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:

add0(x', Cons(x, xs)) → add0(Cons(Cons(Nil, Nil), x'), xs)
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
add0(x, Nil) → x
goal(x, y) → add0(x, y)

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:

add0(x', Cons(x, xs)) → add0(Cons(Cons(Nil, Nil), x'), xs) [1]
notEmpty(Cons(x, xs)) → True [1]
notEmpty(Nil) → False [1]
add0(x, Nil) → x [1]
goal(x, y) → add0(x, y) [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:

add0(x', Cons(x, xs)) → add0(Cons(Cons(Nil, Nil), x'), xs) [1]
notEmpty(Cons(x, xs)) → True [1]
notEmpty(Nil) → False [1]
add0(x, Nil) → x [1]
goal(x, y) → add0(x, y) [1]

The TRS has the following type information:

add0 :: Cons:Nil → Cons:Nil → Cons:Nil Cons :: Cons:Nil → Cons:Nil → Cons:Nil Nil :: Cons:Nil notEmpty :: Cons:Nil → True:False True :: True:False False :: True:False 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:

add0
notEmpty
goal

(c) The following functions are completely defined:
none

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: add0(x', Cons(x, xs)) → add0(Cons(Cons(Nil, Nil), x'), xs) [1] notEmpty(Cons(x, xs)) → True [1] notEmpty(Nil) → False [1] add0(x, Nil) → x [1] goal(x, y) → add0(x, y) [1] The TRS has the following type information: add0 :: Cons:Nil → Cons:Nil → Cons:Nil Cons :: Cons:Nil → Cons:Nil → Cons:Nil Nil :: Cons:Nil notEmpty :: Cons:Nil → True:False True :: True:False False :: True:False 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: add0(x', Cons(x, xs)) → add0(Cons(Cons(Nil, Nil), x'), xs) [1] notEmpty(Cons(x, xs)) → True [1] notEmpty(Nil) → False [1] add0(x, Nil) → x [1] goal(x, y) → add0(x, y) [1] The TRS has the following type information: add0 :: Cons:Nil → Cons:Nil → Cons:Nil Cons :: Cons:Nil → Cons:Nil → Cons:Nil Nil :: Cons:Nil notEmpty :: Cons:Nil → True:False True :: True:False False :: True:False 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:

Nil => 0
True => 1
False => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

Found the following analysis order by SCC decomposition:

{ add0 } { notEmpty } { goal }

(14) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {add0}, {notEmpty}, {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 + 2·z'

(16) Obligation:

Complexity RNTS consisting of the following rules:

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

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

add0: runtime: ?, size: O(n1) [z + 2·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(1 + (1 + 0 + 0) + z, xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
goal(z, z') -{ 1 }→ add0(z, z') :|: z >= 0, z' >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

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

add0: runtime: O(n1) [1 + z'], size: O(n1) [z + 2·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 * (1 + (1 + 0 + 0) + z) + 2 * xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
add0(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
goal(z, z') -{ 2 + z' }→ s' :|: s' >= 0, s' <= 1 * z + 2 * z', z >= 0, z' >= 0
notEmpty(z) -{ 1 }→ 1 :|: z = 1 + x + xs, xs >= 0, x >= 0
notEmpty(z) -{ 1 }→ 0 :|: z = 0

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

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

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

Computed SIZE bound using CoFloCo for: notEmpty
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:

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

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

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

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

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

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

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

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

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

(28) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

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

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

(30) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed:
Previous analysis results are:

add0: runtime: O(n1) [1 + z'], size: O(n1) [z + 2·z'] notEmpty: runtime: O(1) [1], size: O(1) [1] goal: runtime: O(n1) [2 + z'], size: O(n1) [z + 2·z']

(31) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(32) BOUNDS(1, n^1)