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:

addlist(Cons(x, xs'), Cons(S(0), xs)) → Cons(S(x), addlist(xs', xs))
addlist(Cons(S(0), xs'), Cons(x, xs)) → Cons(S(x), addlist(xs', xs))
addlist(Nil, ys) → Nil
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs, ys) → addlist(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^1).


The TRS R consists of the following rules:

addlist(Cons(x, xs'), Cons(S(0), xs)) → Cons(S(x), addlist(xs', xs)) [1]
addlist(Cons(S(0), xs'), Cons(x, xs)) → Cons(S(x), addlist(xs', xs)) [1]
addlist(Nil, ys) → Nil [1]
notEmpty(Cons(x, xs)) → True [1]
notEmpty(Nil) → False [1]
goal(xs, ys) → addlist(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:

addlist(Cons(x, xs'), Cons(S(0), xs)) → Cons(S(x), addlist(xs', xs)) [1]
addlist(Cons(S(0), xs'), Cons(x, xs)) → Cons(S(x), addlist(xs', xs)) [1]
addlist(Nil, ys) → Nil [1]
notEmpty(Cons(x, xs)) → True [1]
notEmpty(Nil) → False [1]
goal(xs, ys) → addlist(xs, ys) [1]

The TRS has the following type information:

addlist :: Cons:Nil → Cons:Nil → Cons:Nil Cons :: 0:S → Cons:Nil → Cons:Nil S :: 0:S → 0:S 0 :: 0:S 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:

addlist
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: addlist(Cons(x, xs'), Cons(S(0), xs)) → Cons(S(x), addlist(xs', xs)) [1] addlist(Cons(S(0), xs'), Cons(x, xs)) → Cons(S(x), addlist(xs', xs)) [1] addlist(Nil, ys) → Nil [1] notEmpty(Cons(x, xs)) → True [1] notEmpty(Nil) → False [1] goal(xs, ys) → addlist(xs, ys) [1] The TRS has the following type information: addlist :: Cons:Nil → Cons:Nil → Cons:Nil Cons :: 0:S → Cons:Nil → Cons:Nil S :: 0:S → 0:S 0 :: 0:S 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: addlist(Cons(x, xs'), Cons(S(0), xs)) → Cons(S(x), addlist(xs', xs)) [1] addlist(Cons(S(0), xs'), Cons(x, xs)) → Cons(S(x), addlist(xs', xs)) [1] addlist(Nil, ys) → Nil [1] notEmpty(Cons(x, xs)) → True [1] notEmpty(Nil) → False [1] goal(xs, ys) → addlist(xs, ys) [1] The TRS has the following type information: addlist :: Cons:Nil → Cons:Nil → Cons:Nil Cons :: 0:S → Cons:Nil → Cons:Nil S :: 0:S → 0:S 0 :: 0:S 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:

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

(10) Obligation:

Complexity RNTS consisting of the following rules:

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

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

{ notEmpty } { addlist } { goal }

(14) Obligation:

Complexity RNTS consisting of the following rules:

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

(15) 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

(16) Obligation:

Complexity RNTS consisting of the following rules:

addlist(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
addlist(z, z') -{ 1 }→ 1 + (1 + x) + addlist(xs', z' - 2) :|: z' - 2 >= 0, x >= 0, xs' >= 0, z = 1 + x + xs'
addlist(z, z') -{ 1 }→ 1 + (1 + x) + addlist(z - 2, xs) :|: xs >= 0, z' = 1 + x + xs, z - 2 >= 0, x >= 0
goal(z, z') -{ 1 }→ addlist(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}, {addlist}, {goal}
Previous analysis results are:

notEmpty: runtime: ?, size: O(1) [1]

(17) 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

(18) Obligation:

Complexity RNTS consisting of the following rules:

addlist(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
addlist(z, z') -{ 1 }→ 1 + (1 + x) + addlist(xs', z' - 2) :|: z' - 2 >= 0, x >= 0, xs' >= 0, z = 1 + x + xs'
addlist(z, z') -{ 1 }→ 1 + (1 + x) + addlist(z - 2, xs) :|: xs >= 0, z' = 1 + x + xs, z - 2 >= 0, x >= 0
goal(z, z') -{ 1 }→ addlist(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: {addlist}, {goal}
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [1]

(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:

addlist(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
addlist(z, z') -{ 1 }→ 1 + (1 + x) + addlist(xs', z' - 2) :|: z' - 2 >= 0, x >= 0, xs' >= 0, z = 1 + x + xs'
addlist(z, z') -{ 1 }→ 1 + (1 + x) + addlist(z - 2, xs) :|: xs >= 0, z' = 1 + x + xs, z - 2 >= 0, x >= 0
goal(z, z') -{ 1 }→ addlist(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: {addlist}, {goal}
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [1]

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

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

addlist(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
addlist(z, z') -{ 1 }→ 1 + (1 + x) + addlist(xs', z' - 2) :|: z' - 2 >= 0, x >= 0, xs' >= 0, z = 1 + x + xs'
addlist(z, z') -{ 1 }→ 1 + (1 + x) + addlist(z - 2, xs) :|: xs >= 0, z' = 1 + x + xs, z - 2 >= 0, x >= 0
goal(z, z') -{ 1 }→ addlist(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: {addlist}, {goal}
Previous analysis results are:

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

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

addlist(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
addlist(z, z') -{ 1 }→ 1 + (1 + x) + addlist(xs', z' - 2) :|: z' - 2 >= 0, x >= 0, xs' >= 0, z = 1 + x + xs'
addlist(z, z') -{ 1 }→ 1 + (1 + x) + addlist(z - 2, xs) :|: xs >= 0, z' = 1 + x + xs, z - 2 >= 0, x >= 0
goal(z, z') -{ 1 }→ addlist(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: {goal}
Previous analysis results are:

notEmpty: runtime: O(1) [1], size: O(1) [1] addlist: runtime: O(n1) [1 + z], size: O(n1) [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:

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

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

(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 + z'

(28) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

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

(31) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(32) BOUNDS(1, n^1)