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:

f(x, nil) → g(nil, x)
f(x, g(y, z)) → g(f(x, y), z)
++(x, nil) → x
++(x, g(y, z)) → g(++(x, y), z)
null(nil) → true
null(g(x, y)) → false
mem(nil, y) → false
mem(g(x, y), z) → or(=(y, z), mem(x, z))
mem(x, max(x)) → not(null(x))
max(g(g(nil, x), y)) → max'(x, y)
max(g(g(g(x, y), z), u)) → max'(max(g(g(x, y), z)), u)

Rewrite Strategy: INNERMOST

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

The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
mem(x, max(x)) → not(null(x))

(2) 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:

f(x, nil) → g(nil, x)
max(g(g(nil, x), y)) → max'(x, y)
max(g(g(g(x, y), z), u)) → max'(max(g(g(x, y), z)), u)
++(x, g(y, z)) → g(++(x, y), z)
null(nil) → true
mem(g(x, y), z) → or(=(y, z), mem(x, z))
mem(nil, y) → false
++(x, nil) → x
null(g(x, y)) → false
f(x, g(y, z)) → g(f(x, y), z)

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^1).


The TRS R consists of the following rules:

f(x, nil) → g(nil, x) [1]
max(g(g(nil, x), y)) → max'(x, y) [1]
max(g(g(g(x, y), z), u)) → max'(max(g(g(x, y), z)), u) [1]
++(x, g(y, z)) → g(++(x, y), z) [1]
null(nil) → true [1]
mem(g(x, y), z) → or(=(y, z), mem(x, z)) [1]
mem(nil, y) → false [1]
++(x, nil) → x [1]
null(g(x, y)) → false [1]
f(x, g(y, z)) → g(f(x, y), z) [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:

f(x, nil) → g(nil, x) [1]
max(g(g(nil, x), y)) → max'(x, y) [1]
max(g(g(g(x, y), z), u)) → max'(max(g(g(x, y), z)), u) [1]
++(x, g(y, z)) → g(++(x, y), z) [1]
null(nil) → true [1]
mem(g(x, y), z) → or(=(y, z), mem(x, z)) [1]
mem(nil, y) → false [1]
++(x, nil) → x [1]
null(g(x, y)) → false [1]
f(x, g(y, z)) → g(f(x, y), z) [1]

The TRS has the following type information:

f :: max':u → nil:g → nil:g nil :: nil:g g :: nil:g → max':u → nil:g max :: nil:g → max':u max' :: max':u → max':u → max':u u :: max':u ++ :: nil:g → nil:g → nil:g null :: nil:g → true:or:false true :: true:or:false mem :: nil:g → a → true:or:false or :: = → true:or:false → true:or:false = :: max':u → a → = false :: true:or:false

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:

f
max
++
null
mem

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:

const, const1

(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: f(x, nil) → g(nil, x) [1] max(g(g(nil, x), y)) → max'(x, y) [1] max(g(g(g(x, y), z), u)) → max'(max(g(g(x, y), z)), u) [1] ++(x, g(y, z)) → g(++(x, y), z) [1] null(nil) → true [1] mem(g(x, y), z) → or(=(y, z), mem(x, z)) [1] mem(nil, y) → false [1] ++(x, nil) → x [1] null(g(x, y)) → false [1] f(x, g(y, z)) → g(f(x, y), z) [1] The TRS has the following type information: f :: max':u → nil:g → nil:g nil :: nil:g g :: nil:g → max':u → nil:g max :: nil:g → max':u max' :: max':u → max':u → max':u u :: max':u ++ :: nil:g → nil:g → nil:g null :: nil:g → true:or:false true :: true:or:false mem :: nil:g → a → true:or:false or :: = → true:or:false → true:or:false = :: max':u → a → = false :: true:or:false const :: a const1 :: = 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: f(x, nil) → g(nil, x) [1] max(g(g(nil, x), y)) → max'(x, y) [1] max(g(g(g(x, y), z), u)) → max'(max(g(g(x, y), z)), u) [1] ++(x, g(y, z)) → g(++(x, y), z) [1] null(nil) → true [1] mem(g(x, y), z) → or(=(y, z), mem(x, z)) [1] mem(nil, y) → false [1] ++(x, nil) → x [1] null(g(x, y)) → false [1] f(x, g(y, z)) → g(f(x, y), z) [1] The TRS has the following type information: f :: max':u → nil:g → nil:g nil :: nil:g g :: nil:g → max':u → nil:g max :: nil:g → max':u max' :: max':u → max':u → max':u u :: max':u ++ :: nil:g → nil:g → nil:g null :: nil:g → true:or:false true :: true:or:false mem :: nil:g → a → true:or:false or :: = → true:or:false → true:or:false = :: max':u → a → = false :: true:or:false const :: a const1 :: = 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
u => 0
true => 1
false => 0
const => 0
const1 => 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

++(z', z'') -{ 1 }→ x :|: z'' = 0, z' = x, x >= 0
++(z', z'') -{ 1 }→ 1 + ++(x, y) + z :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + f(x, y) + z :|: z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + 0 + x :|: z'' = 0, z' = x, x >= 0
max(z') -{ 1 }→ 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
max(z') -{ 1 }→ 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
mem(z', z'') -{ 1 }→ 0 :|: z'' = y, y >= 0, z' = 0
mem(z', z'') -{ 1 }→ 1 + (1 + y + z) + mem(x, z) :|: z'' = z, z >= 0, z' = 1 + x + y, x >= 0, y >= 0
null(z') -{ 1 }→ 1 :|: z' = 0
null(z') -{ 1 }→ 0 :|: z' = 1 + x + y, x >= 0, y >= 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:

++(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
++(z', z'') -{ 1 }→ 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + 0 + z' :|: z'' = 0, z' >= 0
max(z') -{ 1 }→ 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
max(z') -{ 1 }→ 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
mem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
mem(z', z'') -{ 1 }→ 1 + (1 + y + z'') + mem(x, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
null(z') -{ 1 }→ 1 :|: z' = 0
null(z') -{ 1 }→ 0 :|: z' = 1 + x + y, x >= 0, y >= 0

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

Found the following analysis order by SCC decomposition:

{ mem } { max } { f } { null } { ++ }

(16) Obligation:

Complexity RNTS consisting of the following rules:

++(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
++(z', z'') -{ 1 }→ 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + 0 + z' :|: z'' = 0, z' >= 0
max(z') -{ 1 }→ 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
max(z') -{ 1 }→ 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
mem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
mem(z', z'') -{ 1 }→ 1 + (1 + y + z'') + mem(x, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
null(z') -{ 1 }→ 1 :|: z' = 0
null(z') -{ 1 }→ 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {mem}, {max}, {f}, {null}, {++}

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

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

++(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
++(z', z'') -{ 1 }→ 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + 0 + z' :|: z'' = 0, z' >= 0
max(z') -{ 1 }→ 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
max(z') -{ 1 }→ 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
mem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
mem(z', z'') -{ 1 }→ 1 + (1 + y + z'') + mem(x, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
null(z') -{ 1 }→ 1 :|: z' = 0
null(z') -{ 1 }→ 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {mem}, {max}, {f}, {null}, {++}
Previous analysis results are:

mem: runtime: ?, size: O(n2) [z' + z'·z'' + z'2]

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

Computed RUNTIME bound using CoFloCo for: mem
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:

++(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
++(z', z'') -{ 1 }→ 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + 0 + z' :|: z'' = 0, z' >= 0
max(z') -{ 1 }→ 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
max(z') -{ 1 }→ 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
mem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
mem(z', z'') -{ 1 }→ 1 + (1 + y + z'') + mem(x, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
null(z') -{ 1 }→ 1 :|: z' = 0
null(z') -{ 1 }→ 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {max}, {f}, {null}, {++}
Previous analysis results are:

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

(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:

++(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
++(z', z'') -{ 1 }→ 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + 0 + z' :|: z'' = 0, z' >= 0
max(z') -{ 1 }→ 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
max(z') -{ 1 }→ 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
mem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
mem(z', z'') -{ 2 + x }→ 1 + (1 + y + z'') + s :|: s >= 0, s <= 1 * (x * x) + 1 * (x * z'') + 1 * x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
null(z') -{ 1 }→ 1 :|: z' = 0
null(z') -{ 1 }→ 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {max}, {f}, {null}, {++}
Previous analysis results are:

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

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

++(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
++(z', z'') -{ 1 }→ 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + 0 + z' :|: z'' = 0, z' >= 0
max(z') -{ 1 }→ 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
max(z') -{ 1 }→ 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
mem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
mem(z', z'') -{ 2 + x }→ 1 + (1 + y + z'') + s :|: s >= 0, s <= 1 * (x * x) + 1 * (x * z'') + 1 * x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
null(z') -{ 1 }→ 1 :|: z' = 0
null(z') -{ 1 }→ 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {max}, {f}, {null}, {++}
Previous analysis results are:

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

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

Computed RUNTIME bound using KoAT for: max
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:

++(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
++(z', z'') -{ 1 }→ 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + 0 + z' :|: z'' = 0, z' >= 0
max(z') -{ 1 }→ 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
max(z') -{ 1 }→ 1 + max(1 + (1 + x + y) + z) + 0 :|: z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
mem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
mem(z', z'') -{ 2 + x }→ 1 + (1 + y + z'') + s :|: s >= 0, s <= 1 * (x * x) + 1 * (x * z'') + 1 * x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
null(z') -{ 1 }→ 1 :|: z' = 0
null(z') -{ 1 }→ 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {f}, {null}, {++}
Previous analysis results are:

mem: runtime: O(n1) [1 + z'], size: O(n2) [z' + z'·z'' + z'2] max: runtime: O(n1) [1 + z'], size: O(n1) [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:

++(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
++(z', z'') -{ 1 }→ 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + 0 + z' :|: z'' = 0, z' >= 0
max(z') -{ 4 + x + y + z }→ 1 + s' + 0 :|: s' >= 0, s' <= 1 * (1 + (1 + x + y) + z), z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
max(z') -{ 1 }→ 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
mem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
mem(z', z'') -{ 2 + x }→ 1 + (1 + y + z'') + s :|: s >= 0, s <= 1 * (x * x) + 1 * (x * z'') + 1 * x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
null(z') -{ 1 }→ 1 :|: z' = 0
null(z') -{ 1 }→ 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {f}, {null}, {++}
Previous analysis results are:

mem: runtime: O(n1) [1 + z'], size: O(n2) [z' + z'·z'' + z'2] max: runtime: O(n1) [1 + z'], size: O(n1) [z']

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

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

(30) Obligation:

Complexity RNTS consisting of the following rules:

++(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
++(z', z'') -{ 1 }→ 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + 0 + z' :|: z'' = 0, z' >= 0
max(z') -{ 4 + x + y + z }→ 1 + s' + 0 :|: s' >= 0, s' <= 1 * (1 + (1 + x + y) + z), z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
max(z') -{ 1 }→ 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
mem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
mem(z', z'') -{ 2 + x }→ 1 + (1 + y + z'') + s :|: s >= 0, s <= 1 * (x * x) + 1 * (x * z'') + 1 * x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
null(z') -{ 1 }→ 1 :|: z' = 0
null(z') -{ 1 }→ 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {f}, {null}, {++}
Previous analysis results are:

mem: runtime: O(n1) [1 + z'], size: O(n2) [z' + z'·z'' + z'2] max: runtime: O(n1) [1 + z'], size: O(n1) [z'] f: runtime: ?, size: O(n1) [1 + z' + z'']

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

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

(32) Obligation:

Complexity RNTS consisting of the following rules:

++(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
++(z', z'') -{ 1 }→ 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + f(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + 0 + z' :|: z'' = 0, z' >= 0
max(z') -{ 4 + x + y + z }→ 1 + s' + 0 :|: s' >= 0, s' <= 1 * (1 + (1 + x + y) + z), z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
max(z') -{ 1 }→ 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
mem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
mem(z', z'') -{ 2 + x }→ 1 + (1 + y + z'') + s :|: s >= 0, s <= 1 * (x * x) + 1 * (x * z'') + 1 * x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
null(z') -{ 1 }→ 1 :|: z' = 0
null(z') -{ 1 }→ 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {null}, {++}
Previous analysis results are:

mem: runtime: O(n1) [1 + z'], size: O(n2) [z' + z'·z'' + z'2] max: runtime: O(n1) [1 + z'], size: O(n1) [z'] f: runtime: O(n1) [1 + z''], size: O(n1) [1 + z' + z'']

(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:

++(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
++(z', z'') -{ 1 }→ 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 2 + y }→ 1 + s'' + z :|: s'' >= 0, s'' <= 1 * z' + 1 * y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + 0 + z' :|: z'' = 0, z' >= 0
max(z') -{ 4 + x + y + z }→ 1 + s' + 0 :|: s' >= 0, s' <= 1 * (1 + (1 + x + y) + z), z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
max(z') -{ 1 }→ 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
mem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
mem(z', z'') -{ 2 + x }→ 1 + (1 + y + z'') + s :|: s >= 0, s <= 1 * (x * x) + 1 * (x * z'') + 1 * x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
null(z') -{ 1 }→ 1 :|: z' = 0
null(z') -{ 1 }→ 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {null}, {++}
Previous analysis results are:

mem: runtime: O(n1) [1 + z'], size: O(n2) [z' + z'·z'' + z'2] max: runtime: O(n1) [1 + z'], size: O(n1) [z'] f: runtime: O(n1) [1 + z''], size: O(n1) [1 + z' + z'']

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

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

(36) Obligation:

Complexity RNTS consisting of the following rules:

++(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
++(z', z'') -{ 1 }→ 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 2 + y }→ 1 + s'' + z :|: s'' >= 0, s'' <= 1 * z' + 1 * y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + 0 + z' :|: z'' = 0, z' >= 0
max(z') -{ 4 + x + y + z }→ 1 + s' + 0 :|: s' >= 0, s' <= 1 * (1 + (1 + x + y) + z), z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
max(z') -{ 1 }→ 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
mem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
mem(z', z'') -{ 2 + x }→ 1 + (1 + y + z'') + s :|: s >= 0, s <= 1 * (x * x) + 1 * (x * z'') + 1 * x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
null(z') -{ 1 }→ 1 :|: z' = 0
null(z') -{ 1 }→ 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {null}, {++}
Previous analysis results are:

mem: runtime: O(n1) [1 + z'], size: O(n2) [z' + z'·z'' + z'2] max: runtime: O(n1) [1 + z'], size: O(n1) [z'] f: runtime: O(n1) [1 + z''], size: O(n1) [1 + z' + z''] null: runtime: ?, size: O(1) [1]

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

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

(38) Obligation:

Complexity RNTS consisting of the following rules:

++(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
++(z', z'') -{ 1 }→ 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 2 + y }→ 1 + s'' + z :|: s'' >= 0, s'' <= 1 * z' + 1 * y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + 0 + z' :|: z'' = 0, z' >= 0
max(z') -{ 4 + x + y + z }→ 1 + s' + 0 :|: s' >= 0, s' <= 1 * (1 + (1 + x + y) + z), z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
max(z') -{ 1 }→ 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
mem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
mem(z', z'') -{ 2 + x }→ 1 + (1 + y + z'') + s :|: s >= 0, s <= 1 * (x * x) + 1 * (x * z'') + 1 * x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
null(z') -{ 1 }→ 1 :|: z' = 0
null(z') -{ 1 }→ 0 :|: z' = 1 + x + y, x >= 0, y >= 0

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

mem: runtime: O(n1) [1 + z'], size: O(n2) [z' + z'·z'' + z'2] max: runtime: O(n1) [1 + z'], size: O(n1) [z'] f: runtime: O(n1) [1 + z''], size: O(n1) [1 + z' + z''] null: runtime: O(1) [1], size: O(1) [1]

(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:

++(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
++(z', z'') -{ 1 }→ 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 2 + y }→ 1 + s'' + z :|: s'' >= 0, s'' <= 1 * z' + 1 * y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + 0 + z' :|: z'' = 0, z' >= 0
max(z') -{ 4 + x + y + z }→ 1 + s' + 0 :|: s' >= 0, s' <= 1 * (1 + (1 + x + y) + z), z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
max(z') -{ 1 }→ 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
mem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
mem(z', z'') -{ 2 + x }→ 1 + (1 + y + z'') + s :|: s >= 0, s <= 1 * (x * x) + 1 * (x * z'') + 1 * x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
null(z') -{ 1 }→ 1 :|: z' = 0
null(z') -{ 1 }→ 0 :|: z' = 1 + x + y, x >= 0, y >= 0

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

mem: runtime: O(n1) [1 + z'], size: O(n2) [z' + z'·z'' + z'2] max: runtime: O(n1) [1 + z'], size: O(n1) [z'] f: runtime: O(n1) [1 + z''], size: O(n1) [1 + z' + z''] null: runtime: O(1) [1], size: O(1) [1]

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

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

(42) Obligation:

Complexity RNTS consisting of the following rules:

++(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
++(z', z'') -{ 1 }→ 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 2 + y }→ 1 + s'' + z :|: s'' >= 0, s'' <= 1 * z' + 1 * y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + 0 + z' :|: z'' = 0, z' >= 0
max(z') -{ 4 + x + y + z }→ 1 + s' + 0 :|: s' >= 0, s' <= 1 * (1 + (1 + x + y) + z), z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
max(z') -{ 1 }→ 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
mem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
mem(z', z'') -{ 2 + x }→ 1 + (1 + y + z'') + s :|: s >= 0, s <= 1 * (x * x) + 1 * (x * z'') + 1 * x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
null(z') -{ 1 }→ 1 :|: z' = 0
null(z') -{ 1 }→ 0 :|: z' = 1 + x + y, x >= 0, y >= 0

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

mem: runtime: O(n1) [1 + z'], size: O(n2) [z' + z'·z'' + z'2] max: runtime: O(n1) [1 + z'], size: O(n1) [z'] f: runtime: O(n1) [1 + z''], size: O(n1) [1 + z' + z''] null: runtime: O(1) [1], size: O(1) [1] ++: runtime: ?, size: O(n1) [z' + z'']

(43) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(44) Obligation:

Complexity RNTS consisting of the following rules:

++(z', z'') -{ 1 }→ z' :|: z'' = 0, z' >= 0
++(z', z'') -{ 1 }→ 1 + ++(z', y) + z :|: z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 2 + y }→ 1 + s'' + z :|: s'' >= 0, s'' <= 1 * z' + 1 * y + 1, z >= 0, z' >= 0, y >= 0, z'' = 1 + y + z
f(z', z'') -{ 1 }→ 1 + 0 + z' :|: z'' = 0, z' >= 0
max(z') -{ 4 + x + y + z }→ 1 + s' + 0 :|: s' >= 0, s' <= 1 * (1 + (1 + x + y) + z), z' = 1 + (1 + (1 + x + y) + z) + 0, z >= 0, x >= 0, y >= 0
max(z') -{ 1 }→ 1 + x + y :|: z' = 1 + (1 + 0 + x) + y, x >= 0, y >= 0
mem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
mem(z', z'') -{ 2 + x }→ 1 + (1 + y + z'') + s :|: s >= 0, s <= 1 * (x * x) + 1 * (x * z'') + 1 * x, z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
null(z') -{ 1 }→ 1 :|: z' = 0
null(z') -{ 1 }→ 0 :|: z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed:
Previous analysis results are:

mem: runtime: O(n1) [1 + z'], size: O(n2) [z' + z'·z'' + z'2] max: runtime: O(n1) [1 + z'], size: O(n1) [z'] f: runtime: O(n1) [1 + z''], size: O(n1) [1 + z' + z''] null: runtime: O(1) [1], size: O(1) [1] ++: runtime: O(n1) [1 + z''], size: O(n1) [z' + z'']

(45) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(46) BOUNDS(1, n^1)