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

(0) Obligation:

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


The TRS R consists of the following rules:

*(x, *(y, z)) → *(otimes(x, y), z)
*(1, y) → y
*(+(x, y), z) → oplus(*(x, z), *(y, z))
*(x, oplus(y, z)) → oplus(*(x, y), *(x, z))

Rewrite Strategy: INNERMOST

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

The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
*(x, *(y, z)) → *(otimes(x, y), z)

(2) Obligation:

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


The TRS R consists of the following rules:

*(x, oplus(y, z)) → oplus(*(x, y), *(x, z))
*(1, y) → y
*(+(x, y), z) → oplus(*(x, z), *(y, z))

Rewrite Strategy: INNERMOST

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

Transformed TRS to weighted TRS

(4) Obligation:

The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, EXP).


The TRS R consists of the following rules:

*(x, oplus(y, z)) → oplus(*(x, y), *(x, z)) [1]
*(1, y) → y [1]
*(+(x, y), z) → oplus(*(x, z), *(y, z)) [1]

Rewrite Strategy: INNERMOST

(5) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID) transformation)

Renamed defined symbols to avoid conflicts with arithmetic symbols:

* => times

(6) Obligation:

The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, EXP).


The TRS R consists of the following rules:

times(x, oplus(y, z)) → oplus(times(x, y), times(x, z)) [1]
times(1, y) → y [1]
times(+(x, y), z) → oplus(times(x, z), times(y, z)) [1]

Rewrite Strategy: INNERMOST

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

Infered types.

(8) Obligation:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

times(x, oplus(y, z)) → oplus(times(x, y), times(x, z)) [1]
times(1, y) → y [1]
times(+(x, y), z) → oplus(times(x, z), times(y, z)) [1]

The TRS has the following type information:

times :: 1:+ → oplus → oplus oplus :: oplus → oplus → oplus 1 :: 1:+ + :: 1:+ → 1:+ → 1:+

Rewrite Strategy: INNERMOST

(9) 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:

times

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:

const

(10) 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: times(x, oplus(y, z)) → oplus(times(x, y), times(x, z)) [1] times(1, y) → y [1] times(+(x, y), z) → oplus(times(x, z), times(y, z)) [1] The TRS has the following type information: times :: 1:+ → oplus → oplus oplus :: oplus → oplus → oplus 1 :: 1:+ + :: 1:+ → 1:+ → 1:+ const :: oplus Rewrite Strategy: INNERMOST

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

Narrowed the inner basic terms of all right-hand sides by a single narrowing step.

(12) 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: times(x, oplus(y, z)) → oplus(times(x, y), times(x, z)) [1] times(1, y) → y [1] times(+(x, y), z) → oplus(times(x, z), times(y, z)) [1] The TRS has the following type information: times :: 1:+ → oplus → oplus oplus :: oplus → oplus → oplus 1 :: 1:+ + :: 1:+ → 1:+ → 1:+ const :: oplus Rewrite Strategy: INNERMOST

(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

1 => 0
const => 0

(14) Obligation:

Complexity RNTS consisting of the following rules:

times(z', z'') -{ 1 }→ y :|: z'' = y, y >= 0, z' = 0
times(z', z'') -{ 1 }→ 1 + times(x, y) + times(x, z) :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z
times(z', z'') -{ 1 }→ 1 + times(x, z) + times(y, z) :|: z'' = z, z >= 0, z' = 1 + x + y, x >= 0, y >= 0

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

Simplified the RNTS by moving equalities from the constraints into the right-hand sides.

(16) Obligation:

Complexity RNTS consisting of the following rules:

times(z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' = 0
times(z', z'') -{ 1 }→ 1 + times(x, z'') + times(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
times(z', z'') -{ 1 }→ 1 + times(z', y) + times(z', z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z

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

Found the following analysis order by SCC decomposition:

{ times }

(18) Obligation:

Complexity RNTS consisting of the following rules:

times(z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' = 0
times(z', z'') -{ 1 }→ 1 + times(x, z'') + times(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
times(z', z'') -{ 1 }→ 1 + times(z', y) + times(z', z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z

Function symbols to be analyzed: {times}

(19) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using KoAT for: times
after applying outer abstraction to obtain an ITS,
resulting in: EXP with polynomial bound: ?

(20) Obligation:

Complexity RNTS consisting of the following rules:

times(z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' = 0
times(z', z'') -{ 1 }→ 1 + times(x, z'') + times(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
times(z', z'') -{ 1 }→ 1 + times(z', y) + times(z', z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z

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

times: runtime: ?, size: EXP

(21) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using KoAT for: times
after applying outer abstraction to obtain an ITS,
resulting in: EXP with polynomial bound: ?

(22) Obligation:

Complexity RNTS consisting of the following rules:

times(z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' = 0
times(z', z'') -{ 1 }→ 1 + times(x, z'') + times(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
times(z', z'') -{ 1 }→ 1 + times(z', y) + times(z', z) :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z

Function symbols to be analyzed:
Previous analysis results are:

times: runtime: EXP, size: EXP

(23) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(24) BOUNDS(1, EXP)