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:

+(0, y) → y
+(s(x), y) → s(+(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^1).


The TRS R consists of the following rules:

+(0, y) → y [1]
+(s(x), y) → s(+(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:

+ => plus

- => minus

(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:

plus(0, y) → y [1]
plus(s(x), y) → s(plus(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:

plus(0, y) → y [1]
plus(s(x), y) → s(plus(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:

plus :: 0:s → 0:s → 0:s 0 :: 0:s 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:

plus
minus

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

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: plus(0, y) → y [1] plus(s(x), y) → s(plus(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: plus :: 0:s → 0:s → 0:s 0 :: 0:s 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: plus(0, y) → y [1] plus(s(x), y) → s(plus(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: plus :: 0:s → 0:s → 0:s 0 :: 0:s 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:

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

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
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

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

Found the following analysis order by SCC decomposition:

{ minus } { plus }

(16) Obligation:

Complexity RNTS consisting of the following rules:

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
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {minus}, {plus}

(17) 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

(18) Obligation:

Complexity RNTS consisting of the following rules:

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
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

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

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

(19) 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'

(20) Obligation:

Complexity RNTS consisting of the following rules:

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
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

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

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

minus(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 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
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

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

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

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

Computed SIZE bound using CoFloCo for: plus
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:

minus(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 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
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

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

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

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

Computed RUNTIME bound using CoFloCo for: plus
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:

minus(z, z') -{ 1 + z' }→ s :|: s >= 0, s <= 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
plus(z, z') -{ 1 }→ z' :|: z' >= 0, z = 0
plus(z, z') -{ 1 }→ 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed:
Previous analysis results are:

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

(27) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(28) BOUNDS(1, n^1)