → +(*(y, D(x)), *(x, D(y)))
D(-(x, y)) → -(D(x), D(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^1).


The TRS R consists of the following rules:

D(t) → 1 [1]
D(constant) → 0 [1]
D(+(x, y)) → +(D(x), D(y)) [1]
D(*(x, y)) → +(*(y, D(x)), *(x, D(y))) [1]
D(-(x, y)) → -(D(x), D(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:

D(t) → 1 [1]
D(constant) → 0 [1]
D(+(x, y)) → +(D(x), D(y)) [1]
D(*(x, y)) → +(*(y, D(x)), *(x, D(y))) [1]
D(-(x, y)) → -(D(x), D(y)) [1]

The TRS has the following type information:

D :: t:1:constant:0:+:*:- → t:1:constant:0:+:*:- t :: t:1:constant:0:+:*:- 1 :: t:1:constant:0:+:*:- constant :: t:1:constant:0:+:*:- 0 :: t:1:constant:0:+:*:- + :: t:1:constant:0:+:*:- → t:1:constant:0:+:*:- → t:1:constant:0:+:*:- * :: t:1:constant:0:+:*:- → t:1:constant:0:+:*:- → t:1:constant:0:+:*:- - :: t:1:constant:0:+:*:- → t:1:constant:0:+:*:- → t:1:constant: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:

D

(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: D(t) → 1 [1] D(constant) → 0 [1] D(+(x, y)) → +(D(x), D(y)) [1] D(*(x, y)) → +(*(y, D(x)), *(x, D(y))) [1] D(-(x, y)) → -(D(x), D(y)) [1] The TRS has the following type information: D :: t:1:constant:0:+:*:- → t:1:constant:0:+:*:- t :: t:1:constant:0:+:*:- 1 :: t:1:constant:0:+:*:- constant :: t:1:constant:0:+:*:- 0 :: t:1:constant:0:+:*:- + :: t:1:constant:0:+:*:- → t:1:constant:0:+:*:- → t:1:constant:0:+:*:- * :: t:1:constant:0:+:*:- → t:1:constant:0:+:*:- → t:1:constant:0:+:*:- - :: t:1:constant:0:+:*:- → t:1:constant:0:+:*:- → t:1:constant: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: D(t) → 1 [1] D(constant) → 0 [1] D(+(x, y)) → +(D(x), D(y)) [1] D(*(x, y)) → +(*(y, D(x)), *(x, D(y))) [1] D(-(x, y)) → -(D(x), D(y)) [1] The TRS has the following type information: D :: t:1:constant:0:+:*:- → t:1:constant:0:+:*:- t :: t:1:constant:0:+:*:- 1 :: t:1:constant:0:+:*:- constant :: t:1:constant:0:+:*:- 0 :: t:1:constant:0:+:*:- + :: t:1:constant:0:+:*:- → t:1:constant:0:+:*:- → t:1:constant:0:+:*:- * :: t:1:constant:0:+:*:- → t:1:constant:0:+:*:- → t:1:constant:0:+:*:- - :: t:1:constant:0:+:*:- → t:1:constant:0:+:*:- → t:1:constant: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:

t => 3
1 => 1
constant => 2
0 => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

D(z) -{ 1 }→ 1 :|: z = 3
D(z) -{ 1 }→ 0 :|: z = 2
D(z) -{ 1 }→ 1 + D(x) + D(y) :|: z = 1 + x + y, x >= 0, y >= 0
D(z) -{ 1 }→ 1 + (1 + y + D(x)) + (1 + x + D(y)) :|: z = 1 + x + y, x >= 0, y >= 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:

D(z) -{ 1 }→ 1 :|: z = 3
D(z) -{ 1 }→ 0 :|: z = 2
D(z) -{ 1 }→ 1 + D(x) + D(y) :|: z = 1 + x + y, x >= 0, y >= 0
D(z) -{ 1 }→ 1 + (1 + y + D(x)) + (1 + x + D(y)) :|: z = 1 + x + y, x >= 0, y >= 0

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

Found the following analysis order by SCC decomposition:

{ D }

(14) Obligation:

Complexity RNTS consisting of the following rules:

D(z) -{ 1 }→ 1 :|: z = 3
D(z) -{ 1 }→ 0 :|: z = 2
D(z) -{ 1 }→ 1 + D(x) + D(y) :|: z = 1 + x + y, x >= 0, y >= 0
D(z) -{ 1 }→ 1 + (1 + y + D(x)) + (1 + x + D(y)) :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {D}

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

Computed SIZE bound using CoFloCo for: D
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 1 + 2·z + z²

(16) Obligation:

Complexity RNTS consisting of the following rules:

D(z) -{ 1 }→ 1 :|: z = 3
D(z) -{ 1 }→ 0 :|: z = 2
D(z) -{ 1 }→ 1 + D(x) + D(y) :|: z = 1 + x + y, x >= 0, y >= 0
D(z) -{ 1 }→ 1 + (1 + y + D(x)) + (1 + x + D(y)) :|: z = 1 + x + y, x >= 0, y >= 0

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

D: runtime: ?, size: O(n2) [1 + 2·z + z2]

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

Computed RUNTIME bound using KoAT for: D
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 3·z

(18) Obligation:

Complexity RNTS consisting of the following rules:

D(z) -{ 1 }→ 1 :|: z = 3
D(z) -{ 1 }→ 0 :|: z = 2
D(z) -{ 1 }→ 1 + D(x) + D(y) :|: z = 1 + x + y, x >= 0, y >= 0
D(z) -{ 1 }→ 1 + (1 + y + D(x)) + (1 + x + D(y)) :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed:
Previous analysis results are:

D: runtime: O(n1) [3·z], size: O(n2) [1 + 2·z + z2]

(19) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(20) BOUNDS(1, n^1)