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:

norm(nil) → 0
norm(g(x, y)) → s(norm(x))
f(x, nil) → g(nil, x)
f(x, g(y, z)) → g(f(x, y), z)
rem(nil, y) → nil
rem(g(x, y), 0) → g(x, y)
rem(g(x, y), s(z)) → rem(x, z)

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:

norm(nil) → 0 [1]
norm(g(x, y)) → s(norm(x)) [1]
f(x, nil) → g(nil, x) [1]
f(x, g(y, z)) → g(f(x, y), z) [1]
rem(nil, y) → nil [1]
rem(g(x, y), 0) → g(x, y) [1]
rem(g(x, y), s(z)) → rem(x, z) [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:

norm(nil) → 0 [1]
norm(g(x, y)) → s(norm(x)) [1]
f(x, nil) → g(nil, x) [1]
f(x, g(y, z)) → g(f(x, y), z) [1]
rem(nil, y) → nil [1]
rem(g(x, y), 0) → g(x, y) [1]
rem(g(x, y), s(z)) → rem(x, z) [1]

The TRS has the following type information:

norm :: nil:g → 0:s nil :: nil:g 0 :: 0:s g :: nil:g → a → nil:g s :: 0:s → 0:s f :: a → nil:g → nil:g rem :: nil:g → 0:s → nil:g

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:

norm
f
rem

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:

const

(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: norm(nil) → 0 [1] norm(g(x, y)) → s(norm(x)) [1] f(x, nil) → g(nil, x) [1] f(x, g(y, z)) → g(f(x, y), z) [1] rem(nil, y) → nil [1] rem(g(x, y), 0) → g(x, y) [1] rem(g(x, y), s(z)) → rem(x, z) [1] The TRS has the following type information: norm :: nil:g → 0:s nil :: nil:g 0 :: 0:s g :: nil:g → a → nil:g s :: 0:s → 0:s f :: a → nil:g → nil:g rem :: nil:g → 0:s → nil:g const :: a 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: norm(nil) → 0 [1] norm(g(x, y)) → s(norm(x)) [1] f(x, nil) → g(nil, x) [1] f(x, g(y, z)) → g(f(x, y), z) [1] rem(nil, y) → nil [1] rem(g(x, y), 0) → g(x, y) [1] rem(g(x, y), s(z)) → rem(x, z) [1] The TRS has the following type information: norm :: nil:g → 0:s nil :: nil:g 0 :: 0:s g :: nil:g → a → nil:g s :: 0:s → 0:s f :: a → nil:g → nil:g rem :: nil:g → 0:s → nil:g const :: a 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
0 => 0
const => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

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
norm(z') -{ 1 }→ 0 :|: z' = 0
norm(z') -{ 1 }→ 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
rem(z', z'') -{ 1 }→ rem(x, z) :|: z >= 0, z' = 1 + x + y, x >= 0, y >= 0, z'' = 1 + z
rem(z', z'') -{ 1 }→ 0 :|: z'' = y, y >= 0, z' = 0
rem(z', z'') -{ 1 }→ 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 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:

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
norm(z') -{ 1 }→ 0 :|: z' = 0
norm(z') -{ 1 }→ 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
rem(z', z'') -{ 1 }→ rem(x, z'' - 1) :|: z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
rem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
rem(z', z'') -{ 1 }→ 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

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

Found the following analysis order by SCC decomposition:

{ rem } { norm } { f }

(14) Obligation:

Complexity RNTS consisting of the following rules:

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
norm(z') -{ 1 }→ 0 :|: z' = 0
norm(z') -{ 1 }→ 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
rem(z', z'') -{ 1 }→ rem(x, z'' - 1) :|: z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
rem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
rem(z', z'') -{ 1 }→ 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {rem}, {norm}, {f}

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

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

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
norm(z') -{ 1 }→ 0 :|: z' = 0
norm(z') -{ 1 }→ 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
rem(z', z'') -{ 1 }→ rem(x, z'' - 1) :|: z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
rem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
rem(z', z'') -{ 1 }→ 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {rem}, {norm}, {f}
Previous analysis results are:

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

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

Computed RUNTIME bound using CoFloCo for: rem
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:

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
norm(z') -{ 1 }→ 0 :|: z' = 0
norm(z') -{ 1 }→ 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
rem(z', z'') -{ 1 }→ rem(x, z'' - 1) :|: z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
rem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
rem(z', z'') -{ 1 }→ 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

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

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

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
norm(z') -{ 1 }→ 0 :|: z' = 0
norm(z') -{ 1 }→ 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
rem(z', z'') -{ 1 + z'' }→ s :|: s >= 0, s <= 1 * x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
rem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
rem(z', z'') -{ 1 }→ 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

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

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

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

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

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
norm(z') -{ 1 }→ 0 :|: z' = 0
norm(z') -{ 1 }→ 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
rem(z', z'') -{ 1 + z'' }→ s :|: s >= 0, s <= 1 * x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
rem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
rem(z', z'') -{ 1 }→ 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

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

rem: runtime: O(n1) [1 + z''], size: O(n1) [z'] norm: runtime: ?, size: O(n1) [z']

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

Computed RUNTIME bound using CoFloCo for: norm
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:

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
norm(z') -{ 1 }→ 0 :|: z' = 0
norm(z') -{ 1 }→ 1 + norm(x) :|: z' = 1 + x + y, x >= 0, y >= 0
rem(z', z'') -{ 1 + z'' }→ s :|: s >= 0, s <= 1 * x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
rem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
rem(z', z'') -{ 1 }→ 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

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

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

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
norm(z') -{ 1 }→ 0 :|: z' = 0
norm(z') -{ 2 + x }→ 1 + s' :|: s' >= 0, s' <= 1 * x, z' = 1 + x + y, x >= 0, y >= 0
rem(z', z'') -{ 1 + z'' }→ s :|: s >= 0, s <= 1 * x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
rem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
rem(z', z'') -{ 1 }→ 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

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

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

(27) 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''

(28) Obligation:

Complexity RNTS consisting of the following rules:

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
norm(z') -{ 1 }→ 0 :|: z' = 0
norm(z') -{ 2 + x }→ 1 + s' :|: s' >= 0, s' <= 1 * x, z' = 1 + x + y, x >= 0, y >= 0
rem(z', z'') -{ 1 + z'' }→ s :|: s >= 0, s <= 1 * x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
rem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
rem(z', z'') -{ 1 }→ 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

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

rem: runtime: O(n1) [1 + z''], size: O(n1) [z'] norm: runtime: O(n1) [1 + z'], size: O(n1) [z'] f: runtime: ?, size: O(n1) [1 + z' + z'']

(29) 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''

(30) Obligation:

Complexity RNTS consisting of the following rules:

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
norm(z') -{ 1 }→ 0 :|: z' = 0
norm(z') -{ 2 + x }→ 1 + s' :|: s' >= 0, s' <= 1 * x, z' = 1 + x + y, x >= 0, y >= 0
rem(z', z'') -{ 1 + z'' }→ s :|: s >= 0, s <= 1 * x, z'' - 1 >= 0, z' = 1 + x + y, x >= 0, y >= 0
rem(z', z'') -{ 1 }→ 0 :|: z'' >= 0, z' = 0
rem(z', z'') -{ 1 }→ 1 + x + y :|: z'' = 0, z' = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed:
Previous analysis results are:

rem: runtime: O(n1) [1 + z''], size: O(n1) [z'] norm: runtime: O(n1) [1 + z'], size: O(n1) [z'] f: runtime: O(n1) [1 + z''], size: O(n1) [1 + z' + z'']

(31) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(32) BOUNDS(1, n^1)