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:

cond(true, x, y) → cond(gr(x, y), p(x), s(y))
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
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^2).


The TRS R consists of the following rules:

cond(true, x, y) → cond(gr(x, y), p(x), s(y)) [1]
gr(0, x) → false [1]
gr(s(x), 0) → true [1]
gr(s(x), s(y)) → gr(x, y) [1]
p(0) → 0 [1]
p(s(x)) → x [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:

cond(true, x, y) → cond(gr(x, y), p(x), s(y)) [1]
gr(0, x) → false [1]
gr(s(x), 0) → true [1]
gr(s(x), s(y)) → gr(x, y) [1]
p(0) → 0 [1]
p(s(x)) → x [1]

The TRS has the following type information:

cond :: true:false → s:0 → s:0 → cond true :: true:false gr :: s:0 → s:0 → true:false p :: s:0 → s:0 s :: s:0 → s:0 0 :: s:0 false :: true:false

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:

cond

(c) The following functions are completely defined:

gr
p

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: cond(true, x, y) → cond(gr(x, y), p(x), s(y)) [1] gr(0, x) → false [1] gr(s(x), 0) → true [1] gr(s(x), s(y)) → gr(x, y) [1] p(0) → 0 [1] p(s(x)) → x [1] The TRS has the following type information: cond :: true:false → s:0 → s:0 → cond true :: true:false gr :: s:0 → s:0 → true:false p :: s:0 → s:0 s :: s:0 → s:0 0 :: s:0 false :: true:false const :: cond 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: cond(true, 0, y) → cond(false, 0, s(y)) [3] cond(true, s(x'), 0) → cond(true, x', s(0)) [3] cond(true, s(x''), s(y')) → cond(gr(x'', y'), x'', s(s(y'))) [3] gr(0, x) → false [1] gr(s(x), 0) → true [1] gr(s(x), s(y)) → gr(x, y) [1] p(0) → 0 [1] p(s(x)) → x [1] The TRS has the following type information: cond :: true:false → s:0 → s:0 → cond true :: true:false gr :: s:0 → s:0 → true:false p :: s:0 → s:0 s :: s:0 → s:0 0 :: s:0 false :: true:false const :: cond 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:

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

(10) Obligation:

Complexity RNTS consisting of the following rules:

cond(z, z', z'') -{ 3 }→ cond(gr(x'', y'), x'', 1 + (1 + y')) :|: z' = 1 + x'', z = 1, y' >= 0, x'' >= 0, z'' = 1 + y'
cond(z, z', z'') -{ 3 }→ cond(1, x', 1 + 0) :|: z'' = 0, z' = 1 + x', z = 1, x' >= 0
cond(z, z', z'') -{ 3 }→ cond(0, 0, 1 + y) :|: z'' = y, z = 1, y >= 0, z' = 0
gr(z, z') -{ 1 }→ gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
gr(z, z') -{ 1 }→ 1 :|: x >= 0, z = 1 + x, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' = 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:

cond(z, z', z'') -{ 3 }→ cond(gr(z' - 1, z'' - 1), z' - 1, 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond(z, z', z'') -{ 3 }→ cond(1, z' - 1, 1 + 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond(z, z', z'') -{ 3 }→ cond(0, 0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 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:

{ p } { gr } { cond }

(14) Obligation:

Complexity RNTS consisting of the following rules:

cond(z, z', z'') -{ 3 }→ cond(gr(z' - 1, z'' - 1), z' - 1, 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond(z, z', z'') -{ 3 }→ cond(1, z' - 1, 1 + 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond(z, z', z'') -{ 3 }→ cond(0, 0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {gr}, {cond}

(15) 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

(16) Obligation:

Complexity RNTS consisting of the following rules:

cond(z, z', z'') -{ 3 }→ cond(gr(z' - 1, z'' - 1), z' - 1, 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond(z, z', z'') -{ 3 }→ cond(1, z' - 1, 1 + 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond(z, z', z'') -{ 3 }→ cond(0, 0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

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

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

(17) 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

(18) Obligation:

Complexity RNTS consisting of the following rules:

cond(z, z', z'') -{ 3 }→ cond(gr(z' - 1, z'' - 1), z' - 1, 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond(z, z', z'') -{ 3 }→ cond(1, z' - 1, 1 + 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond(z, z', z'') -{ 3 }→ cond(0, 0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {gr}, {cond}
Previous analysis results are:

p: runtime: O(1) [1], 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:

cond(z, z', z'') -{ 3 }→ cond(gr(z' - 1, z'' - 1), z' - 1, 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond(z, z', z'') -{ 3 }→ cond(1, z' - 1, 1 + 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond(z, z', z'') -{ 3 }→ cond(0, 0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {gr}, {cond}
Previous analysis results are:

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

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

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

cond(z, z', z'') -{ 3 }→ cond(gr(z' - 1, z'' - 1), z' - 1, 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond(z, z', z'') -{ 3 }→ cond(1, z' - 1, 1 + 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond(z, z', z'') -{ 3 }→ cond(0, 0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {gr}, {cond}
Previous analysis results are:

p: runtime: O(1) [1], size: O(n1) [z] gr: runtime: ?, size: O(1) [1]

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

Computed RUNTIME bound using PUBS for: gr
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:

cond(z, z', z'') -{ 3 }→ cond(gr(z' - 1, z'' - 1), z' - 1, 1 + (1 + (z'' - 1))) :|: z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond(z, z', z'') -{ 3 }→ cond(1, z' - 1, 1 + 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond(z, z', z'') -{ 3 }→ cond(0, 0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
gr(z, z') -{ 1 }→ gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

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

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

(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:

cond(z, z', z'') -{ 3 + z'' }→ cond(s, z' - 1, 1 + (1 + (z'' - 1))) :|: s >= 0, s <= 1, z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond(z, z', z'') -{ 3 }→ cond(1, z' - 1, 1 + 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond(z, z', z'') -{ 3 }→ cond(0, 0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
gr(z, z') -{ 1 + z' }→ s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

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

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

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

Computed SIZE bound using CoFloCo for: cond
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

(28) Obligation:

Complexity RNTS consisting of the following rules:

cond(z, z', z'') -{ 3 + z'' }→ cond(s, z' - 1, 1 + (1 + (z'' - 1))) :|: s >= 0, s <= 1, z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond(z, z', z'') -{ 3 }→ cond(1, z' - 1, 1 + 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond(z, z', z'') -{ 3 }→ cond(0, 0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
gr(z, z') -{ 1 + z' }→ s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 0 :|: z = 0
p(z) -{ 1 }→ z - 1 :|: z - 1 >= 0

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

p: runtime: O(1) [1], size: O(n1) [z] gr: runtime: O(n1) [1 + z'], size: O(1) [1] cond: runtime: ?, size: O(1) [0]

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

Computed RUNTIME bound using CoFloCo for: cond
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 8 + 3·z' + z'·z'' + z'²

(30) Obligation:

Complexity RNTS consisting of the following rules:

cond(z, z', z'') -{ 3 + z'' }→ cond(s, z' - 1, 1 + (1 + (z'' - 1))) :|: s >= 0, s <= 1, z = 1, z'' - 1 >= 0, z' - 1 >= 0
cond(z, z', z'') -{ 3 }→ cond(1, z' - 1, 1 + 0) :|: z'' = 0, z = 1, z' - 1 >= 0
cond(z, z', z'') -{ 3 }→ cond(0, 0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
gr(z, z') -{ 1 + z' }→ s' :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
gr(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0
gr(z, z') -{ 1 }→ 0 :|: z' >= 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:

p: runtime: O(1) [1], size: O(n1) [z] gr: runtime: O(n1) [1 + z'], size: O(1) [1] cond: runtime: O(n2) [8 + 3·z' + z'·z'' + z'2], size: O(1) [0]

(31) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(32) BOUNDS(1, n^2)