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:

p(0) → 0
p(s(x)) → x
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
minus(x, y) → if(le(x, y), x, y)
if(true, x, y) → 0
if(false, x, y) → s(minus(p(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:

p(0) → 0 [1]
p(s(x)) → x [1]
le(0, y) → true [1]
le(s(x), 0) → false [1]
le(s(x), s(y)) → le(x, y) [1]
minus(x, y) → if(le(x, y), x, y) [1]
if(true, x, y) → 0 [1]
if(false, x, y) → s(minus(p(x), y)) [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:

p(0) → 0 [1]
p(s(x)) → x [1]
le(0, y) → true [1]
le(s(x), 0) → false [1]
le(s(x), s(y)) → le(x, y) [1]
minus(x, y) → if(le(x, y), x, y) [1]
if(true, x, y) → 0 [1]
if(false, x, y) → s(minus(p(x), y)) [1]

The TRS has the following type information:

p :: 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s le :: 0:s → 0:s → true:false true :: true:false false :: true:false minus :: 0:s → 0:s → 0:s if :: true:false → 0:s → 0:s → 0:s

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:

minus
if

(c) The following functions are completely defined:

p
le

Due to the following rules being added:
none

And the following fresh constants: none

(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: p(0) → 0 [1] p(s(x)) → x [1] le(0, y) → true [1] le(s(x), 0) → false [1] le(s(x), s(y)) → le(x, y) [1] minus(x, y) → if(le(x, y), x, y) [1] if(true, x, y) → 0 [1] if(false, x, y) → s(minus(p(x), y)) [1] The TRS has the following type information: p :: 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s le :: 0:s → 0:s → true:false true :: true:false false :: true:false minus :: 0:s → 0:s → 0:s if :: true:false → 0:s → 0:s → 0:s 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: p(0) → 0 [1] p(s(x)) → x [1] le(0, y) → true [1] le(s(x), 0) → false [1] le(s(x), s(y)) → le(x, y) [1] minus(0, y) → if(true, 0, y) [2] minus(s(x'), 0) → if(false, s(x'), 0) [2] minus(s(x''), s(y')) → if(le(x'', y'), s(x''), s(y')) [2] if(true, x, y) → 0 [1] if(false, 0, y) → s(minus(0, y)) [2] if(false, s(x1), y) → s(minus(x1, y)) [2] The TRS has the following type information: p :: 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s le :: 0:s → 0:s → true:false true :: true:false false :: true:false minus :: 0:s → 0:s → 0:s if :: true:false → 0:s → 0:s → 0:s 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:

0 => 0
true => 1
false => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

if(z, z', z'') -{ 1 }→ 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
if(z, z', z'') -{ 2 }→ 1 + minus(x1, y) :|: x1 >= 0, z'' = y, y >= 0, z' = 1 + x1, z = 0
if(z, z', z'') -{ 2 }→ 1 + minus(0, y) :|: z'' = y, y >= 0, z = 0, z' = 0
le(z, z') -{ 1 }→ le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
le(z, z') -{ 1 }→ 1 :|: y >= 0, z = 0, z' = y
le(z, z') -{ 1 }→ 0 :|: x >= 0, z = 1 + x, z' = 0
minus(z, z') -{ 2 }→ if(le(x'', y'), 1 + x'', 1 + y') :|: z = 1 + x'', y' >= 0, z' = 1 + y', x'' >= 0
minus(z, z') -{ 2 }→ if(1, 0, y) :|: y >= 0, z = 0, z' = y
minus(z, z') -{ 2 }→ if(0, 1 + x', 0) :|: z = 1 + x', x' >= 0, z' = 0
p(z) -{ 1 }→ x :|: x >= 0, z = 1 + x
p(z) -{ 1 }→ 0 :|: z = 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:

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

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

Found the following analysis order by SCC decomposition:

{ le } { p } { if, minus }

(14) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

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

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

le: runtime: ?, size: O(1) [1]

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

Computed RUNTIME bound using PUBS for: le
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:

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

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

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

(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:

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

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

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

(21) 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

(22) Obligation:

Complexity RNTS consisting of the following rules:

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

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

le: runtime: O(n1) [1 + z'], size: O(1) [1] p: runtime: ?, size: O(n1) [z]

(23) 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

(24) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

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

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

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

(28) Obligation:

Complexity RNTS consisting of the following rules:

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

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

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

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

Computed RUNTIME bound using CoFloCo for: if
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 9 + 4·z' + z'·z'' + z''

Computed RUNTIME bound using PUBS for: minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 11 + 4·z + z·z' + 2·z'

(30) Obligation:

Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed:
Previous analysis results are:

le: runtime: O(n1) [1 + z'], size: O(1) [1] p: runtime: O(1) [1], size: O(n1) [z] if: runtime: O(n2) [9 + 4·z' + z'·z'' + z''], size: O(n1) [1 + z'] minus: runtime: O(n2) [11 + 4·z + z·z' + 2·z'], size: O(n1) [1 + z]

(31) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(32) BOUNDS(1, n^2)