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:

cond(true, x, y) → cond(and(gr(x, 0), gr(y, 0)), p(x), p(y))
and(true, true) → true
and(x, false) → false
and(false, x) → false
gr(0, 0) → false
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^1).


The TRS R consists of the following rules:

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

cond

(c) The following functions are completely defined:

and
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(and(gr(x, 0), gr(y, 0)), p(x), p(y)) [1] and(true, true) → true [1] and(x, false) → false [1] and(false, x) → false [1] gr(0, 0) → false [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 → 0:s → 0:s → cond true :: true:false and :: true:false → true:false → true:false gr :: 0:s → 0:s → true:false 0 :: 0:s p :: 0:s → 0:s false :: true:false s :: 0:s → 0:s 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, 0) → cond(and(false, false), 0, 0) [5] cond(true, 0, 0) → cond(and(false, false), 0, 0) [5] cond(true, 0, s(x'')) → cond(and(false, true), 0, x'') [5] cond(true, 0, 0) → cond(and(false, false), 0, 0) [5] cond(true, 0, 0) → cond(and(false, false), 0, 0) [5] cond(true, 0, s(x1)) → cond(and(false, true), 0, x1) [5] cond(true, s(x'), 0) → cond(and(true, false), x', 0) [5] cond(true, s(x'), 0) → cond(and(true, false), x', 0) [5] cond(true, s(x'), s(x2)) → cond(and(true, true), x', x2) [5] and(true, true) → true [1] and(x, false) → false [1] and(false, x) → false [1] gr(0, 0) → false [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 → 0:s → 0:s → cond true :: true:false and :: true:false → true:false → true:false gr :: 0:s → 0:s → true:false 0 :: 0:s p :: 0:s → 0:s false :: true:false s :: 0:s → 0:s 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:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: x >= 0, z = x, z' = 0
and(z, z') -{ 1 }→ 0 :|: z' = x, x >= 0, z = 0
cond(z, z', z'') -{ 5 }→ cond(and(1, 1), x', x2) :|: z' = 1 + x', z = 1, x' >= 0, z'' = 1 + x2, x2 >= 0
cond(z, z', z'') -{ 5 }→ cond(and(1, 0), x', 0) :|: z'' = 0, z' = 1 + x', z = 1, x' >= 0
cond(z, z', z'') -{ 5 }→ cond(and(0, 1), 0, x'') :|: z = 1, z'' = 1 + x'', x'' >= 0, z' = 0
cond(z, z', z'') -{ 5 }→ cond(and(0, 1), 0, x1) :|: x1 >= 0, z = 1, z'' = 1 + x1, z' = 0
cond(z, z', z'') -{ 5 }→ cond(and(0, 0), 0, 0) :|: z'' = 0, z = 1, 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 = 0, 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) InliningProof (UPPER BOUND(ID) transformation)

Inlined the following terminating rules on right-hand sides where appropriate:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: x >= 0, z = x, z' = 0
and(z, z') -{ 1 }→ 0 :|: z' = x, x >= 0, z = 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: x >= 0, z = x, z' = 0
and(z, z') -{ 1 }→ 0 :|: z' = x, x >= 0, z = 0
cond(z, z', z'') -{ 6 }→ cond(1, x', x2) :|: z' = 1 + x', z = 1, x' >= 0, z'' = 1 + x2, x2 >= 0, 1 = 1
cond(z, z', z'') -{ 6 }→ cond(0, x', 0) :|: z'' = 0, z' = 1 + x', z = 1, x' >= 0, x >= 0, 1 = x, 0 = 0
cond(z, z', z'') -{ 6 }→ cond(0, 0, x'') :|: z = 1, z'' = 1 + x'', x'' >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
cond(z, z', z'') -{ 6 }→ cond(0, 0, x1) :|: x1 >= 0, z = 1, z'' = 1 + x1, z' = 0, 1 = x, x >= 0, 0 = 0
cond(z, z', z'') -{ 6 }→ cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 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 = 0, 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

(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:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
cond(z, z', z'') -{ 6 }→ cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
cond(z, z', z'') -{ 6 }→ cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
cond(z, z', z'') -{ 6 }→ cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
cond(z, z', z'') -{ 6 }→ cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 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
gr(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
p(z) -{ 1 }→ 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:

{ and } { cond } { p } { gr }

(16) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
cond(z, z', z'') -{ 6 }→ cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
cond(z, z', z'') -{ 6 }→ cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
cond(z, z', z'') -{ 6 }→ cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
cond(z, z', z'') -{ 6 }→ cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 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
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: {and}, {cond}, {p}, {gr}

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

Computed SIZE bound using CoFloCo for: and
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:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
cond(z, z', z'') -{ 6 }→ cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
cond(z, z', z'') -{ 6 }→ cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
cond(z, z', z'') -{ 6 }→ cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
cond(z, z', z'') -{ 6 }→ cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 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
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: {and}, {cond}, {p}, {gr}
Previous analysis results are:

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

(19) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(20) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
cond(z, z', z'') -{ 6 }→ cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
cond(z, z', z'') -{ 6 }→ cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
cond(z, z', z'') -{ 6 }→ cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
cond(z, z', z'') -{ 6 }→ cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 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
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}, {p}, {gr}
Previous analysis results are:

and: runtime: O(1) [1], size: O(1) [1]

(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:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
cond(z, z', z'') -{ 6 }→ cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
cond(z, z', z'') -{ 6 }→ cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
cond(z, z', z'') -{ 6 }→ cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
cond(z, z', z'') -{ 6 }→ cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 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
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}, {p}, {gr}
Previous analysis results are:

and: runtime: O(1) [1], size: O(1) [1]

(23) 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

(24) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
cond(z, z', z'') -{ 6 }→ cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
cond(z, z', z'') -{ 6 }→ cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
cond(z, z', z'') -{ 6 }→ cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
cond(z, z', z'') -{ 6 }→ cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 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
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}, {p}, {gr}
Previous analysis results are:

and: runtime: O(1) [1], size: O(1) [1] cond: runtime: ?, size: O(1) [0]

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

Computed RUNTIME bound using KoAT for: cond
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 24·z + 24·z'

(26) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
cond(z, z', z'') -{ 6 }→ cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
cond(z, z', z'') -{ 6 }→ cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
cond(z, z', z'') -{ 6 }→ cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
cond(z, z', z'') -{ 6 }→ cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 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
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}
Previous analysis results are:

and: runtime: O(1) [1], size: O(1) [1] cond: runtime: O(n1) [24·z + 24·z'], size: O(1) [0]

(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:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
cond(z, z', z'') -{ 6 }→ s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
cond(z, z', z'') -{ 6 }→ s' :|: s' >= 0, s' <= 0, z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
cond(z, z', z'') -{ -18 + 24·z' }→ s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
cond(z, z', z'') -{ 6 + 24·z' }→ s1 :|: s1 >= 0, s1 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
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
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}
Previous analysis results are:

and: runtime: O(1) [1], size: O(1) [1] cond: runtime: O(n1) [24·z + 24·z'], size: O(1) [0]

(29) 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

(30) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
cond(z, z', z'') -{ 6 }→ s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
cond(z, z', z'') -{ 6 }→ s' :|: s' >= 0, s' <= 0, z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
cond(z, z', z'') -{ -18 + 24·z' }→ s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
cond(z, z', z'') -{ 6 + 24·z' }→ s1 :|: s1 >= 0, s1 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
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
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}
Previous analysis results are:

and: runtime: O(1) [1], size: O(1) [1] cond: runtime: O(n1) [24·z + 24·z'], size: O(1) [0] p: runtime: ?, size: O(n1) [z]

(31) 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

(32) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
cond(z, z', z'') -{ 6 }→ s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
cond(z, z', z'') -{ 6 }→ s' :|: s' >= 0, s' <= 0, z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
cond(z, z', z'') -{ -18 + 24·z' }→ s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
cond(z, z', z'') -{ 6 + 24·z' }→ s1 :|: s1 >= 0, s1 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
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
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}
Previous analysis results are:

and: runtime: O(1) [1], size: O(1) [1] cond: runtime: O(n1) [24·z + 24·z'], size: O(1) [0] p: runtime: O(1) [1], size: O(n1) [z]

(33) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(34) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
cond(z, z', z'') -{ 6 }→ s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
cond(z, z', z'') -{ 6 }→ s' :|: s' >= 0, s' <= 0, z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
cond(z, z', z'') -{ -18 + 24·z' }→ s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
cond(z, z', z'') -{ 6 + 24·z' }→ s1 :|: s1 >= 0, s1 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
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
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}
Previous analysis results are:

and: runtime: O(1) [1], size: O(1) [1] cond: runtime: O(n1) [24·z + 24·z'], size: O(1) [0] p: runtime: O(1) [1], size: O(n1) [z]

(35) 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

(36) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
cond(z, z', z'') -{ 6 }→ s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
cond(z, z', z'') -{ 6 }→ s' :|: s' >= 0, s' <= 0, z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
cond(z, z', z'') -{ -18 + 24·z' }→ s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
cond(z, z', z'') -{ 6 + 24·z' }→ s1 :|: s1 >= 0, s1 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
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
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}
Previous analysis results are:

and: runtime: O(1) [1], size: O(1) [1] cond: runtime: O(n1) [24·z + 24·z'], size: O(1) [0] p: runtime: O(1) [1], size: O(n1) [z] gr: runtime: ?, size: O(1) [1]

(37) 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'

(38) Obligation:

Complexity RNTS consisting of the following rules:

and(z, z') -{ 1 }→ 1 :|: z = 1, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' = 0
and(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
cond(z, z', z'') -{ 6 }→ s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
cond(z, z', z'') -{ 6 }→ s' :|: s' >= 0, s' <= 0, z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
cond(z, z', z'') -{ -18 + 24·z' }→ s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
cond(z, z', z'') -{ 6 + 24·z' }→ s1 :|: s1 >= 0, s1 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
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
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:

and: runtime: O(1) [1], size: O(1) [1] cond: runtime: O(n1) [24·z + 24·z'], size: O(1) [0] p: runtime: O(1) [1], size: O(n1) [z] gr: runtime: O(n1) [1 + z'], size: O(1) [1]

(39) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(40) BOUNDS(1, n^1)