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:

f(0, y) → y
f(x, 0) → x
f(i(x), y) → i(x)
f(f(x, y), z) → f(x, f(y, z))
f(g(x, y), z) → g(f(x, z), f(y, z))
f(1, g(x, y)) → x
f(2, g(x, y)) → y

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:
f(f(x, y), z) → f(x, f(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(g(x, y), z) → g(f(x, z), f(y, z))
f(2, g(x, y)) → y
f(1, g(x, y)) → x
f(i(x), y) → i(x)
f(0, y) → y
f(x, 0) → x

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(g(x, y), z) → g(f(x, z), f(y, z)) [1]
f(2, g(x, y)) → y [1]
f(1, g(x, y)) → x [1]
f(i(x), y) → i(x) [1]
f(0, y) → y [1]
f(x, 0) → x [1]

Rewrite Strategy: INNERMOST

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

Infered types.

(6) Obligation:

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

f(g(x, y), z) → g(f(x, z), f(y, z)) [1]
f(2, g(x, y)) → y [1]
f(1, g(x, y)) → x [1]
f(i(x), y) → i(x) [1]
f(0, y) → y [1]
f(x, 0) → x [1]

The TRS has the following type information:

f :: g:2:1:i:0 → g:2:1:i:0 → g:2:1:i:0 g :: g:2:1:i:0 → g:2:1:i:0 → g:2:1:i:0 2 :: g:2:1:i:0 1 :: g:2:1:i:0 i :: a → g:2:1:i:0 0 :: g:2:1:i:0

Rewrite Strategy: INNERMOST

(7) 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:

f

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:

const

(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: f(g(x, y), z) → g(f(x, z), f(y, z)) [1] f(2, g(x, y)) → y [1] f(1, g(x, y)) → x [1] f(i(x), y) → i(x) [1] f(0, y) → y [1] f(x, 0) → x [1] The TRS has the following type information: f :: g:2:1:i:0 → g:2:1:i:0 → g:2:1:i:0 g :: g:2:1:i:0 → g:2:1:i:0 → g:2:1:i:0 2 :: g:2:1:i:0 1 :: g:2:1:i:0 i :: a → g:2:1:i:0 0 :: g:2:1:i:0 const :: a Rewrite Strategy: INNERMOST

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

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

(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: f(g(x, y), z) → g(f(x, z), f(y, z)) [1] f(2, g(x, y)) → y [1] f(1, g(x, y)) → x [1] f(i(x), y) → i(x) [1] f(0, y) → y [1] f(x, 0) → x [1] The TRS has the following type information: f :: g:2:1:i:0 → g:2:1:i:0 → g:2:1:i:0 g :: g:2:1:i:0 → g:2:1:i:0 → g:2:1:i:0 2 :: g:2:1:i:0 1 :: g:2:1:i:0 i :: a → g:2:1:i:0 0 :: g:2:1:i:0 const :: a Rewrite Strategy: INNERMOST

(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

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

2 => 2
1 => 1
0 => 0
const => 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

f(z', z'') -{ 1 }→ x :|: z'' = 1 + x + y, x >= 0, y >= 0, z' = 1
f(z', z'') -{ 1 }→ x :|: z'' = 0, z' = x, x >= 0
f(z', z'') -{ 1 }→ y :|: z' = 2, z'' = 1 + x + y, x >= 0, y >= 0
f(z', z'') -{ 1 }→ y :|: z'' = y, y >= 0, z' = 0
f(z', z'') -{ 1 }→ 1 + x :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
f(z', z'') -{ 1 }→ 1 + f(x, z) + f(y, z) :|: z'' = z, z >= 0, z' = 1 + x + y, x >= 0, y >= 0

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

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

(14) Obligation:

Complexity RNTS consisting of the following rules:

f(z', z'') -{ 1 }→ x :|: z'' = 1 + x + y, x >= 0, y >= 0, z' = 1
f(z', z'') -{ 1 }→ y :|: z' = 2, z'' = 1 + x + y, x >= 0, y >= 0
f(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
f(z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' = 0
f(z', z'') -{ 1 }→ 1 + (z' - 1) :|: z' - 1 >= 0, z'' >= 0
f(z', z'') -{ 1 }→ 1 + f(x, z'') + f(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0

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

Found the following analysis order by SCC decomposition:

{ f }

(16) Obligation:

Complexity RNTS consisting of the following rules:

f(z', z'') -{ 1 }→ x :|: z'' = 1 + x + y, x >= 0, y >= 0, z' = 1
f(z', z'') -{ 1 }→ y :|: z' = 2, z'' = 1 + x + y, x >= 0, y >= 0
f(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
f(z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' = 0
f(z', z'') -{ 1 }→ 1 + (z' - 1) :|: z' - 1 >= 0, z'' >= 0
f(z', z'') -{ 1 }→ 1 + f(x, z'') + f(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {f}

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

Computed SIZE bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 2 + 8·z' + 4·z'·z'' + 2·z''

(18) Obligation:

Complexity RNTS consisting of the following rules:

f(z', z'') -{ 1 }→ x :|: z'' = 1 + x + y, x >= 0, y >= 0, z' = 1
f(z', z'') -{ 1 }→ y :|: z' = 2, z'' = 1 + x + y, x >= 0, y >= 0
f(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
f(z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' = 0
f(z', z'') -{ 1 }→ 1 + (z' - 1) :|: z' - 1 >= 0, z'' >= 0
f(z', z'') -{ 1 }→ 1 + f(x, z'') + f(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0

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

f: runtime: ?, size: O(n2) [2 + 8·z' + 4·z'·z'' + 2·z'']

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

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

(20) Obligation:

Complexity RNTS consisting of the following rules:

f(z', z'') -{ 1 }→ x :|: z'' = 1 + x + y, x >= 0, y >= 0, z' = 1
f(z', z'') -{ 1 }→ y :|: z' = 2, z'' = 1 + x + y, x >= 0, y >= 0
f(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
f(z', z'') -{ 1 }→ z'' :|: z'' >= 0, z' = 0
f(z', z'') -{ 1 }→ 1 + (z' - 1) :|: z' - 1 >= 0, z'' >= 0
f(z', z'') -{ 1 }→ 1 + f(x, z'') + f(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed:
Previous analysis results are:

f: runtime: O(n1) [1 + 2·z'], size: O(n2) [2 + 8·z' + 4·z'·z'' + 2·z'']

(21) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(22) BOUNDS(1, n^1)