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

(0) Obligation:

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


The TRS R consists of the following rules:

map(Cons(x, xs)) → Cons(f(x), map(xs))
map(Nil) → Nil
goal(xs) → map(xs)
f(x) → *(x, x)
+Full(S(x), y) → +Full(x, S(y))
+Full(0, y) → y

The (relative) TRS S consists of the following rules:

*(x, S(S(y))) → +(x, *(x, S(y)))
*(x, S(0)) → x
*(x, 0) → 0
*(0, y) → 0

Rewrite Strategy: INNERMOST

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

Transformed relative 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:

map(Cons(x, xs)) → Cons(f(x), map(xs)) [1]
map(Nil) → Nil [1]
goal(xs) → map(xs) [1]
f(x) → *(x, x) [1]
+Full(S(x), y) → +Full(x, S(y)) [1]
+Full(0, y) → y [1]
*(x, S(S(y))) → +(x, *(x, S(y))) [0]
*(x, S(0)) → x [0]
*(x, 0) → 0 [0]
*(0, y) → 0 [0]

Rewrite Strategy: INNERMOST

(3) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID) transformation)

Renamed defined symbols to avoid conflicts with arithmetic symbols:

* => times

(4) 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:

map(Cons(x, xs)) → Cons(f(x), map(xs)) [1]
map(Nil) → Nil [1]
goal(xs) → map(xs) [1]
f(x) → times(x, x) [1]
+Full(S(x), y) → +Full(x, S(y)) [1]
+Full(0, y) → y [1]
times(x, S(S(y))) → +(x, times(x, S(y))) [0]
times(x, S(0)) → x [0]
times(x, 0) → 0 [0]
times(0, y) → 0 [0]

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:

map(Cons(x, xs)) → Cons(f(x), map(xs)) [1]
map(Nil) → Nil [1]
goal(xs) → map(xs) [1]
f(x) → times(x, x) [1]
+Full(S(x), y) → +Full(x, S(y)) [1]
+Full(0, y) → y [1]
times(x, S(S(y))) → +(x, times(x, S(y))) [0]
times(x, S(0)) → x [0]
times(x, 0) → 0 [0]
times(0, y) → 0 [0]

The TRS has the following type information:

map :: Cons:Nil → Cons:Nil Cons :: S:0:+ → Cons:Nil → Cons:Nil f :: S:0:+ → S:0:+ Nil :: Cons:Nil goal :: Cons:Nil → Cons:Nil times :: S:0:+ → S:0:+ → S:0:+ +Full :: S:0:+ → S:0:+ → S:0:+ S :: S:0:+ → S:0:+ 0 :: S:0:+ + :: S:0:+ → S: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:

map
goal
f
+Full

(c) The following functions are completely defined:

times

Due to the following rules being added:

times(v0, v1) → 0 [0]

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: map(Cons(x, xs)) → Cons(f(x), map(xs)) [1] map(Nil) → Nil [1] goal(xs) → map(xs) [1] f(x) → times(x, x) [1] +Full(S(x), y) → +Full(x, S(y)) [1] +Full(0, y) → y [1] times(x, S(S(y))) → +(x, times(x, S(y))) [0] times(x, S(0)) → x [0] times(x, 0) → 0 [0] times(0, y) → 0 [0] times(v0, v1) → 0 [0] The TRS has the following type information: map :: Cons:Nil → Cons:Nil Cons :: S:0:+ → Cons:Nil → Cons:Nil f :: S:0:+ → S:0:+ Nil :: Cons:Nil goal :: Cons:Nil → Cons:Nil times :: S:0:+ → S:0:+ → S:0:+ +Full :: S:0:+ → S:0:+ → S:0:+ S :: S:0:+ → S:0:+ 0 :: S:0:+ + :: S:0:+ → S: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: map(Cons(x, xs)) → Cons(f(x), map(xs)) [1] map(Nil) → Nil [1] goal(xs) → map(xs) [1] f(x) → times(x, x) [1] +Full(S(x), y) → +Full(x, S(y)) [1] +Full(0, y) → y [1] times(x, S(S(y))) → +(x, times(x, S(y))) [0] times(x, S(0)) → x [0] times(x, 0) → 0 [0] times(0, y) → 0 [0] times(v0, v1) → 0 [0] The TRS has the following type information: map :: Cons:Nil → Cons:Nil Cons :: S:0:+ → Cons:Nil → Cons:Nil f :: S:0:+ → S:0:+ Nil :: Cons:Nil goal :: Cons:Nil → Cons:Nil times :: S:0:+ → S:0:+ → S:0:+ +Full :: S:0:+ → S:0:+ → S:0:+ S :: S:0:+ → S:0:+ 0 :: S:0:+ + :: S:0:+ → S: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:

+Full(z, z') -{ 1 }→ y :|: y >= 0, z = 0, z' = y
+Full(z, z') -{ 1 }→ +Full(x, 1 + y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
f(z) -{ 1 }→ times(x, x) :|: x >= 0, z = x
goal(z) -{ 1 }→ map(xs) :|: xs >= 0, z = xs
map(z) -{ 1 }→ 0 :|: z = 0
map(z) -{ 1 }→ 1 + f(x) + map(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0
times(z, z') -{ 0 }→ x :|: x >= 0, z' = 1 + 0, z = x
times(z, z') -{ 0 }→ 0 :|: x >= 0, z = x, z' = 0
times(z, z') -{ 0 }→ 0 :|: y >= 0, z = 0, z' = y
times(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
times(z, z') -{ 0 }→ 1 + x + times(x, 1 + y) :|: z' = 1 + (1 + y), x >= 0, y >= 0, z = x

(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:

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

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

Found the following analysis order by SCC decomposition:

{ times } { +Full } { f } { map } { goal }

(16) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {times}, {+Full}, {f}, {map}, {goal}

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

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {times}, {+Full}, {f}, {map}, {goal}
Previous analysis results are:

times: runtime: ?, size: O(n2) [z + z·z' + z']

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

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

(20) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {+Full}, {f}, {map}, {goal}
Previous analysis results are:

times: runtime: O(1) [0], size: O(n2) [z + z·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:

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

Function symbols to be analyzed: {+Full}, {f}, {map}, {goal}
Previous analysis results are:

times: runtime: O(1) [0], size: O(n2) [z + z·z' + z']

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

Computed SIZE bound using CoFloCo for: +Full
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:

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

Function symbols to be analyzed: {+Full}, {f}, {map}, {goal}
Previous analysis results are:

times: runtime: O(1) [0], size: O(n2) [z + z·z' + z'] +Full: runtime: ?, size: O(n1) [z + z']

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

Computed RUNTIME bound using CoFloCo for: +Full
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:

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

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

times: runtime: O(1) [0], size: O(n2) [z + z·z' + z'] +Full: 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:

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

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

times: runtime: O(1) [0], size: O(n2) [z + z·z' + z'] +Full: runtime: O(n1) [1 + z], size: O(n1) [z + z']

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

Computed SIZE bound using KoAT for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 2·z + z²

(30) Obligation:

Complexity RNTS consisting of the following rules:

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

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

times: runtime: O(1) [0], size: O(n2) [z + z·z' + z'] +Full: runtime: O(n1) [1 + z], size: O(n1) [z + z'] f: runtime: ?, size: O(n2) [2·z + z2]

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

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

(32) Obligation:

Complexity RNTS consisting of the following rules:

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

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

times: runtime: O(1) [0], size: O(n2) [z + z·z' + z'] +Full: runtime: O(n1) [1 + z], size: O(n1) [z + z'] f: runtime: O(1) [1], size: O(n2) [2·z + z2]

(33) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(34) Obligation:

Complexity RNTS consisting of the following rules:

+Full(z, z') -{ 1 + z }→ s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 1 * (1 + z'), z - 1 >= 0, z' >= 0
+Full(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
f(z) -{ 1 }→ s :|: s >= 0, s <= 1 * (z * z) + 1 * z + 1 * z, z >= 0
goal(z) -{ 1 }→ map(z) :|: z >= 0
map(z) -{ 1 }→ 0 :|: z = 0
map(z) -{ 2 }→ 1 + s1 + map(xs) :|: s1 >= 0, s1 <= 2 * x + 1 * (x * x), z = 1 + x + xs, xs >= 0, x >= 0
times(z, z') -{ 0 }→ z :|: z >= 0, z' = 1 + 0
times(z, z') -{ 0 }→ 0 :|: z >= 0, z' = 0
times(z, z') -{ 0 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
times(z, z') -{ 0 }→ 1 + z + s' :|: s' >= 0, s' <= 1 * ((1 + (z' - 2)) * z) + 1 * (1 + (z' - 2)) + 1 * z, z >= 0, z' - 2 >= 0

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

times: runtime: O(1) [0], size: O(n2) [z + z·z' + z'] +Full: runtime: O(n1) [1 + z], size: O(n1) [z + z'] f: runtime: O(1) [1], size: O(n2) [2·z + z2]

(35) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: map
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

(36) Obligation:

Complexity RNTS consisting of the following rules:

+Full(z, z') -{ 1 + z }→ s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 1 * (1 + z'), z - 1 >= 0, z' >= 0
+Full(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
f(z) -{ 1 }→ s :|: s >= 0, s <= 1 * (z * z) + 1 * z + 1 * z, z >= 0
goal(z) -{ 1 }→ map(z) :|: z >= 0
map(z) -{ 1 }→ 0 :|: z = 0
map(z) -{ 2 }→ 1 + s1 + map(xs) :|: s1 >= 0, s1 <= 2 * x + 1 * (x * x), z = 1 + x + xs, xs >= 0, x >= 0
times(z, z') -{ 0 }→ z :|: z >= 0, z' = 1 + 0
times(z, z') -{ 0 }→ 0 :|: z >= 0, z' = 0
times(z, z') -{ 0 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
times(z, z') -{ 0 }→ 1 + z + s' :|: s' >= 0, s' <= 1 * ((1 + (z' - 2)) * z) + 1 * (1 + (z' - 2)) + 1 * z, z >= 0, z' - 2 >= 0

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

times: runtime: O(1) [0], size: O(n2) [z + z·z' + z'] +Full: runtime: O(n1) [1 + z], size: O(n1) [z + z'] f: runtime: O(1) [1], size: O(n2) [2·z + z2] map: runtime: ?, size: INF

(37) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using CoFloCo for: map
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 1 + 2·z

(38) Obligation:

Complexity RNTS consisting of the following rules:

+Full(z, z') -{ 1 + z }→ s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 1 * (1 + z'), z - 1 >= 0, z' >= 0
+Full(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
f(z) -{ 1 }→ s :|: s >= 0, s <= 1 * (z * z) + 1 * z + 1 * z, z >= 0
goal(z) -{ 1 }→ map(z) :|: z >= 0
map(z) -{ 1 }→ 0 :|: z = 0
map(z) -{ 2 }→ 1 + s1 + map(xs) :|: s1 >= 0, s1 <= 2 * x + 1 * (x * x), z = 1 + x + xs, xs >= 0, x >= 0
times(z, z') -{ 0 }→ z :|: z >= 0, z' = 1 + 0
times(z, z') -{ 0 }→ 0 :|: z >= 0, z' = 0
times(z, z') -{ 0 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
times(z, z') -{ 0 }→ 1 + z + s' :|: s' >= 0, s' <= 1 * ((1 + (z' - 2)) * z) + 1 * (1 + (z' - 2)) + 1 * z, z >= 0, z' - 2 >= 0

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

times: runtime: O(1) [0], size: O(n2) [z + z·z' + z'] +Full: runtime: O(n1) [1 + z], size: O(n1) [z + z'] f: runtime: O(1) [1], size: O(n2) [2·z + z2] map: runtime: O(n1) [1 + 2·z], size: INF

(39) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(40) Obligation:

Complexity RNTS consisting of the following rules:

+Full(z, z') -{ 1 + z }→ s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 1 * (1 + z'), z - 1 >= 0, z' >= 0
+Full(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
f(z) -{ 1 }→ s :|: s >= 0, s <= 1 * (z * z) + 1 * z + 1 * z, z >= 0
goal(z) -{ 2 + 2·z }→ s3 :|: s3 >= 0, s3 <= inf', z >= 0
map(z) -{ 1 }→ 0 :|: z = 0
map(z) -{ 3 + 2·xs }→ 1 + s1 + s2 :|: s2 >= 0, s2 <= inf, s1 >= 0, s1 <= 2 * x + 1 * (x * x), z = 1 + x + xs, xs >= 0, x >= 0
times(z, z') -{ 0 }→ z :|: z >= 0, z' = 1 + 0
times(z, z') -{ 0 }→ 0 :|: z >= 0, z' = 0
times(z, z') -{ 0 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
times(z, z') -{ 0 }→ 1 + z + s' :|: s' >= 0, s' <= 1 * ((1 + (z' - 2)) * z) + 1 * (1 + (z' - 2)) + 1 * z, z >= 0, z' - 2 >= 0

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

times: runtime: O(1) [0], size: O(n2) [z + z·z' + z'] +Full: runtime: O(n1) [1 + z], size: O(n1) [z + z'] f: runtime: O(1) [1], size: O(n2) [2·z + z2] map: runtime: O(n1) [1 + 2·z], size: INF

(41) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using CoFloCo for: goal
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

(42) Obligation:

Complexity RNTS consisting of the following rules:

+Full(z, z') -{ 1 + z }→ s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 1 * (1 + z'), z - 1 >= 0, z' >= 0
+Full(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
f(z) -{ 1 }→ s :|: s >= 0, s <= 1 * (z * z) + 1 * z + 1 * z, z >= 0
goal(z) -{ 2 + 2·z }→ s3 :|: s3 >= 0, s3 <= inf', z >= 0
map(z) -{ 1 }→ 0 :|: z = 0
map(z) -{ 3 + 2·xs }→ 1 + s1 + s2 :|: s2 >= 0, s2 <= inf, s1 >= 0, s1 <= 2 * x + 1 * (x * x), z = 1 + x + xs, xs >= 0, x >= 0
times(z, z') -{ 0 }→ z :|: z >= 0, z' = 1 + 0
times(z, z') -{ 0 }→ 0 :|: z >= 0, z' = 0
times(z, z') -{ 0 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
times(z, z') -{ 0 }→ 1 + z + s' :|: s' >= 0, s' <= 1 * ((1 + (z' - 2)) * z) + 1 * (1 + (z' - 2)) + 1 * z, z >= 0, z' - 2 >= 0

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

times: runtime: O(1) [0], size: O(n2) [z + z·z' + z'] +Full: runtime: O(n1) [1 + z], size: O(n1) [z + z'] f: runtime: O(1) [1], size: O(n2) [2·z + z2] map: runtime: O(n1) [1 + 2·z], size: INF goal: runtime: ?, size: INF

(43) 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 + 2·z

(44) Obligation:

Complexity RNTS consisting of the following rules:

+Full(z, z') -{ 1 + z }→ s'' :|: s'' >= 0, s'' <= 1 * (z - 1) + 1 * (1 + z'), z - 1 >= 0, z' >= 0
+Full(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
f(z) -{ 1 }→ s :|: s >= 0, s <= 1 * (z * z) + 1 * z + 1 * z, z >= 0
goal(z) -{ 2 + 2·z }→ s3 :|: s3 >= 0, s3 <= inf', z >= 0
map(z) -{ 1 }→ 0 :|: z = 0
map(z) -{ 3 + 2·xs }→ 1 + s1 + s2 :|: s2 >= 0, s2 <= inf, s1 >= 0, s1 <= 2 * x + 1 * (x * x), z = 1 + x + xs, xs >= 0, x >= 0
times(z, z') -{ 0 }→ z :|: z >= 0, z' = 1 + 0
times(z, z') -{ 0 }→ 0 :|: z >= 0, z' = 0
times(z, z') -{ 0 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
times(z, z') -{ 0 }→ 1 + z + s' :|: s' >= 0, s' <= 1 * ((1 + (z' - 2)) * z) + 1 * (1 + (z' - 2)) + 1 * z, z >= 0, z' - 2 >= 0

Function symbols to be analyzed:
Previous analysis results are:

times: runtime: O(1) [0], size: O(n2) [z + z·z' + z'] +Full: runtime: O(n1) [1 + z], size: O(n1) [z + z'] f: runtime: O(1) [1], size: O(n2) [2·z + z2] map: runtime: O(n1) [1 + 2·z], size: INF goal: runtime: O(n1) [2 + 2·z], size: INF

(45) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(46) BOUNDS(1, n^1)