Runtime Complexity (innermost) proof of /tmp/tmpk8buqo/polo2.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), 0 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), 5440 ms)

↳16 CpxRNTS

↳17 IntTrsBoundProof (UPPER BOUND(ID), 146 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:

dx(X) → one
dx(a) → zero
dx(plus(ALPHA, BETA)) → plus(dx(ALPHA), dx(BETA))
dx(times(ALPHA, BETA)) → plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA)))
dx(minus(ALPHA, BETA)) → minus(dx(ALPHA), dx(BETA))
dx(neg(ALPHA)) → neg(dx(ALPHA))
dx(div(ALPHA, BETA)) → minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two))))
dx(ln(ALPHA)) → div(dx(ALPHA), ALPHA)
dx(exp(ALPHA, BETA)) → plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA))))

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:

dx(X) → one [1]
dx(a) → zero [1]
dx(plus(ALPHA, BETA)) → plus(dx(ALPHA), dx(BETA)) [1]
dx(times(ALPHA, BETA)) → plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) [1]
dx(minus(ALPHA, BETA)) → minus(dx(ALPHA), dx(BETA)) [1]
dx(neg(ALPHA)) → neg(dx(ALPHA)) [1]
dx(div(ALPHA, BETA)) → minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) [1]
dx(ln(ALPHA)) → div(dx(ALPHA), ALPHA) [1]
dx(exp(ALPHA, BETA)) → plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) [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:

dx(X) → one [1]
dx(a) → zero [1]
dx(plus(ALPHA, BETA)) → plus(dx(ALPHA), dx(BETA)) [1]
dx(times(ALPHA, BETA)) → plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) [1]
dx(minus(ALPHA, BETA)) → minus(dx(ALPHA), dx(BETA)) [1]
dx(neg(ALPHA)) → neg(dx(ALPHA)) [1]
dx(div(ALPHA, BETA)) → minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) [1]
dx(ln(ALPHA)) → div(dx(ALPHA), ALPHA) [1]
dx(exp(ALPHA, BETA)) → plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) [1]

The TRS has the following type information:

dx :: one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln
one :: one:a:zero:plus:times:minus:neg:div:two:exp:ln
a :: one:a:zero:plus:times:minus:neg:div:two:exp:ln
zero :: one:a:zero:plus:times:minus:neg:div:two:exp:ln
plus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln
times :: one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln
minus :: one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln
neg :: one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln
div :: one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln
exp :: one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp:ln
two :: one:a:zero:plus:times:minus:neg:div:two:exp:ln
ln :: one:a:zero:plus:times:minus:neg:div:two:exp:ln → one:a:zero:plus:times:minus:neg:div:two:exp: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:

dx

(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:

dx(X) → one [1]
dx(a) → zero [1]
dx(plus(ALPHA, BETA)) → plus(dx(ALPHA), dx(BETA)) [1]
dx(times(ALPHA, BETA)) → plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) [1]
dx(minus(ALPHA, BETA)) → minus(dx(ALPHA), dx(BETA)) [1]
dx(neg(ALPHA)) → neg(dx(ALPHA)) [1]
dx(div(ALPHA, BETA)) → minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) [1]
dx(ln(ALPHA)) → div(dx(ALPHA), ALPHA) [1]
dx(exp(ALPHA, BETA)) → plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) [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:

dx(X) → one [1]
dx(a) → zero [1]
dx(plus(ALPHA, BETA)) → plus(dx(ALPHA), dx(BETA)) [1]
dx(times(ALPHA, BETA)) → plus(times(BETA, dx(ALPHA)), times(ALPHA, dx(BETA))) [1]
dx(minus(ALPHA, BETA)) → minus(dx(ALPHA), dx(BETA)) [1]
dx(neg(ALPHA)) → neg(dx(ALPHA)) [1]
dx(div(ALPHA, BETA)) → minus(div(dx(ALPHA), BETA), times(ALPHA, div(dx(BETA), exp(BETA, two)))) [1]
dx(ln(ALPHA)) → div(dx(ALPHA), ALPHA) [1]
dx(exp(ALPHA, BETA)) → plus(times(BETA, times(exp(ALPHA, minus(BETA, one)), dx(ALPHA))), times(exp(ALPHA, BETA), times(ln(ALPHA), dx(BETA)))) [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:

one => 1
a => 0
zero => 3
two => 2

(10) Obligation:

Complexity RNTS consisting of the following rules:

dx(z) -{ 1 }→ 3 :|: z = 0
dx(z) -{ 1 }→ 1 :|: X >= 0, z = X
dx(z) -{ 1 }→ 1 + dx(ALPHA) :|: ALPHA >= 0, z = 1 + ALPHA
dx(z) -{ 1 }→ 1 + dx(ALPHA) + ALPHA :|: ALPHA >= 0, z = 1 + ALPHA
dx(z) -{ 1 }→ 1 + dx(ALPHA) + dx(BETA) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA
dx(z) -{ 1 }→ 1 + (1 + BETA + dx(ALPHA)) + (1 + ALPHA + dx(BETA)) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA
dx(z) -{ 1 }→ 1 + (1 + BETA + (1 + (1 + ALPHA + (1 + BETA + 1)) + dx(ALPHA))) + (1 + (1 + ALPHA + BETA) + (1 + (1 + ALPHA) + dx(BETA))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA
dx(z) -{ 1 }→ 1 + (1 + dx(ALPHA) + BETA) + (1 + ALPHA + (1 + dx(BETA) + (1 + BETA + 2))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA

(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:

dx(z) -{ 1 }→ 3 :|: z = 0
dx(z) -{ 1 }→ 1 :|: z >= 0
dx(z) -{ 1 }→ 1 + dx(z - 1) :|: z - 1 >= 0
dx(z) -{ 1 }→ 1 + dx(ALPHA) + dx(BETA) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA
dx(z) -{ 1 }→ 1 + dx(z - 1) + (z - 1) :|: z - 1 >= 0
dx(z) -{ 1 }→ 1 + (1 + BETA + dx(ALPHA)) + (1 + ALPHA + dx(BETA)) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA
dx(z) -{ 1 }→ 1 + (1 + BETA + (1 + (1 + ALPHA + (1 + BETA + 1)) + dx(ALPHA))) + (1 + (1 + ALPHA + BETA) + (1 + (1 + ALPHA) + dx(BETA))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA
dx(z) -{ 1 }→ 1 + (1 + dx(ALPHA) + BETA) + (1 + ALPHA + (1 + dx(BETA) + (1 + BETA + 2))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA

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

Found the following analysis order by SCC decomposition:

{ dx }

(14) Obligation:

Complexity RNTS consisting of the following rules:

dx(z) -{ 1 }→ 3 :|: z = 0
dx(z) -{ 1 }→ 1 :|: z >= 0
dx(z) -{ 1 }→ 1 + dx(z - 1) :|: z - 1 >= 0
dx(z) -{ 1 }→ 1 + dx(ALPHA) + dx(BETA) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA
dx(z) -{ 1 }→ 1 + dx(z - 1) + (z - 1) :|: z - 1 >= 0
dx(z) -{ 1 }→ 1 + (1 + BETA + dx(ALPHA)) + (1 + ALPHA + dx(BETA)) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA
dx(z) -{ 1 }→ 1 + (1 + BETA + (1 + (1 + ALPHA + (1 + BETA + 1)) + dx(ALPHA))) + (1 + (1 + ALPHA + BETA) + (1 + (1 + ALPHA) + dx(BETA))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA
dx(z) -{ 1 }→ 1 + (1 + dx(ALPHA) + BETA) + (1 + ALPHA + (1 + dx(BETA) + (1 + BETA + 2))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA

Function symbols to be analyzed: {dx}

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

Computed SIZE bound using CoFloCo for: dx
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 3 + 10·z + 3·z²

(16) Obligation:

Complexity RNTS consisting of the following rules:

dx(z) -{ 1 }→ 3 :|: z = 0
dx(z) -{ 1 }→ 1 :|: z >= 0
dx(z) -{ 1 }→ 1 + dx(z - 1) :|: z - 1 >= 0
dx(z) -{ 1 }→ 1 + dx(ALPHA) + dx(BETA) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA
dx(z) -{ 1 }→ 1 + dx(z - 1) + (z - 1) :|: z - 1 >= 0
dx(z) -{ 1 }→ 1 + (1 + BETA + dx(ALPHA)) + (1 + ALPHA + dx(BETA)) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA
dx(z) -{ 1 }→ 1 + (1 + BETA + (1 + (1 + ALPHA + (1 + BETA + 1)) + dx(ALPHA))) + (1 + (1 + ALPHA + BETA) + (1 + (1 + ALPHA) + dx(BETA))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA
dx(z) -{ 1 }→ 1 + (1 + dx(ALPHA) + BETA) + (1 + ALPHA + (1 + dx(BETA) + (1 + BETA + 2))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA

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

dx: runtime: ?, size: O(n²) [3 + 10·z + 3·z²]

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

Computed RUNTIME bound using CoFloCo for: dx
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + 2·z

(18) Obligation:

Complexity RNTS consisting of the following rules:

dx(z) -{ 1 }→ 3 :|: z = 0
dx(z) -{ 1 }→ 1 :|: z >= 0
dx(z) -{ 1 }→ 1 + dx(z - 1) :|: z - 1 >= 0
dx(z) -{ 1 }→ 1 + dx(ALPHA) + dx(BETA) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA
dx(z) -{ 1 }→ 1 + dx(z - 1) + (z - 1) :|: z - 1 >= 0
dx(z) -{ 1 }→ 1 + (1 + BETA + dx(ALPHA)) + (1 + ALPHA + dx(BETA)) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA
dx(z) -{ 1 }→ 1 + (1 + BETA + (1 + (1 + ALPHA + (1 + BETA + 1)) + dx(ALPHA))) + (1 + (1 + ALPHA + BETA) + (1 + (1 + ALPHA) + dx(BETA))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA
dx(z) -{ 1 }→ 1 + (1 + dx(ALPHA) + BETA) + (1 + ALPHA + (1 + dx(BETA) + (1 + BETA + 2))) :|: ALPHA >= 0, BETA >= 0, z = 1 + ALPHA + BETA

Function symbols to be analyzed:
Previous analysis results are:

dx: runtime: O(n¹) [1 + 2·z], size: O(n²) [3 + 10·z + 3·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)