Runtime Complexity (innermost) proof of /tmp/tmpVqYvOT/11.xml

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

0 CpxTRS

↳1 TrsToWeightedTrsProof (BOTH BOUNDS(ID, ID), 2 ms)

↳2 CpxWeightedTrs

↳3 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)

↳4 CpxTypedWeightedTrs

↳5 CompletionProof (UPPER BOUND(ID), 0 ms)

↳6 CpxTypedWeightedCompleteTrs

↳7 NarrowingProof (BOTH BOUNDS(ID, ID), 0 ms)

↳8 CpxTypedWeightedCompleteTrs

↳9 CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID), 0 ms)

↳10 CpxRNTS

↳11 SimplificationProof (BOTH BOUNDS(ID, ID), 0 ms)

↳12 CpxRNTS

↳13 CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID), 0 ms)

↳14 CpxRNTS

↳15 IntTrsBoundProof (UPPER BOUND(ID), 5332 ms)

↳16 CpxRNTS

↳17 IntTrsBoundProof (UPPER BOUND(ID), 115 ms)

↳18 CpxRNTS

↳19 FinalProof (⇔, 0 ms)

↳20 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))
D(minus(x)) → minus(D(x))
D(div(x, y)) → -(div(D(x), y), div(*(x, D(y)), pow(y, 2)))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(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]
D(minus(x)) → minus(D(x)) [1]
D(div(x, y)) → -(div(D(x), y), div(*(x, D(y)), pow(y, 2))) [1]
D(ln(x)) → div(D(x), x) [1]
D(pow(x, y)) → +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(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]
D(minus(x)) → minus(D(x)) [1]
D(div(x, y)) → -(div(D(x), y), div(*(x, D(y)), pow(y, 2))) [1]
D(ln(x)) → div(D(x), x) [1]
D(pow(x, y)) → +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y))) [1]

The TRS has the following type information:

D :: t:1:constant:0:+:*:-:minus:div:2:pow:ln → t:1:constant:0:+:*:-:minus:div:2:pow:ln
t :: t:1:constant:0:+:*:-:minus:div:2:pow:ln
1 :: t:1:constant:0:+:*:-:minus:div:2:pow:ln
constant :: t:1:constant:0:+:*:-:minus:div:2:pow:ln
0 :: t:1:constant:0:+:*:-:minus:div:2:pow:ln
+ :: t:1:constant:0:+:*:-:minus:div:2:pow:ln → t:1:constant:0:+:*:-:minus:div:2:pow:ln → t:1:constant:0:+:*:-:minus:div:2:pow:ln
* :: t:1:constant:0:+:*:-:minus:div:2:pow:ln → t:1:constant:0:+:*:-:minus:div:2:pow:ln → t:1:constant:0:+:*:-:minus:div:2:pow:ln
- :: t:1:constant:0:+:*:-:minus:div:2:pow:ln → t:1:constant:0:+:*:-:minus:div:2:pow:ln → t:1:constant:0:+:*:-:minus:div:2:pow:ln
minus :: t:1:constant:0:+:*:-:minus:div:2:pow:ln → t:1:constant:0:+:*:-:minus:div:2:pow:ln
div :: t:1:constant:0:+:*:-:minus:div:2:pow:ln → t:1:constant:0:+:*:-:minus:div:2:pow:ln → t:1:constant:0:+:*:-:minus:div:2:pow:ln
pow :: t:1:constant:0:+:*:-:minus:div:2:pow:ln → t:1:constant:0:+:*:-:minus:div:2:pow:ln → t:1:constant:0:+:*:-:minus:div:2:pow:ln
2 :: t:1:constant:0:+:*:-:minus:div:2:pow:ln
ln :: t:1:constant:0:+:*:-:minus:div:2:pow:ln → t:1:constant:0:+:*:-:minus:div:2:pow:ln

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]
D(minus(x)) → minus(D(x)) [1]
D(div(x, y)) → -(div(D(x), y), div(*(x, D(y)), pow(y, 2))) [1]
D(ln(x)) → div(D(x), x) [1]
D(pow(x, y)) → +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y))) [1]

The TRS has the following type information:

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]
D(minus(x)) → minus(D(x)) [1]
D(div(x, y)) → -(div(D(x), y), div(*(x, D(y)), pow(y, 2))) [1]
D(ln(x)) → div(D(x), x) [1]
D(pow(x, y)) → +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y))) [1]

The TRS has the following type information:

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 => 4
1 => 1
constant => 3
0 => 0
2 => 2

(10) Obligation:

Complexity RNTS consisting of the following rules:

D(z) -{ 1 }→ 1 :|: z = 4
D(z) -{ 1 }→ 0 :|: z = 3
D(z) -{ 1 }→ 1 + D(x) :|: x >= 0, z = 1 + x
D(z) -{ 1 }→ 1 + D(x) + x :|: x >= 0, z = 1 + x
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
D(z) -{ 1 }→ 1 + (1 + D(x) + y) + (1 + (1 + x + D(y)) + (1 + y + 2)) :|: z = 1 + x + y, x >= 0, y >= 0
D(z) -{ 1 }→ 1 + (1 + (1 + y + (1 + x + (1 + y + 1))) + D(x)) + (1 + (1 + (1 + x + y) + (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 = 4
D(z) -{ 1 }→ 0 :|: z = 3
D(z) -{ 1 }→ 1 + D(z - 1) :|: z - 1 >= 0
D(z) -{ 1 }→ 1 + D(x) + D(y) :|: z = 1 + x + y, x >= 0, y >= 0
D(z) -{ 1 }→ 1 + D(z - 1) + (z - 1) :|: z - 1 >= 0
D(z) -{ 1 }→ 1 + (1 + y + D(x)) + (1 + x + D(y)) :|: z = 1 + x + y, x >= 0, y >= 0
D(z) -{ 1 }→ 1 + (1 + D(x) + y) + (1 + (1 + x + D(y)) + (1 + y + 2)) :|: z = 1 + x + y, x >= 0, y >= 0
D(z) -{ 1 }→ 1 + (1 + (1 + y + (1 + x + (1 + y + 1))) + D(x)) + (1 + (1 + (1 + x + y) + (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 = 4
D(z) -{ 1 }→ 0 :|: z = 3
D(z) -{ 1 }→ 1 + D(z - 1) :|: z - 1 >= 0
D(z) -{ 1 }→ 1 + D(x) + D(y) :|: z = 1 + x + y, x >= 0, y >= 0
D(z) -{ 1 }→ 1 + D(z - 1) + (z - 1) :|: z - 1 >= 0
D(z) -{ 1 }→ 1 + (1 + y + D(x)) + (1 + x + D(y)) :|: z = 1 + x + y, x >= 0, y >= 0
D(z) -{ 1 }→ 1 + (1 + D(x) + y) + (1 + (1 + x + D(y)) + (1 + y + 2)) :|: z = 1 + x + y, x >= 0, y >= 0
D(z) -{ 1 }→ 1 + (1 + (1 + y + (1 + x + (1 + y + 1))) + D(x)) + (1 + (1 + (1 + x + y) + (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 KoAT for: D
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 23·z + 12·z²

(16) Obligation:

Complexity RNTS consisting of the following rules:

D(z) -{ 1 }→ 1 :|: z = 4
D(z) -{ 1 }→ 0 :|: z = 3
D(z) -{ 1 }→ 1 + D(z - 1) :|: z - 1 >= 0
D(z) -{ 1 }→ 1 + D(x) + D(y) :|: z = 1 + x + y, x >= 0, y >= 0
D(z) -{ 1 }→ 1 + D(z - 1) + (z - 1) :|: z - 1 >= 0
D(z) -{ 1 }→ 1 + (1 + y + D(x)) + (1 + x + D(y)) :|: z = 1 + x + y, x >= 0, y >= 0
D(z) -{ 1 }→ 1 + (1 + D(x) + y) + (1 + (1 + x + D(y)) + (1 + y + 2)) :|: z = 1 + x + y, x >= 0, y >= 0
D(z) -{ 1 }→ 1 + (1 + (1 + y + (1 + x + (1 + y + 1))) + D(x)) + (1 + (1 + (1 + x + y) + (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) [23·z + 12·z²]

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

Computed RUNTIME bound using CoFloCo for: D
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:

D(z) -{ 1 }→ 1 :|: z = 4
D(z) -{ 1 }→ 0 :|: z = 3
D(z) -{ 1 }→ 1 + D(z - 1) :|: z - 1 >= 0
D(z) -{ 1 }→ 1 + D(x) + D(y) :|: z = 1 + x + y, x >= 0, y >= 0
D(z) -{ 1 }→ 1 + D(z - 1) + (z - 1) :|: z - 1 >= 0
D(z) -{ 1 }→ 1 + (1 + y + D(x)) + (1 + x + D(y)) :|: z = 1 + x + y, x >= 0, y >= 0
D(z) -{ 1 }→ 1 + (1 + D(x) + y) + (1 + (1 + x + D(y)) + (1 + y + 2)) :|: z = 1 + x + y, x >= 0, y >= 0
D(z) -{ 1 }→ 1 + (1 + (1 + y + (1 + x + (1 + y + 1))) + D(x)) + (1 + (1 + (1 + x + y) + (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) [1 + z], size: O(n²) [23·z + 12·z²]

(19) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(0) Obligation:

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

(2) Obligation:

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

(4) Obligation:

(5) CompletionProof (UPPER BOUND(ID) transformation)

(6) Obligation:

(7) NarrowingProof (BOTH BOUNDS(ID, ID) transformation)

(8) Obligation:

(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

(10) Obligation:

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

(12) Obligation:

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

(14) Obligation:

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

(16) Obligation:

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

(18) Obligation:

(19) FinalProof (EQUIVALENT transformation)

(20) BOUNDS(1, n^1)