The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

mul0(Cons(x, xs), y) → add0(mul0(xs, y), y)
add0(Cons(x, xs), y) → add0(xs, Cons(S, y))
mul0(Nil, y) → Nil
add0(Nil, y) → y
goal(xs, ys) → mul0(xs, ys)

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^3).


The TRS R consists of the following rules:

mul0(Cons(x, xs), y) → add0(mul0(xs, y), y) [1]
add0(Cons(x, xs), y) → add0(xs, Cons(S, y)) [1]
mul0(Nil, y) → Nil [1]
add0(Nil, y) → y [1]
goal(xs, ys) → mul0(xs, ys) [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:

mul0(Cons(x, xs), y) → add0(mul0(xs, y), y) [1]
add0(Cons(x, xs), y) → add0(xs, Cons(S, y)) [1]
mul0(Nil, y) → Nil [1]
add0(Nil, y) → y [1]
goal(xs, ys) → mul0(xs, ys) [1]

The TRS has the following type information:

mul0 :: Cons:Nil → Cons:Nil → Cons:Nil Cons :: S → Cons:Nil → Cons:Nil add0 :: Cons:Nil → Cons:Nil → Cons:Nil S :: S Nil :: Cons:Nil goal :: Cons:Nil → Cons:Nil → Cons:Nil

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:

goal

(c) The following functions are completely defined:

mul0
add0

Due to the following rules being added:
none

And the following fresh constants: none

(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: mul0(Cons(x, xs), y) → add0(mul0(xs, y), y) [1] add0(Cons(x, xs), y) → add0(xs, Cons(S, y)) [1] mul0(Nil, y) → Nil [1] add0(Nil, y) → y [1] goal(xs, ys) → mul0(xs, ys) [1] The TRS has the following type information: mul0 :: Cons:Nil → Cons:Nil → Cons:Nil Cons :: S → Cons:Nil → Cons:Nil add0 :: Cons:Nil → Cons:Nil → Cons:Nil S :: S Nil :: Cons:Nil goal :: Cons:Nil → Cons:Nil → Cons:Nil 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: mul0(Cons(x, Cons(x', xs')), y) → add0(add0(mul0(xs', y), y), y) [2] mul0(Cons(x, Nil), y) → add0(Nil, y) [2] add0(Cons(x, xs), y) → add0(xs, Cons(S, y)) [1] mul0(Nil, y) → Nil [1] add0(Nil, y) → y [1] goal(xs, ys) → mul0(xs, ys) [1] The TRS has the following type information: mul0 :: Cons:Nil → Cons:Nil → Cons:Nil Cons :: S → Cons:Nil → Cons:Nil add0 :: Cons:Nil → Cons:Nil → Cons:Nil S :: S Nil :: Cons:Nil goal :: Cons:Nil → Cons:Nil → Cons:Nil 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:

S => 0
Nil => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

add0(z, z') -{ 1 }→ y :|: y >= 0, z = 0, z' = y
add0(z, z') -{ 1 }→ add0(xs, 1 + 0 + y) :|: z = 1 + x + xs, xs >= 0, x >= 0, y >= 0, z' = y
goal(z, z') -{ 1 }→ mul0(xs, ys) :|: xs >= 0, z = xs, z' = ys, ys >= 0
mul0(z, z') -{ 2 }→ add0(add0(mul0(xs', y), y), y) :|: x >= 0, x' >= 0, xs' >= 0, y >= 0, z = 1 + x + (1 + x' + xs'), z' = y
mul0(z, z') -{ 2 }→ add0(0, y) :|: x >= 0, y >= 0, z' = y, z = 1 + x + 0
mul0(z, z') -{ 1 }→ 0 :|: y >= 0, z = 0, z' = y

(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:

add0(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add0(z, z') -{ 1 }→ add0(xs, 1 + 0 + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
goal(z, z') -{ 1 }→ mul0(z, z') :|: z >= 0, z' >= 0
mul0(z, z') -{ 2 }→ add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs')
mul0(z, z') -{ 2 }→ add0(0, z') :|: z - 1 >= 0, z' >= 0
mul0(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

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

Found the following analysis order by SCC decomposition:

{ add0 } { mul0 } { goal }

(14) Obligation:

Complexity RNTS consisting of the following rules:

add0(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add0(z, z') -{ 1 }→ add0(xs, 1 + 0 + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
goal(z, z') -{ 1 }→ mul0(z, z') :|: z >= 0, z' >= 0
mul0(z, z') -{ 2 }→ add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs')
mul0(z, z') -{ 2 }→ add0(0, z') :|: z - 1 >= 0, z' >= 0
mul0(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {add0}, {mul0}, {goal}

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

Computed SIZE bound using CoFloCo for: add0
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: z + z'

(16) Obligation:

Complexity RNTS consisting of the following rules:

add0(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add0(z, z') -{ 1 }→ add0(xs, 1 + 0 + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
goal(z, z') -{ 1 }→ mul0(z, z') :|: z >= 0, z' >= 0
mul0(z, z') -{ 2 }→ add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs')
mul0(z, z') -{ 2 }→ add0(0, z') :|: z - 1 >= 0, z' >= 0
mul0(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {add0}, {mul0}, {goal}
Previous analysis results are:

add0: runtime: ?, size: O(n1) [z + z']

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

Computed RUNTIME bound using CoFloCo for: add0
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + z

(18) Obligation:

Complexity RNTS consisting of the following rules:

add0(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
add0(z, z') -{ 1 }→ add0(xs, 1 + 0 + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
goal(z, z') -{ 1 }→ mul0(z, z') :|: z >= 0, z' >= 0
mul0(z, z') -{ 2 }→ add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs')
mul0(z, z') -{ 2 }→ add0(0, z') :|: z - 1 >= 0, z' >= 0
mul0(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {mul0}, {goal}
Previous analysis results are:

add0: runtime: O(n1) [1 + z], size: O(n1) [z + z']

(19) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(20) Obligation:

Complexity RNTS consisting of the following rules:

add0(z, z') -{ 2 + xs }→ s' :|: s' >= 0, s' <= 1 * xs + 1 * (1 + 0 + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
add0(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
goal(z, z') -{ 1 }→ mul0(z, z') :|: z >= 0, z' >= 0
mul0(z, z') -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * z', z - 1 >= 0, z' >= 0
mul0(z, z') -{ 2 }→ add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs')
mul0(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {mul0}, {goal}
Previous analysis results are:

add0: runtime: O(n1) [1 + z], size: O(n1) [z + z']

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

Computed SIZE bound using KoAT for: mul0
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 2·z·z' + z'

(22) Obligation:

Complexity RNTS consisting of the following rules:

add0(z, z') -{ 2 + xs }→ s' :|: s' >= 0, s' <= 1 * xs + 1 * (1 + 0 + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
add0(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
goal(z, z') -{ 1 }→ mul0(z, z') :|: z >= 0, z' >= 0
mul0(z, z') -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * z', z - 1 >= 0, z' >= 0
mul0(z, z') -{ 2 }→ add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs')
mul0(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {mul0}, {goal}
Previous analysis results are:

add0: runtime: O(n1) [1 + z], size: O(n1) [z + z'] mul0: runtime: ?, size: O(n2) [2·z·z' + z']

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

Computed RUNTIME bound using KoAT for: mul0
after applying outer abstraction to obtain an ITS,
resulting in: O(n³) with polynomial bound: 4 + 4·z + 3·z·z' + 4·z²·z'

(24) Obligation:

Complexity RNTS consisting of the following rules:

add0(z, z') -{ 2 + xs }→ s' :|: s' >= 0, s' <= 1 * xs + 1 * (1 + 0 + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
add0(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
goal(z, z') -{ 1 }→ mul0(z, z') :|: z >= 0, z' >= 0
mul0(z, z') -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * z', z - 1 >= 0, z' >= 0
mul0(z, z') -{ 2 }→ add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs')
mul0(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

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

add0: runtime: O(n1) [1 + z], size: O(n1) [z + z'] mul0: runtime: O(n3) [4 + 4·z + 3·z·z' + 4·z2·z'], size: O(n2) [2·z·z' + z']

(25) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(26) Obligation:

Complexity RNTS consisting of the following rules:

add0(z, z') -{ 2 + xs }→ s' :|: s' >= 0, s' <= 1 * xs + 1 * (1 + 0 + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
add0(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
goal(z, z') -{ 5 + 4·z + 3·z·z' + 4·z²·z' }→ s3 :|: s3 >= 0, s3 <= 2 * (z' * z) + 1 * z', z >= 0, z' >= 0
mul0(z, z') -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * z', z - 1 >= 0, z' >= 0
mul0(z, z') -{ 8 + s'' + s1 + 4·xs' + 3·xs'·z' + 4·xs'²·z' }→ s2 :|: s'' >= 0, s'' <= 2 * (z' * xs') + 1 * z', s1 >= 0, s1 <= 1 * s'' + 1 * z', s2 >= 0, s2 <= 1 * s1 + 1 * z', x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs')
mul0(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

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

add0: runtime: O(n1) [1 + z], size: O(n1) [z + z'] mul0: runtime: O(n3) [4 + 4·z + 3·z·z' + 4·z2·z'], size: O(n2) [2·z·z' + z']

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

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

(28) Obligation:

Complexity RNTS consisting of the following rules:

add0(z, z') -{ 2 + xs }→ s' :|: s' >= 0, s' <= 1 * xs + 1 * (1 + 0 + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
add0(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
goal(z, z') -{ 5 + 4·z + 3·z·z' + 4·z²·z' }→ s3 :|: s3 >= 0, s3 <= 2 * (z' * z) + 1 * z', z >= 0, z' >= 0
mul0(z, z') -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * z', z - 1 >= 0, z' >= 0
mul0(z, z') -{ 8 + s'' + s1 + 4·xs' + 3·xs'·z' + 4·xs'²·z' }→ s2 :|: s'' >= 0, s'' <= 2 * (z' * xs') + 1 * z', s1 >= 0, s1 <= 1 * s'' + 1 * z', s2 >= 0, s2 <= 1 * s1 + 1 * z', x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs')
mul0(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

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

add0: runtime: O(n1) [1 + z], size: O(n1) [z + z'] mul0: runtime: O(n3) [4 + 4·z + 3·z·z' + 4·z2·z'], size: O(n2) [2·z·z' + z'] goal: runtime: ?, size: O(n2) [2·z·z' + z']

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

Computed RUNTIME bound using KoAT for: goal
after applying outer abstraction to obtain an ITS,
resulting in: O(n³) with polynomial bound: 5 + 4·z + 3·z·z' + 4·z²·z'

(30) Obligation:

Complexity RNTS consisting of the following rules:

add0(z, z') -{ 2 + xs }→ s' :|: s' >= 0, s' <= 1 * xs + 1 * (1 + 0 + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
add0(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
goal(z, z') -{ 5 + 4·z + 3·z·z' + 4·z²·z' }→ s3 :|: s3 >= 0, s3 <= 2 * (z' * z) + 1 * z', z >= 0, z' >= 0
mul0(z, z') -{ 3 }→ s :|: s >= 0, s <= 1 * 0 + 1 * z', z - 1 >= 0, z' >= 0
mul0(z, z') -{ 8 + s'' + s1 + 4·xs' + 3·xs'·z' + 4·xs'²·z' }→ s2 :|: s'' >= 0, s'' <= 2 * (z' * xs') + 1 * z', s1 >= 0, s1 <= 1 * s'' + 1 * z', s2 >= 0, s2 <= 1 * s1 + 1 * z', x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs')
mul0(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0

Function symbols to be analyzed:
Previous analysis results are:

add0: runtime: O(n1) [1 + z], size: O(n1) [z + z'] mul0: runtime: O(n3) [4 + 4·z + 3·z·z' + 4·z2·z'], size: O(n2) [2·z·z' + z'] goal: runtime: O(n3) [5 + 4·z + 3·z·z' + 4·z2·z'], size: O(n2) [2·z·z' + z']

(31) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(32) BOUNDS(1, n^3)