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:

leq(0, y) → true
leq(s(x), 0) → false
leq(s(x), s(y)) → leq(x, y)
if(true, x, y) → x
if(false, x, y) → y
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
mod(0, y) → 0
mod(s(x), 0) → 0
mod(s(x), s(y)) → if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(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:

leq(0, y) → true [1]
leq(s(x), 0) → false [1]
leq(s(x), s(y)) → leq(x, y) [1]
if(true, x, y) → x [1]
if(false, x, y) → y [1]
-(x, 0) → x [1]
-(s(x), s(y)) → -(x, y) [1]
mod(0, y) → 0 [1]
mod(s(x), 0) → 0 [1]
mod(s(x), s(y)) → if(leq(y, x), mod(-(s(x), s(y)), s(y)), s(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^2).


The TRS R consists of the following rules:

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

leq(0, y) → true [1]
leq(s(x), 0) → false [1]
leq(s(x), s(y)) → leq(x, y) [1]
if(true, x, y) → x [1]
if(false, x, y) → y [1]
minus(x, 0) → x [1]
minus(s(x), s(y)) → minus(x, y) [1]
mod(0, y) → 0 [1]
mod(s(x), 0) → 0 [1]
mod(s(x), s(y)) → if(leq(y, x), mod(minus(s(x), s(y)), s(y)), s(x)) [1]

The TRS has the following type information:

leq :: 0:s → 0:s → true:false 0 :: 0:s true :: true:false s :: 0:s → 0:s false :: true:false if :: true:false → 0:s → 0:s → 0:s minus :: 0:s → 0:s → 0:s mod :: 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:
none

(c) The following functions are completely defined:

leq
mod
minus
if

Due to the following rules being added:

minus(v0, v1) → 0 [0]

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: leq(0, y) → true [1] leq(s(x), 0) → false [1] leq(s(x), s(y)) → leq(x, y) [1] if(true, x, y) → x [1] if(false, x, y) → y [1] minus(x, 0) → x [1] minus(s(x), s(y)) → minus(x, y) [1] mod(0, y) → 0 [1] mod(s(x), 0) → 0 [1] mod(s(x), s(y)) → if(leq(y, x), mod(minus(s(x), s(y)), s(y)), s(x)) [1] minus(v0, v1) → 0 [0] The TRS has the following type information: leq :: 0:s → 0:s → true:false 0 :: 0:s true :: true:false s :: 0:s → 0:s false :: true:false if :: true:false → 0:s → 0:s → 0:s minus :: 0:s → 0:s → 0:s mod :: 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: leq(0, y) → true [1] leq(s(x), 0) → false [1] leq(s(x), s(y)) → leq(x, y) [1] if(true, x, y) → x [1] if(false, x, y) → y [1] minus(x, 0) → x [1] minus(s(x), s(y)) → minus(x, y) [1] mod(0, y) → 0 [1] mod(s(x), 0) → 0 [1] mod(s(x), s(0)) → if(true, mod(minus(x, 0), s(0)), s(x)) [3] mod(s(x), s(0)) → if(true, mod(0, s(0)), s(x)) [2] mod(s(0), s(s(x'))) → if(false, mod(minus(0, s(x')), s(s(x'))), s(0)) [3] mod(s(0), s(s(x'))) → if(false, mod(0, s(s(x'))), s(0)) [2] mod(s(s(y')), s(s(x''))) → if(leq(x'', y'), mod(minus(s(y'), s(x'')), s(s(x''))), s(s(y'))) [3] mod(s(s(y')), s(s(x''))) → if(leq(x'', y'), mod(0, s(s(x''))), s(s(y'))) [2] minus(v0, v1) → 0 [0] The TRS has the following type information: leq :: 0:s → 0:s → true:false 0 :: 0:s true :: true:false s :: 0:s → 0:s false :: true:false if :: true:false → 0:s → 0:s → 0:s minus :: 0:s → 0:s → 0:s mod :: 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
true => 1
false => 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

if(z, z', z'') -{ 1 }→ x :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
if(z, z', z'') -{ 1 }→ y :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
leq(z, z') -{ 1 }→ leq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
leq(z, z') -{ 1 }→ 1 :|: y >= 0, z = 0, z' = y
leq(z, z') -{ 1 }→ 0 :|: x >= 0, z = 1 + x, z' = 0
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') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
mod(z, z') -{ 3 }→ if(leq(x'', y'), mod(minus(1 + y', 1 + x''), 1 + (1 + x'')), 1 + (1 + y')) :|: z' = 1 + (1 + x''), z = 1 + (1 + y'), y' >= 0, x'' >= 0
mod(z, z') -{ 2 }→ if(leq(x'', y'), mod(0, 1 + (1 + x'')), 1 + (1 + y')) :|: z' = 1 + (1 + x''), z = 1 + (1 + y'), y' >= 0, x'' >= 0
mod(z, z') -{ 3 }→ if(1, mod(minus(x, 0), 1 + 0), 1 + x) :|: x >= 0, z' = 1 + 0, z = 1 + x
mod(z, z') -{ 2 }→ if(1, mod(0, 1 + 0), 1 + x) :|: x >= 0, z' = 1 + 0, z = 1 + x
mod(z, z') -{ 3 }→ if(0, mod(minus(0, 1 + x'), 1 + (1 + x')), 1 + 0) :|: z' = 1 + (1 + x'), z = 1 + 0, x' >= 0
mod(z, z') -{ 2 }→ if(0, mod(0, 1 + (1 + x')), 1 + 0) :|: z' = 1 + (1 + x'), z = 1 + 0, x' >= 0
mod(z, z') -{ 1 }→ 0 :|: y >= 0, z = 0, z' = y
mod(z, z') -{ 1 }→ 0 :|: x >= 0, z = 1 + x, z' = 0

(13) InliningProof (UPPER BOUND(ID) transformation)

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

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

(14) Obligation:

Complexity RNTS consisting of the following rules:

if(z, z', z'') -{ 1 }→ x :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
if(z, z', z'') -{ 1 }→ y :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
leq(z, z') -{ 1 }→ leq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
leq(z, z') -{ 1 }→ 1 :|: y >= 0, z = 0, z' = y
leq(z, z') -{ 1 }→ 0 :|: x >= 0, z = 1 + x, z' = 0
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') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
mod(z, z') -{ 3 }→ if(leq(x'', y'), mod(minus(1 + y', 1 + x''), 1 + (1 + x'')), 1 + (1 + y')) :|: z' = 1 + (1 + x''), z = 1 + (1 + y'), y' >= 0, x'' >= 0
mod(z, z') -{ 2 }→ if(leq(x'', y'), mod(0, 1 + (1 + x'')), 1 + (1 + y')) :|: z' = 1 + (1 + x''), z = 1 + (1 + y'), y' >= 0, x'' >= 0
mod(z, z') -{ 3 }→ if(1, mod(minus(x, 0), 1 + 0), 1 + x) :|: x >= 0, z' = 1 + 0, z = 1 + x
mod(z, z') -{ 2 }→ if(1, mod(0, 1 + 0), 1 + x) :|: x >= 0, z' = 1 + 0, z = 1 + x
mod(z, z') -{ 3 }→ if(0, mod(minus(0, 1 + x'), 1 + (1 + x')), 1 + 0) :|: z' = 1 + (1 + x'), z = 1 + 0, x' >= 0
mod(z, z') -{ 2 }→ if(0, mod(0, 1 + (1 + x')), 1 + 0) :|: z' = 1 + (1 + x'), z = 1 + 0, x' >= 0
mod(z, z') -{ 1 }→ 0 :|: y >= 0, z = 0, z' = y
mod(z, z') -{ 1 }→ 0 :|: x >= 0, z = 1 + x, z' = 0

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

Simplified the RNTS by moving equalities from the constraints into the right-hand sides.

(16) Obligation:

Complexity RNTS consisting of the following rules:

if(z, z', z'') -{ 1 }→ z' :|: z = 1, z' >= 0, z'' >= 0
if(z, z', z'') -{ 1 }→ z'' :|: z' >= 0, z'' >= 0, z = 0
leq(z, z') -{ 1 }→ leq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
leq(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
leq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
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') -{ 0 }→ 0 :|: z >= 0, z' >= 0
mod(z, z') -{ 3 }→ if(leq(z' - 2, z - 2), mod(minus(1 + (z - 2), 1 + (z' - 2)), 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 2 }→ if(leq(z' - 2, z - 2), mod(0, 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 3 }→ if(1, mod(minus(z - 1, 0), 1 + 0), 1 + (z - 1)) :|: z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 2 }→ if(1, mod(0, 1 + 0), 1 + (z - 1)) :|: z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 3 }→ if(0, mod(minus(0, 1 + (z' - 2)), 1 + (1 + (z' - 2))), 1 + 0) :|: z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 2 }→ if(0, mod(0, 1 + (1 + (z' - 2))), 1 + 0) :|: z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
mod(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0

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

Found the following analysis order by SCC decomposition:

{ minus } { leq } { if } { mod }

(18) Obligation:

Complexity RNTS consisting of the following rules:

if(z, z', z'') -{ 1 }→ z' :|: z = 1, z' >= 0, z'' >= 0
if(z, z', z'') -{ 1 }→ z'' :|: z' >= 0, z'' >= 0, z = 0
leq(z, z') -{ 1 }→ leq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
leq(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
leq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
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') -{ 0 }→ 0 :|: z >= 0, z' >= 0
mod(z, z') -{ 3 }→ if(leq(z' - 2, z - 2), mod(minus(1 + (z - 2), 1 + (z' - 2)), 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 2 }→ if(leq(z' - 2, z - 2), mod(0, 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 3 }→ if(1, mod(minus(z - 1, 0), 1 + 0), 1 + (z - 1)) :|: z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 2 }→ if(1, mod(0, 1 + 0), 1 + (z - 1)) :|: z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 3 }→ if(0, mod(minus(0, 1 + (z' - 2)), 1 + (1 + (z' - 2))), 1 + 0) :|: z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 2 }→ if(0, mod(0, 1 + (1 + (z' - 2))), 1 + 0) :|: z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
mod(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0

Function symbols to be analyzed: {minus}, {leq}, {if}, {mod}

(19) 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

(20) Obligation:

Complexity RNTS consisting of the following rules:

if(z, z', z'') -{ 1 }→ z' :|: z = 1, z' >= 0, z'' >= 0
if(z, z', z'') -{ 1 }→ z'' :|: z' >= 0, z'' >= 0, z = 0
leq(z, z') -{ 1 }→ leq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
leq(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
leq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
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') -{ 0 }→ 0 :|: z >= 0, z' >= 0
mod(z, z') -{ 3 }→ if(leq(z' - 2, z - 2), mod(minus(1 + (z - 2), 1 + (z' - 2)), 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 2 }→ if(leq(z' - 2, z - 2), mod(0, 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 3 }→ if(1, mod(minus(z - 1, 0), 1 + 0), 1 + (z - 1)) :|: z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 2 }→ if(1, mod(0, 1 + 0), 1 + (z - 1)) :|: z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 3 }→ if(0, mod(minus(0, 1 + (z' - 2)), 1 + (1 + (z' - 2))), 1 + 0) :|: z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 2 }→ if(0, mod(0, 1 + (1 + (z' - 2))), 1 + 0) :|: z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
mod(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0

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

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

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

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

if(z, z', z'') -{ 1 }→ z' :|: z = 1, z' >= 0, z'' >= 0
if(z, z', z'') -{ 1 }→ z'' :|: z' >= 0, z'' >= 0, z = 0
leq(z, z') -{ 1 }→ leq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
leq(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
leq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
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') -{ 0 }→ 0 :|: z >= 0, z' >= 0
mod(z, z') -{ 3 }→ if(leq(z' - 2, z - 2), mod(minus(1 + (z - 2), 1 + (z' - 2)), 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 2 }→ if(leq(z' - 2, z - 2), mod(0, 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 3 }→ if(1, mod(minus(z - 1, 0), 1 + 0), 1 + (z - 1)) :|: z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 2 }→ if(1, mod(0, 1 + 0), 1 + (z - 1)) :|: z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 3 }→ if(0, mod(minus(0, 1 + (z' - 2)), 1 + (1 + (z' - 2))), 1 + 0) :|: z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 2 }→ if(0, mod(0, 1 + (1 + (z' - 2))), 1 + 0) :|: z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
mod(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0

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

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

(23) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

if(z, z', z'') -{ 1 }→ z' :|: z = 1, z' >= 0, z'' >= 0
if(z, z', z'') -{ 1 }→ z'' :|: z' >= 0, z'' >= 0, z = 0
leq(z, z') -{ 1 }→ leq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
leq(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
leq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
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') -{ 0 }→ 0 :|: z >= 0, z' >= 0
mod(z, z') -{ 3 + z' }→ if(leq(z' - 2, z - 2), mod(s1, 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: s1 >= 0, s1 <= 1 * (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 2 }→ if(leq(z' - 2, z - 2), mod(0, 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 4 }→ if(1, mod(s', 1 + 0), 1 + (z - 1)) :|: s' >= 0, s' <= 1 * (z - 1), z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 2 }→ if(1, mod(0, 1 + 0), 1 + (z - 1)) :|: z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 3 + z' }→ if(0, mod(s'', 1 + (1 + (z' - 2))), 1 + 0) :|: s'' >= 0, s'' <= 1 * 0, z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 2 }→ if(0, mod(0, 1 + (1 + (z' - 2))), 1 + 0) :|: z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
mod(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0

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

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

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

Computed SIZE bound using CoFloCo for: leq
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:

if(z, z', z'') -{ 1 }→ z' :|: z = 1, z' >= 0, z'' >= 0
if(z, z', z'') -{ 1 }→ z'' :|: z' >= 0, z'' >= 0, z = 0
leq(z, z') -{ 1 }→ leq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
leq(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
leq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
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') -{ 0 }→ 0 :|: z >= 0, z' >= 0
mod(z, z') -{ 3 + z' }→ if(leq(z' - 2, z - 2), mod(s1, 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: s1 >= 0, s1 <= 1 * (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 2 }→ if(leq(z' - 2, z - 2), mod(0, 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 4 }→ if(1, mod(s', 1 + 0), 1 + (z - 1)) :|: s' >= 0, s' <= 1 * (z - 1), z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 2 }→ if(1, mod(0, 1 + 0), 1 + (z - 1)) :|: z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 3 + z' }→ if(0, mod(s'', 1 + (1 + (z' - 2))), 1 + 0) :|: s'' >= 0, s'' <= 1 * 0, z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 2 }→ if(0, mod(0, 1 + (1 + (z' - 2))), 1 + 0) :|: z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
mod(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0

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

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

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

Computed RUNTIME bound using PUBS for: leq
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 }→ z' :|: z = 1, z' >= 0, z'' >= 0
if(z, z', z'') -{ 1 }→ z'' :|: z' >= 0, z'' >= 0, z = 0
leq(z, z') -{ 1 }→ leq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
leq(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
leq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
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') -{ 0 }→ 0 :|: z >= 0, z' >= 0
mod(z, z') -{ 3 + z' }→ if(leq(z' - 2, z - 2), mod(s1, 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: s1 >= 0, s1 <= 1 * (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 2 }→ if(leq(z' - 2, z - 2), mod(0, 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 4 }→ if(1, mod(s', 1 + 0), 1 + (z - 1)) :|: s' >= 0, s' <= 1 * (z - 1), z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 2 }→ if(1, mod(0, 1 + 0), 1 + (z - 1)) :|: z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 3 + z' }→ if(0, mod(s'', 1 + (1 + (z' - 2))), 1 + 0) :|: s'' >= 0, s'' <= 1 * 0, z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 2 }→ if(0, mod(0, 1 + (1 + (z' - 2))), 1 + 0) :|: z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
mod(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0

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

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

(29) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(30) Obligation:

Complexity RNTS consisting of the following rules:

if(z, z', z'') -{ 1 }→ z' :|: z = 1, z' >= 0, z'' >= 0
if(z, z', z'') -{ 1 }→ z'' :|: z' >= 0, z'' >= 0, z = 0
leq(z, z') -{ 1 + z' }→ s2 :|: s2 >= 0, s2 <= 1, z - 1 >= 0, z' - 1 >= 0
leq(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
leq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
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') -{ 0 }→ 0 :|: z >= 0, z' >= 0
mod(z, z') -{ 2 + z + z' }→ if(s3, mod(s1, 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: s3 >= 0, s3 <= 1, s1 >= 0, s1 <= 1 * (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 1 + z }→ if(s4, mod(0, 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: s4 >= 0, s4 <= 1, z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 4 }→ if(1, mod(s', 1 + 0), 1 + (z - 1)) :|: s' >= 0, s' <= 1 * (z - 1), z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 2 }→ if(1, mod(0, 1 + 0), 1 + (z - 1)) :|: z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 3 + z' }→ if(0, mod(s'', 1 + (1 + (z' - 2))), 1 + 0) :|: s'' >= 0, s'' <= 1 * 0, z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 2 }→ if(0, mod(0, 1 + (1 + (z' - 2))), 1 + 0) :|: z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
mod(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0

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

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

(31) 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: z' + z''

(32) Obligation:

Complexity RNTS consisting of the following rules:

if(z, z', z'') -{ 1 }→ z' :|: z = 1, z' >= 0, z'' >= 0
if(z, z', z'') -{ 1 }→ z'' :|: z' >= 0, z'' >= 0, z = 0
leq(z, z') -{ 1 + z' }→ s2 :|: s2 >= 0, s2 <= 1, z - 1 >= 0, z' - 1 >= 0
leq(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
leq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
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') -{ 0 }→ 0 :|: z >= 0, z' >= 0
mod(z, z') -{ 2 + z + z' }→ if(s3, mod(s1, 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: s3 >= 0, s3 <= 1, s1 >= 0, s1 <= 1 * (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 1 + z }→ if(s4, mod(0, 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: s4 >= 0, s4 <= 1, z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 4 }→ if(1, mod(s', 1 + 0), 1 + (z - 1)) :|: s' >= 0, s' <= 1 * (z - 1), z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 2 }→ if(1, mod(0, 1 + 0), 1 + (z - 1)) :|: z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 3 + z' }→ if(0, mod(s'', 1 + (1 + (z' - 2))), 1 + 0) :|: s'' >= 0, s'' <= 1 * 0, z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 2 }→ if(0, mod(0, 1 + (1 + (z' - 2))), 1 + 0) :|: z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
mod(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0

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

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

(33) IntTrsBoundProof (UPPER BOUND(ID) transformation)

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

(34) Obligation:

Complexity RNTS consisting of the following rules:

if(z, z', z'') -{ 1 }→ z' :|: z = 1, z' >= 0, z'' >= 0
if(z, z', z'') -{ 1 }→ z'' :|: z' >= 0, z'' >= 0, z = 0
leq(z, z') -{ 1 + z' }→ s2 :|: s2 >= 0, s2 <= 1, z - 1 >= 0, z' - 1 >= 0
leq(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
leq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
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') -{ 0 }→ 0 :|: z >= 0, z' >= 0
mod(z, z') -{ 2 + z + z' }→ if(s3, mod(s1, 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: s3 >= 0, s3 <= 1, s1 >= 0, s1 <= 1 * (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 1 + z }→ if(s4, mod(0, 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: s4 >= 0, s4 <= 1, z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 4 }→ if(1, mod(s', 1 + 0), 1 + (z - 1)) :|: s' >= 0, s' <= 1 * (z - 1), z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 2 }→ if(1, mod(0, 1 + 0), 1 + (z - 1)) :|: z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 3 + z' }→ if(0, mod(s'', 1 + (1 + (z' - 2))), 1 + 0) :|: s'' >= 0, s'' <= 1 * 0, z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 2 }→ if(0, mod(0, 1 + (1 + (z' - 2))), 1 + 0) :|: z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
mod(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0

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

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

(35) ResultPropagationProof (UPPER BOUND(ID) transformation)

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

(36) Obligation:

Complexity RNTS consisting of the following rules:

if(z, z', z'') -{ 1 }→ z' :|: z = 1, z' >= 0, z'' >= 0
if(z, z', z'') -{ 1 }→ z'' :|: z' >= 0, z'' >= 0, z = 0
leq(z, z') -{ 1 + z' }→ s2 :|: s2 >= 0, s2 <= 1, z - 1 >= 0, z' - 1 >= 0
leq(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
leq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
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') -{ 0 }→ 0 :|: z >= 0, z' >= 0
mod(z, z') -{ 2 + z + z' }→ if(s3, mod(s1, 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: s3 >= 0, s3 <= 1, s1 >= 0, s1 <= 1 * (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 1 + z }→ if(s4, mod(0, 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: s4 >= 0, s4 <= 1, z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 4 }→ if(1, mod(s', 1 + 0), 1 + (z - 1)) :|: s' >= 0, s' <= 1 * (z - 1), z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 2 }→ if(1, mod(0, 1 + 0), 1 + (z - 1)) :|: z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 3 + z' }→ if(0, mod(s'', 1 + (1 + (z' - 2))), 1 + 0) :|: s'' >= 0, s'' <= 1 * 0, z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 2 }→ if(0, mod(0, 1 + (1 + (z' - 2))), 1 + 0) :|: z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
mod(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0

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

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

(37) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed SIZE bound using PUBS for: mod
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: z + z²

(38) Obligation:

Complexity RNTS consisting of the following rules:

if(z, z', z'') -{ 1 }→ z' :|: z = 1, z' >= 0, z'' >= 0
if(z, z', z'') -{ 1 }→ z'' :|: z' >= 0, z'' >= 0, z = 0
leq(z, z') -{ 1 + z' }→ s2 :|: s2 >= 0, s2 <= 1, z - 1 >= 0, z' - 1 >= 0
leq(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
leq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
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') -{ 0 }→ 0 :|: z >= 0, z' >= 0
mod(z, z') -{ 2 + z + z' }→ if(s3, mod(s1, 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: s3 >= 0, s3 <= 1, s1 >= 0, s1 <= 1 * (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 1 + z }→ if(s4, mod(0, 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: s4 >= 0, s4 <= 1, z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 4 }→ if(1, mod(s', 1 + 0), 1 + (z - 1)) :|: s' >= 0, s' <= 1 * (z - 1), z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 2 }→ if(1, mod(0, 1 + 0), 1 + (z - 1)) :|: z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 3 + z' }→ if(0, mod(s'', 1 + (1 + (z' - 2))), 1 + 0) :|: s'' >= 0, s'' <= 1 * 0, z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 2 }→ if(0, mod(0, 1 + (1 + (z' - 2))), 1 + 0) :|: z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
mod(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0

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

minus: runtime: O(n1) [1 + z'], size: O(n1) [z] leq: runtime: O(n1) [1 + z'], size: O(1) [1] if: runtime: O(1) [1], size: O(n1) [z' + z''] mod: runtime: ?, size: O(n2) [z + z2]

(39) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using PUBS for: mod
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 1 + 5·z + z·z' + z²

(40) Obligation:

Complexity RNTS consisting of the following rules:

if(z, z', z'') -{ 1 }→ z' :|: z = 1, z' >= 0, z'' >= 0
if(z, z', z'') -{ 1 }→ z'' :|: z' >= 0, z'' >= 0, z = 0
leq(z, z') -{ 1 + z' }→ s2 :|: s2 >= 0, s2 <= 1, z - 1 >= 0, z' - 1 >= 0
leq(z, z') -{ 1 }→ 1 :|: z' >= 0, z = 0
leq(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0
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') -{ 0 }→ 0 :|: z >= 0, z' >= 0
mod(z, z') -{ 2 + z + z' }→ if(s3, mod(s1, 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: s3 >= 0, s3 <= 1, s1 >= 0, s1 <= 1 * (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 1 + z }→ if(s4, mod(0, 1 + (1 + (z' - 2))), 1 + (1 + (z - 2))) :|: s4 >= 0, s4 <= 1, z - 2 >= 0, z' - 2 >= 0
mod(z, z') -{ 4 }→ if(1, mod(s', 1 + 0), 1 + (z - 1)) :|: s' >= 0, s' <= 1 * (z - 1), z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 2 }→ if(1, mod(0, 1 + 0), 1 + (z - 1)) :|: z - 1 >= 0, z' = 1 + 0
mod(z, z') -{ 3 + z' }→ if(0, mod(s'', 1 + (1 + (z' - 2))), 1 + 0) :|: s'' >= 0, s'' <= 1 * 0, z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 2 }→ if(0, mod(0, 1 + (1 + (z' - 2))), 1 + 0) :|: z = 1 + 0, z' - 2 >= 0
mod(z, z') -{ 1 }→ 0 :|: z' >= 0, z = 0
mod(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0

Function symbols to be analyzed:
Previous analysis results are:

minus: runtime: O(n1) [1 + z'], size: O(n1) [z] leq: runtime: O(n1) [1 + z'], size: O(1) [1] if: runtime: O(1) [1], size: O(n1) [z' + z''] mod: runtime: O(n2) [1 + 5·z + z·z' + z2], size: O(n2) [z + z2]

(41) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(42) BOUNDS(1, n^2)