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:

implies(not(x), y) → or(x, y)
implies(not(x), or(y, z)) → implies(y, or(x, z))
implies(x, or(y, z)) → or(y, implies(x, z))

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:

implies(not(x), y) → or(x, y) [1]
implies(not(x), or(y, z)) → implies(y, or(x, z)) [1]
implies(x, or(y, z)) → or(y, implies(x, z)) [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:

implies(not(x), y) → or(x, y) [1]
implies(not(x), or(y, z)) → implies(y, or(x, z)) [1]
implies(x, or(y, z)) → or(y, implies(x, z)) [1]

The TRS has the following type information:

implies :: not → or → or not :: not → not or :: not → or → or

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:

implies

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:

const, const1

(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: implies(not(x), y) → or(x, y) [1] implies(not(x), or(y, z)) → implies(y, or(x, z)) [1] implies(x, or(y, z)) → or(y, implies(x, z)) [1] The TRS has the following type information: implies :: not → or → or not :: not → not or :: not → or → or const :: or const1 :: not 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: implies(not(x), y) → or(x, y) [1] implies(not(x), or(y, z)) → implies(y, or(x, z)) [1] implies(x, or(y, z)) → or(y, implies(x, z)) [1] The TRS has the following type information: implies :: not → or → or not :: not → not or :: not → or → or const :: or const1 :: not 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:

const => 0
const1 => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

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

(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:

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

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

Found the following analysis order by SCC decomposition:

{ implies }

(14) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {implies}

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

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

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

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

implies: runtime: ?, size: O(n1) [z' + z'']

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

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed:
Previous analysis results are:

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

(19) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(20) BOUNDS(1, n^1)