The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

exp(x, 0) → s(0)
exp(x, s(y)) → *(x, exp(x, y))
*(0, y) → 0
*(s(x), y) → +(y, *(x, y))
-(0, y) → 0
-(x, 0) → x
-(s(x), s(y)) → -(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^2).


The TRS R consists of the following rules:

exp(x, 0) → s(0) [1]
exp(x, s(y)) → *(x, exp(x, y)) [1]
*(0, y) → 0 [1]
*(s(x), y) → +(y, *(x, y)) [1]
-(0, y) → 0 [1]
-(x, 0) → x [1]
-(s(x), s(y)) → -(x, y) [1]

Rewrite Strategy: INNERMOST

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

Renamed defined symbols to avoid conflicts with arithmetic symbols:

* => times

- => minus

(4) Obligation:

The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

exp(x, 0) → s(0) [1]
exp(x, s(y)) → times(x, exp(x, y)) [1]
times(0, y) → 0 [1]
times(s(x), y) → +(y, times(x, y)) [1]
minus(0, y) → 0 [1]
minus(x, 0) → x [1]
minus(s(x), s(y)) → minus(x, y) [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:

exp(x, 0) → s(0) [1]
exp(x, s(y)) → times(x, exp(x, y)) [1]
times(0, y) → 0 [1]
times(s(x), y) → +(y, times(x, y)) [1]
minus(0, y) → 0 [1]
minus(x, 0) → x [1]
minus(s(x), s(y)) → minus(x, y) [1]

The TRS has the following type information:

exp :: 0:s:+ → 0:s:+ → 0:s:+ 0 :: 0:s:+ s :: 0:s:+ → 0:s:+ times :: 0:s:+ → 0:s:+ → 0:s:+ + :: 0:s:+ → 0:s:+ → 0:s:+ minus :: 0:s:+ → 0:s:+ → 0:s:+

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:

minus

(c) The following functions are completely defined:

exp
times

Due to the following rules being added:

exp(v0, v1) → 0 [0]
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: exp(x, 0) → s(0) [1] exp(x, s(y)) → times(x, exp(x, y)) [1] times(0, y) → 0 [1] times(s(x), y) → +(y, times(x, y)) [1] minus(0, y) → 0 [1] minus(x, 0) → x [1] minus(s(x), s(y)) → minus(x, y) [1] exp(v0, v1) → 0 [0] times(v0, v1) → 0 [0] The TRS has the following type information: exp :: 0:s:+ → 0:s:+ → 0:s:+ 0 :: 0:s:+ s :: 0:s:+ → 0:s:+ times :: 0:s:+ → 0:s:+ → 0:s:+ + :: 0:s:+ → 0:s:+ → 0:s:+ minus :: 0:s:+ → 0:s:+ → 0:s:+ 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: exp(x, 0) → s(0) [1] exp(x, s(0)) → times(x, s(0)) [2] exp(x, s(s(y'))) → times(x, times(x, exp(x, y'))) [2] exp(x, s(y)) → times(x, 0) [1] times(0, y) → 0 [1] times(s(x), y) → +(y, times(x, y)) [1] minus(0, y) → 0 [1] minus(x, 0) → x [1] minus(s(x), s(y)) → minus(x, y) [1] exp(v0, v1) → 0 [0] times(v0, v1) → 0 [0] The TRS has the following type information: exp :: 0:s:+ → 0:s:+ → 0:s:+ 0 :: 0:s:+ s :: 0:s:+ → 0:s:+ times :: 0:s:+ → 0:s:+ → 0:s:+ + :: 0:s:+ → 0:s:+ → 0:s:+ minus :: 0:s:+ → 0:s:+ → 0:s:+ 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:

0 => 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

exp(z, z') -{ 2 }→ times(x, times(x, exp(x, y'))) :|: z' = 1 + (1 + y'), x >= 0, y' >= 0, z = x
exp(z, z') -{ 1 }→ times(x, 0) :|: z' = 1 + y, x >= 0, y >= 0, z = x
exp(z, z') -{ 2 }→ times(x, 1 + 0) :|: x >= 0, z' = 1 + 0, z = x
exp(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
exp(z, z') -{ 1 }→ 1 + 0 :|: x >= 0, z = x, z' = 0
minus(z, z') -{ 1 }→ x :|: x >= 0, z = x, z' = 0
minus(z, z') -{ 1 }→ minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
minus(z, z') -{ 1 }→ 0 :|: y >= 0, z = 0, z' = y
times(z, z') -{ 1 }→ 0 :|: y >= 0, z = 0, z' = y
times(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
times(z, z') -{ 1 }→ 1 + y + times(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y

(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:

exp(z, z') -{ 2 }→ times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
exp(z, z') -{ 1 }→ times(z, 0) :|: z >= 0, z' - 1 >= 0
exp(z, z') -{ 2 }→ times(z, 1 + 0) :|: z >= 0, z' = 1 + 0
exp(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
exp(z, z') -{ 1 }→ 1 + 0 :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
times(z, z') -{ 1 }→ 1 + z' + times(z - 1, z') :|: z - 1 >= 0, z' >= 0

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

Found the following analysis order by SCC decomposition:

{ times } { minus } { exp }

(16) Obligation:

Complexity RNTS consisting of the following rules:

exp(z, z') -{ 2 }→ times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
exp(z, z') -{ 1 }→ times(z, 0) :|: z >= 0, z' - 1 >= 0
exp(z, z') -{ 2 }→ times(z, 1 + 0) :|: z >= 0, z' = 1 + 0
exp(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
exp(z, z') -{ 1 }→ 1 + 0 :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
times(z, z') -{ 1 }→ 1 + z' + times(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {times}, {minus}, {exp}

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

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

(18) Obligation:

Complexity RNTS consisting of the following rules:

exp(z, z') -{ 2 }→ times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
exp(z, z') -{ 1 }→ times(z, 0) :|: z >= 0, z' - 1 >= 0
exp(z, z') -{ 2 }→ times(z, 1 + 0) :|: z >= 0, z' = 1 + 0
exp(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
exp(z, z') -{ 1 }→ 1 + 0 :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
times(z, z') -{ 1 }→ 1 + z' + times(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {times}, {minus}, {exp}
Previous analysis results are:

times: runtime: ?, size: O(n2) [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(n¹) with polynomial bound: 1 + z

(20) Obligation:

Complexity RNTS consisting of the following rules:

exp(z, z') -{ 2 }→ times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
exp(z, z') -{ 1 }→ times(z, 0) :|: z >= 0, z' - 1 >= 0
exp(z, z') -{ 2 }→ times(z, 1 + 0) :|: z >= 0, z' = 1 + 0
exp(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
exp(z, z') -{ 1 }→ 1 + 0 :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
times(z, z') -{ 1 }→ 1 + z' + times(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {minus}, {exp}
Previous analysis results are:

times: runtime: O(n1) [1 + z], size: O(n2) [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:

exp(z, z') -{ 3 + z }→ s :|: s >= 0, s <= 1 * (z * (1 + 0)) + 1 * z, z >= 0, z' = 1 + 0
exp(z, z') -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * (z * 0) + 1 * z, z >= 0, z' - 1 >= 0
exp(z, z') -{ 2 }→ times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
exp(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
exp(z, z') -{ 1 }→ 1 + 0 :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
times(z, z') -{ 1 + z }→ 1 + z' + s'' :|: s'' >= 0, s'' <= 1 * ((z - 1) * z') + 1 * (z - 1), z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {minus}, {exp}
Previous analysis results are:

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

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

Computed SIZE bound using KoAT for: minus
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:

exp(z, z') -{ 3 + z }→ s :|: s >= 0, s <= 1 * (z * (1 + 0)) + 1 * z, z >= 0, z' = 1 + 0
exp(z, z') -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * (z * 0) + 1 * z, z >= 0, z' - 1 >= 0
exp(z, z') -{ 2 }→ times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
exp(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
exp(z, z') -{ 1 }→ 1 + 0 :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
times(z, z') -{ 1 + z }→ 1 + z' + s'' :|: s'' >= 0, s'' <= 1 * ((z - 1) * z') + 1 * (z - 1), z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {minus}, {exp}
Previous analysis results are:

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

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

Computed RUNTIME bound using PUBS for: minus
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:

exp(z, z') -{ 3 + z }→ s :|: s >= 0, s <= 1 * (z * (1 + 0)) + 1 * z, z >= 0, z' = 1 + 0
exp(z, z') -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * (z * 0) + 1 * z, z >= 0, z' - 1 >= 0
exp(z, z') -{ 2 }→ times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
exp(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
exp(z, z') -{ 1 }→ 1 + 0 :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
times(z, z') -{ 1 + z }→ 1 + z' + s'' :|: s'' >= 0, s'' <= 1 * ((z - 1) * z') + 1 * (z - 1), z - 1 >= 0, z' >= 0

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

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

exp(z, z') -{ 3 + z }→ s :|: s >= 0, s <= 1 * (z * (1 + 0)) + 1 * z, z >= 0, z' = 1 + 0
exp(z, z') -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * (z * 0) + 1 * z, z >= 0, z' - 1 >= 0
exp(z, z') -{ 2 }→ times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
exp(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
exp(z, z') -{ 1 }→ 1 + 0 :|: z >= 0, z' = 0
minus(z, z') -{ 1 + z' }→ s1 :|: s1 >= 0, s1 <= 1 * (z - 1), z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
times(z, z') -{ 1 + z }→ 1 + z' + s'' :|: s'' >= 0, s'' <= 1 * ((z - 1) * z') + 1 * (z - 1), z - 1 >= 0, z' >= 0

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

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

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

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

(30) Obligation:

Complexity RNTS consisting of the following rules:

exp(z, z') -{ 3 + z }→ s :|: s >= 0, s <= 1 * (z * (1 + 0)) + 1 * z, z >= 0, z' = 1 + 0
exp(z, z') -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * (z * 0) + 1 * z, z >= 0, z' - 1 >= 0
exp(z, z') -{ 2 }→ times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
exp(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
exp(z, z') -{ 1 }→ 1 + 0 :|: z >= 0, z' = 0
minus(z, z') -{ 1 + z' }→ s1 :|: s1 >= 0, s1 <= 1 * (z - 1), z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
times(z, z') -{ 1 + z }→ 1 + z' + s'' :|: s'' >= 0, s'' <= 1 * ((z - 1) * z') + 1 * (z - 1), z - 1 >= 0, z' >= 0

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

times: runtime: O(n1) [1 + z], size: O(n2) [z + z·z'] minus: runtime: O(n1) [1 + z'], size: O(n1) [z] exp: runtime: ?, size: INF

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

Computed RUNTIME bound using KoAT for: exp
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 6 + 2·z + 2·z·z' + 4·z'

(32) Obligation:

Complexity RNTS consisting of the following rules:

exp(z, z') -{ 3 + z }→ s :|: s >= 0, s <= 1 * (z * (1 + 0)) + 1 * z, z >= 0, z' = 1 + 0
exp(z, z') -{ 2 + z }→ s' :|: s' >= 0, s' <= 1 * (z * 0) + 1 * z, z >= 0, z' - 1 >= 0
exp(z, z') -{ 2 }→ times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
exp(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
exp(z, z') -{ 1 }→ 1 + 0 :|: z >= 0, z' = 0
minus(z, z') -{ 1 + z' }→ s1 :|: s1 >= 0, s1 <= 1 * (z - 1), z - 1 >= 0, z' - 1 >= 0
minus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
minus(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
times(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
times(z, z') -{ 1 + z }→ 1 + z' + s'' :|: s'' >= 0, s'' <= 1 * ((z - 1) * z') + 1 * (z - 1), z - 1 >= 0, z' >= 0

Function symbols to be analyzed:
Previous analysis results are:

times: runtime: O(n1) [1 + z], size: O(n2) [z + z·z'] minus: runtime: O(n1) [1 + z'], size: O(n1) [z] exp: runtime: O(n2) [6 + 2·z + 2·z·z' + 4·z'], size: INF

(33) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(34) BOUNDS(1, n^2)