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:

+(+(x, y), z) → +(x, +(y, z))
+(f(x), f(y)) → f(+(x, y))
+(f(x), +(f(y), z)) → +(f(+(x, y)), 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) → +(x, +(y, z))
+(f(x), +(f(y), z)) → +(f(+(x, y)), z)

(2) 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:

+(f(x), f(y)) → f(+(x, y))

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, n^1).


The TRS R consists of the following rules:

+(f(x), f(y)) → f(+(x, y)) [1]

Rewrite Strategy: INNERMOST

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

Renamed defined symbols to avoid conflicts with arithmetic symbols:

+ => plus

(6) 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:

plus(f(x), f(y)) → f(plus(x, y)) [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:

plus(f(x), f(y)) → f(plus(x, y)) [1]

The TRS has the following type information:

plus :: f → f → f f :: f → f

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:

plus

(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: plus(f(x), f(y)) → f(plus(x, y)) [1] The TRS has the following type information: plus :: f → f → f f :: f → f const :: f 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: plus(f(x), f(y)) → f(plus(x, y)) [1] The TRS has the following type information: plus :: f → f → f f :: f → f const :: f 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:

const => 0

(14) Obligation:

Complexity RNTS consisting of the following rules:

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

(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:

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

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

Found the following analysis order by SCC decomposition:

{ plus }

(18) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {plus}

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

Computed SIZE bound using CoFloCo for: plus
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

(20) Obligation:

Complexity RNTS consisting of the following rules:

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

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

plus: runtime: ?, size: O(1) [0]

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

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed:
Previous analysis results are:

plus: runtime: O(n1) [z'], size: O(1) [0]

(23) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(24) BOUNDS(1, n^1)