The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

app(nil, k) → k
app(l, nil) → l
app(cons(x, l), k) → cons(x, app(l, k))
sum(cons(x, nil)) → cons(x, nil)
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))

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:
sum(app(l, cons(x, cons(y, k)))) → sum(app(l, sum(cons(x, cons(y, k)))))

(2) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

app(cons(x, l), k) → cons(x, app(l, k))
plus(s(x), y) → s(plus(x, y))
sum(cons(x, nil)) → cons(x, nil)
app(nil, k) → k
app(l, nil) → l
plus(0, y) → y
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l))

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^2).


The TRS R consists of the following rules:

app(cons(x, l), k) → cons(x, app(l, k)) [1]
plus(s(x), y) → s(plus(x, y)) [1]
sum(cons(x, nil)) → cons(x, nil) [1]
app(nil, k) → k [1]
app(l, nil) → l [1]
plus(0, y) → y [1]
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l)) [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:

app(cons(x, l), k) → cons(x, app(l, k)) [1]
plus(s(x), y) → s(plus(x, y)) [1]
sum(cons(x, nil)) → cons(x, nil) [1]
app(nil, k) → k [1]
app(l, nil) → l [1]
plus(0, y) → y [1]
sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l)) [1]

The TRS has the following type information:

app :: cons:nil → cons:nil → cons:nil cons :: s:0 → cons:nil → cons:nil plus :: s:0 → s:0 → s:0 s :: s:0 → s:0 sum :: cons:nil → cons:nil nil :: cons:nil 0 :: s: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:

app
sum

(c) The following functions are completely defined:

plus

Due to the following rules being added:
none

And the following fresh constants: none

(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: app(cons(x, l), k) → cons(x, app(l, k)) [1] plus(s(x), y) → s(plus(x, y)) [1] sum(cons(x, nil)) → cons(x, nil) [1] app(nil, k) → k [1] app(l, nil) → l [1] plus(0, y) → y [1] sum(cons(x, cons(y, l))) → sum(cons(plus(x, y), l)) [1] The TRS has the following type information: app :: cons:nil → cons:nil → cons:nil cons :: s:0 → cons:nil → cons:nil plus :: s:0 → s:0 → s:0 s :: s:0 → s:0 sum :: cons:nil → cons:nil nil :: cons:nil 0 :: s:0 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: app(cons(x, l), k) → cons(x, app(l, k)) [1] plus(s(x), y) → s(plus(x, y)) [1] sum(cons(x, nil)) → cons(x, nil) [1] app(nil, k) → k [1] app(l, nil) → l [1] plus(0, y) → y [1] sum(cons(s(x'), cons(y, l))) → sum(cons(s(plus(x', y)), l)) [2] sum(cons(0, cons(y, l))) → sum(cons(y, l)) [2] The TRS has the following type information: app :: cons:nil → cons:nil → cons:nil cons :: s:0 → cons:nil → cons:nil plus :: s:0 → s:0 → s:0 s :: s:0 → s:0 sum :: cons:nil → cons:nil nil :: cons:nil 0 :: s:0 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:

nil => 0
0 => 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ k :|: k >= 0, z' = k, z = 0
app(z, z') -{ 1 }→ l :|: z = l, l >= 0, z' = 0
app(z, z') -{ 1 }→ 1 + x + app(l, k) :|: x >= 0, l >= 0, z = 1 + x + l, k >= 0, z' = k
plus(z, z') -{ 1 }→ y :|: y >= 0, z = 0, z' = y
plus(z, z') -{ 1 }→ 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
sum(z) -{ 2 }→ sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
sum(z) -{ 2 }→ sum(1 + (1 + plus(x', y)) + l) :|: x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
sum(z) -{ 1 }→ 1 + x + 0 :|: x >= 0, z = 1 + x + 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:

app(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 1 }→ 1 + x + app(l, z') :|: x >= 0, l >= 0, z = 1 + x + l, z' >= 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
sum(z) -{ 2 }→ sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
sum(z) -{ 2 }→ sum(1 + (1 + plus(x', y)) + l) :|: x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
sum(z) -{ 1 }→ 1 + (z - 1) + 0 :|: z - 1 >= 0

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

Found the following analysis order by SCC decomposition:

{ app } { plus } { sum }

(16) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 1 }→ 1 + x + app(l, z') :|: x >= 0, l >= 0, z = 1 + x + l, z' >= 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
sum(z) -{ 2 }→ sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
sum(z) -{ 2 }→ sum(1 + (1 + plus(x', y)) + l) :|: x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
sum(z) -{ 1 }→ 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed: {app}, {plus}, {sum}

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

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 1 }→ 1 + x + app(l, z') :|: x >= 0, l >= 0, z = 1 + x + l, z' >= 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
sum(z) -{ 2 }→ sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
sum(z) -{ 2 }→ sum(1 + (1 + plus(x', y)) + l) :|: x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
sum(z) -{ 1 }→ 1 + (z - 1) + 0 :|: z - 1 >= 0

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

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

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

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

(20) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 1 }→ 1 + x + app(l, z') :|: x >= 0, l >= 0, z = 1 + x + l, z' >= 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
sum(z) -{ 2 }→ sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
sum(z) -{ 2 }→ sum(1 + (1 + plus(x', y)) + l) :|: x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
sum(z) -{ 1 }→ 1 + (z - 1) + 0 :|: z - 1 >= 0

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

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

(21) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 2 + l }→ 1 + x + s :|: s >= 0, s <= 1 * l + 1 * z', x >= 0, l >= 0, z = 1 + x + l, z' >= 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
sum(z) -{ 2 }→ sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
sum(z) -{ 2 }→ sum(1 + (1 + plus(x', y)) + l) :|: x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
sum(z) -{ 1 }→ 1 + (z - 1) + 0 :|: z - 1 >= 0

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

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

(23) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 2 + l }→ 1 + x + s :|: s >= 0, s <= 1 * l + 1 * z', x >= 0, l >= 0, z = 1 + x + l, z' >= 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
sum(z) -{ 2 }→ sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
sum(z) -{ 2 }→ sum(1 + (1 + plus(x', y)) + l) :|: x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
sum(z) -{ 1 }→ 1 + (z - 1) + 0 :|: z - 1 >= 0

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

app: runtime: O(n1) [1 + z], size: O(n1) [z + z'] plus: runtime: ?, size: O(n1) [z + z']

(25) 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: 1 + z

(26) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 2 + l }→ 1 + x + s :|: s >= 0, s <= 1 * l + 1 * z', x >= 0, l >= 0, z = 1 + x + l, z' >= 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
sum(z) -{ 2 }→ sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
sum(z) -{ 2 }→ sum(1 + (1 + plus(x', y)) + l) :|: x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
sum(z) -{ 1 }→ 1 + (z - 1) + 0 :|: z - 1 >= 0

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

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

(27) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(28) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 2 + l }→ 1 + x + s :|: s >= 0, s <= 1 * l + 1 * z', x >= 0, l >= 0, z = 1 + x + l, z' >= 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 + z }→ 1 + s' :|: s' >= 0, s' <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0
sum(z) -{ 2 }→ sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
sum(z) -{ 3 + x' }→ sum(1 + (1 + s'') + l) :|: s'' >= 0, s'' <= 1 * x' + 1 * y, x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
sum(z) -{ 1 }→ 1 + (z - 1) + 0 :|: z - 1 >= 0

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(30) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 2 + l }→ 1 + x + s :|: s >= 0, s <= 1 * l + 1 * z', x >= 0, l >= 0, z = 1 + x + l, z' >= 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 + z }→ 1 + s' :|: s' >= 0, s' <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0
sum(z) -{ 2 }→ sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
sum(z) -{ 3 + x' }→ sum(1 + (1 + s'') + l) :|: s'' >= 0, s'' <= 1 * x' + 1 * y, x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
sum(z) -{ 1 }→ 1 + (z - 1) + 0 :|: z - 1 >= 0

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

app: runtime: O(n1) [1 + z], size: O(n1) [z + z'] plus: runtime: O(n1) [1 + z], size: O(n1) [z + z'] sum: runtime: ?, size: O(n1) [z]

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

Computed RUNTIME bound using CoFloCo for: sum
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 3 + 2·z + z²

(32) Obligation:

Complexity RNTS consisting of the following rules:

app(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
app(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
app(z, z') -{ 2 + l }→ 1 + x + s :|: s >= 0, s <= 1 * l + 1 * z', x >= 0, l >= 0, z = 1 + x + l, z' >= 0
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 + z }→ 1 + s' :|: s' >= 0, s' <= 1 * (z - 1) + 1 * z', z - 1 >= 0, z' >= 0
sum(z) -{ 2 }→ sum(1 + y + l) :|: y >= 0, l >= 0, z = 1 + 0 + (1 + y + l)
sum(z) -{ 3 + x' }→ sum(1 + (1 + s'') + l) :|: s'' >= 0, s'' <= 1 * x' + 1 * y, x' >= 0, y >= 0, z = 1 + (1 + x') + (1 + y + l), l >= 0
sum(z) -{ 1 }→ 1 + (z - 1) + 0 :|: z - 1 >= 0

Function symbols to be analyzed:
Previous analysis results are:

app: runtime: O(n1) [1 + z], size: O(n1) [z + z'] plus: runtime: O(n1) [1 + z], size: O(n1) [z + z'] sum: runtime: O(n2) [3 + 2·z + z2], size: O(n1) [z]

(33) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(34) BOUNDS(1, n^2)