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) → 0
-(x, 0) → x
-(x, s(y)) → if(greater(x, s(y)), s(-(x, p(s(y)))), 0)
p(0) → 0
p(s(x)) → x

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) → 0 [1]
-(x, 0) → x [1]
-(x, s(y)) → if(greater(x, s(y)), s(-(x, p(s(y)))), 0) [1]
p(0) → 0 [1]
p(s(x)) → x [1]

Rewrite Strategy: INNERMOST

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

Renamed defined symbols to avoid conflicts with arithmetic symbols:

- => 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:

minus(0, y) → 0 [1]
minus(x, 0) → x [1]
minus(x, s(y)) → if(greater(x, s(y)), s(minus(x, p(s(y)))), 0) [1]
p(0) → 0 [1]
p(s(x)) → x [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:

minus(0, y) → 0 [1]
minus(x, 0) → x [1]
minus(x, s(y)) → if(greater(x, s(y)), s(minus(x, p(s(y)))), 0) [1]
p(0) → 0 [1]
p(s(x)) → x [1]

The TRS has the following type information:

minus :: 0:s:if → 0:s:if → 0:s:if 0 :: 0:s:if s :: 0:s:if → 0:s:if if :: greater → 0:s:if → 0:s:if → 0:s:if greater :: 0:s:if → 0:s:if → greater p :: 0:s:if → 0:s:if

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:

p

Due to the following rules being added:

p(v0) → 0 [0]

And the following fresh constants:

const

(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: minus(0, y) → 0 [1] minus(x, 0) → x [1] minus(x, s(y)) → if(greater(x, s(y)), s(minus(x, p(s(y)))), 0) [1] p(0) → 0 [1] p(s(x)) → x [1] p(v0) → 0 [0] The TRS has the following type information: minus :: 0:s:if → 0:s:if → 0:s:if 0 :: 0:s:if s :: 0:s:if → 0:s:if if :: greater → 0:s:if → 0:s:if → 0:s:if greater :: 0:s:if → 0:s:if → greater p :: 0:s:if → 0:s:if const :: greater 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: minus(0, y) → 0 [1] minus(x, 0) → x [1] minus(x, s(y)) → if(greater(x, s(y)), s(minus(x, y)), 0) [2] minus(x, s(y)) → if(greater(x, s(y)), s(minus(x, 0)), 0) [1] p(0) → 0 [1] p(s(x)) → x [1] p(v0) → 0 [0] The TRS has the following type information: minus :: 0:s:if → 0:s:if → 0:s:if 0 :: 0:s:if s :: 0:s:if → 0:s:if if :: greater → 0:s:if → 0:s:if → 0:s:if greater :: 0:s:if → 0:s:if → greater p :: 0:s:if → 0:s:if const :: greater 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
const => 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 }→ 0 :|: y >= 0, z = 0, z' = y
minus(z, z') -{ 2 }→ 1 + (1 + x + (1 + y)) + (1 + minus(x, y)) + 0 :|: z' = 1 + y, x >= 0, y >= 0, z = x
minus(z, z') -{ 1 }→ 1 + (1 + x + (1 + y)) + (1 + minus(x, 0)) + 0 :|: z' = 1 + y, x >= 0, y >= 0, z = x
p(z) -{ 1 }→ x :|: x >= 0, z = 1 + x
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0

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

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

Found the following analysis order by SCC decomposition:

{ minus } { p }

(16) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

Computed SIZE bound using PUBS for: minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: z + z·z' + 3·z' + z'²

(18) Obligation:

Complexity RNTS consisting of the following rules:

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

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

minus: runtime: ?, size: O(n2) [z + z·z' + 3·z' + z'2]

(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 + 2·z'

(20) Obligation:

Complexity RNTS consisting of the following rules:

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

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

minus: runtime: O(n1) [1 + 2·z'], size: O(n2) [z + z·z' + 3·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:

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

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

minus: runtime: O(n1) [1 + 2·z'], size: O(n2) [z + z·z' + 3·z' + z'2]

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

Computed SIZE bound using KoAT for: p
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:

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

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

minus: runtime: O(n1) [1 + 2·z'], size: O(n2) [z + z·z' + 3·z' + z'2] p: runtime: ?, size: O(n1) [z]

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

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

(26) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed:
Previous analysis results are:

minus: runtime: O(n1) [1 + 2·z'], size: O(n2) [z + z·z' + 3·z' + z'2] p: runtime: O(1) [1], size: O(n1) [z]

(27) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(28) BOUNDS(1, n^1)