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:

flatten(nil) → nil
flatten(unit(x)) → flatten(x)
flatten(++(x, y)) → ++(flatten(x), flatten(y))
flatten(++(unit(x), y)) → ++(flatten(x), flatten(y))
flatten(flatten(x)) → flatten(x)
rev(nil) → nil
rev(unit(x)) → unit(x)
rev(++(x, y)) → ++(rev(y), rev(x))
rev(rev(x)) → x
++(x, nil) → x
++(nil, y) → y
++(++(x, y), z) → ++(x, ++(y, z))

Rewrite Strategy: INNERMOST

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

The following rules are not reachable from basic terms in the dependency graph and can be removed:
rev(++(x, y)) → ++(rev(y), rev(x))
rev(rev(x)) → x

(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:

flatten(nil) → nil
rev(unit(x)) → unit(x)
flatten(++(unit(x), y)) → ++(flatten(x), flatten(y))
flatten(++(x, y)) → ++(flatten(x), flatten(y))
++(++(x, y), z) → ++(x, ++(y, z))
rev(nil) → nil
flatten(flatten(x)) → flatten(x)
flatten(unit(x)) → flatten(x)
++(x, nil) → x
++(nil, y) → y

Rewrite Strategy: INNERMOST

(3) 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:
flatten(++(unit(x), y)) → ++(flatten(x), flatten(y))
flatten(++(x, y)) → ++(flatten(x), flatten(y))
++(++(x, y), z) → ++(x, ++(y, z))
flatten(flatten(x)) → flatten(x)

(4) 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:

flatten(nil) → nil
rev(unit(x)) → unit(x)
rev(nil) → nil
flatten(unit(x)) → flatten(x)
++(x, nil) → x
++(nil, y) → y

Rewrite Strategy: INNERMOST

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

Transformed TRS to weighted TRS

(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:

flatten(nil) → nil [1]
rev(unit(x)) → unit(x) [1]
rev(nil) → nil [1]
flatten(unit(x)) → flatten(x) [1]
++(x, nil) → x [1]
++(nil, y) → 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:

flatten(nil) → nil [1]
rev(unit(x)) → unit(x) [1]
rev(nil) → nil [1]
flatten(unit(x)) → flatten(x) [1]
++(x, nil) → x [1]
++(nil, y) → y [1]

The TRS has the following type information:

flatten :: nil:unit → nil:unit nil :: nil:unit rev :: nil:unit → nil:unit unit :: nil:unit → nil:unit ++ :: nil:unit → nil:unit → nil:unit

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:

flatten
rev
++

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants: none

(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: flatten(nil) → nil [1] rev(unit(x)) → unit(x) [1] rev(nil) → nil [1] flatten(unit(x)) → flatten(x) [1] ++(x, nil) → x [1] ++(nil, y) → y [1] The TRS has the following type information: flatten :: nil:unit → nil:unit nil :: nil:unit rev :: nil:unit → nil:unit unit :: nil:unit → nil:unit ++ :: nil:unit → nil:unit → nil:unit 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: flatten(nil) → nil [1] rev(unit(x)) → unit(x) [1] rev(nil) → nil [1] flatten(unit(x)) → flatten(x) [1] ++(x, nil) → x [1] ++(nil, y) → y [1] The TRS has the following type information: flatten :: nil:unit → nil:unit nil :: nil:unit rev :: nil:unit → nil:unit unit :: nil:unit → nil:unit ++ :: nil:unit → nil:unit → nil:unit 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:

nil => 0

(14) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

Found the following analysis order by SCC decomposition:

{ rev } { flatten } { ++ }

(18) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {rev}, {flatten}, {++}

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

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

(20) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {rev}, {flatten}, {++}
Previous analysis results are:

rev: runtime: ?, size: O(n1) [z]

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

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {flatten}, {++}
Previous analysis results are:

rev: runtime: O(1) [1], size: O(n1) [z]

(23) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(24) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {flatten}, {++}
Previous analysis results are:

rev: runtime: O(1) [1], size: O(n1) [z]

(25) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(26) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {flatten}, {++}
Previous analysis results are:

rev: runtime: O(1) [1], size: O(n1) [z] flatten: runtime: ?, size: O(1) [0]

(27) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(28) Obligation:

Complexity RNTS consisting of the following rules:

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

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

rev: runtime: O(1) [1], size: O(n1) [z] flatten: runtime: O(n1) [1 + z], size: O(1) [0]

(29) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(30) Obligation:

Complexity RNTS consisting of the following rules:

++(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
++(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
flatten(z) -{ 1 + z }→ s :|: s >= 0, s <= 0, z - 1 >= 0
flatten(z) -{ 1 }→ 0 :|: z = 0
rev(z) -{ 1 }→ 0 :|: z = 0
rev(z) -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0

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

rev: runtime: O(1) [1], size: O(n1) [z] flatten: runtime: O(n1) [1 + z], size: O(1) [0]

(31) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(32) Obligation:

Complexity RNTS consisting of the following rules:

++(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
++(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
flatten(z) -{ 1 + z }→ s :|: s >= 0, s <= 0, z - 1 >= 0
flatten(z) -{ 1 }→ 0 :|: z = 0
rev(z) -{ 1 }→ 0 :|: z = 0
rev(z) -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0

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

rev: runtime: O(1) [1], size: O(n1) [z] flatten: runtime: O(n1) [1 + z], size: O(1) [0] ++: runtime: ?, size: O(n1) [z + z']

(33) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(34) Obligation:

Complexity RNTS consisting of the following rules:

++(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
++(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
flatten(z) -{ 1 + z }→ s :|: s >= 0, s <= 0, z - 1 >= 0
flatten(z) -{ 1 }→ 0 :|: z = 0
rev(z) -{ 1 }→ 0 :|: z = 0
rev(z) -{ 1 }→ 1 + (z - 1) :|: z - 1 >= 0

Function symbols to be analyzed:
Previous analysis results are:

rev: runtime: O(1) [1], size: O(n1) [z] flatten: runtime: O(n1) [1 + z], size: O(1) [0] ++: runtime: O(1) [1], size: O(n1) [z + z']

(35) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(36) BOUNDS(1, n^1)