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:

g(f(x, y), z) → f(x, g(y, z))
g(h(x, y), z) → g(x, f(y, z))
g(x, h(y, z)) → h(g(x, y), 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:

g(f(x, y), z) → f(x, g(y, z)) [1]
g(h(x, y), z) → g(x, f(y, z)) [1]
g(x, h(y, z)) → h(g(x, y), 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:

g(f(x, y), z) → f(x, g(y, z)) [1]
g(h(x, y), z) → g(x, f(y, z)) [1]
g(x, h(y, z)) → h(g(x, y), z) [1]

The TRS has the following type information:

g :: f:h → f:h → f:h f :: a → f:h → f:h h :: f:h → a → f:h

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:

g

(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: g(f(x, y), z) → f(x, g(y, z)) [1] g(h(x, y), z) → g(x, f(y, z)) [1] g(x, h(y, z)) → h(g(x, y), z) [1] The TRS has the following type information: g :: f:h → f:h → f:h f :: a → f:h → f:h h :: f:h → a → f:h const :: f:h const1 :: a 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: g(f(x, y), z) → f(x, g(y, z)) [1] g(h(x, y), z) → g(x, f(y, z)) [1] g(x, h(y, z)) → h(g(x, y), z) [1] The TRS has the following type information: g :: f:h → f:h → f:h f :: a → f:h → f:h h :: f:h → a → f:h const :: f:h const1 :: a 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:

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

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

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

Found the following analysis order by SCC decomposition:

{ g }

(14) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {g}

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

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

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

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed:
Previous analysis results are:

g: runtime: O(n1) [2·z' + z''], size: O(1) [0]

(19) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(20) BOUNDS(1, n^1)