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:

sqr(0) → 0
sqr(s(x)) → +(sqr(x), s(double(x)))
double(0) → 0
double(s(x)) → s(s(double(x)))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
sqr(s(x)) → s(+(sqr(x), double(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:

sqr(0) → 0 [1]
sqr(s(x)) → +(sqr(x), s(double(x))) [1]
double(0) → 0 [1]
double(s(x)) → s(s(double(x))) [1]
+(x, 0) → x [1]
+(x, s(y)) → s(+(x, y)) [1]
sqr(s(x)) → s(+(sqr(x), double(x))) [1]

Rewrite Strategy: INNERMOST

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

Renamed defined symbols to avoid conflicts with arithmetic symbols:

+ => plus

(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:

sqr(0) → 0 [1]
sqr(s(x)) → plus(sqr(x), s(double(x))) [1]
double(0) → 0 [1]
double(s(x)) → s(s(double(x))) [1]
plus(x, 0) → x [1]
plus(x, s(y)) → s(plus(x, y)) [1]
sqr(s(x)) → s(plus(sqr(x), double(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:

sqr(0) → 0 [1]
sqr(s(x)) → plus(sqr(x), s(double(x))) [1]
double(0) → 0 [1]
double(s(x)) → s(s(double(x))) [1]
plus(x, 0) → x [1]
plus(x, s(y)) → s(plus(x, y)) [1]
sqr(s(x)) → s(plus(sqr(x), double(x))) [1]

The TRS has the following type information:

sqr :: 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s plus :: 0:s → 0:s → 0:s double :: 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:

sqr
double
plus

Due to the following rules being added:
none

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: sqr(0) → 0 [1] sqr(s(x)) → plus(sqr(x), s(double(x))) [1] double(0) → 0 [1] double(s(x)) → s(s(double(x))) [1] plus(x, 0) → x [1] plus(x, s(y)) → s(plus(x, y)) [1] sqr(s(x)) → s(plus(sqr(x), double(x))) [1] The TRS has the following type information: sqr :: 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s plus :: 0:s → 0:s → 0:s double :: 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: sqr(0) → 0 [1] sqr(s(0)) → plus(0, s(0)) [3] sqr(s(s(x'))) → plus(plus(sqr(x'), s(double(x'))), s(s(s(double(x'))))) [3] sqr(s(s(x''))) → plus(s(plus(sqr(x''), double(x''))), s(s(s(double(x''))))) [3] double(0) → 0 [1] double(s(x)) → s(s(double(x))) [1] plus(x, 0) → x [1] plus(x, s(y)) → s(plus(x, y)) [1] sqr(s(0)) → s(plus(0, 0)) [3] sqr(s(s(x1))) → s(plus(plus(sqr(x1), s(double(x1))), s(s(double(x1))))) [3] sqr(s(s(x2))) → s(plus(s(plus(sqr(x2), double(x2))), s(s(double(x2))))) [3] The TRS has the following type information: sqr :: 0:s → 0:s 0 :: 0:s s :: 0:s → 0:s plus :: 0:s → 0:s → 0:s double :: 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

(12) Obligation:

Complexity RNTS consisting of the following rules:

double(z) -{ 1 }→ 0 :|: z = 0
double(z) -{ 1 }→ 1 + (1 + double(x)) :|: x >= 0, z = 1 + x
plus(z, z') -{ 1 }→ x :|: x >= 0, z = x, z' = 0
plus(z, z') -{ 1 }→ 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
sqr(z) -{ 3 }→ plus(plus(sqr(x'), 1 + double(x')), 1 + (1 + (1 + double(x')))) :|: x' >= 0, z = 1 + (1 + x')
sqr(z) -{ 3 }→ plus(0, 1 + 0) :|: z = 1 + 0
sqr(z) -{ 3 }→ plus(1 + plus(sqr(x''), double(x'')), 1 + (1 + (1 + double(x'')))) :|: x'' >= 0, z = 1 + (1 + x'')
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 3 }→ 1 + plus(plus(sqr(x1), 1 + double(x1)), 1 + (1 + double(x1))) :|: z = 1 + (1 + x1), x1 >= 0
sqr(z) -{ 3 }→ 1 + plus(0, 0) :|: z = 1 + 0
sqr(z) -{ 3 }→ 1 + plus(1 + plus(sqr(x2), double(x2)), 1 + (1 + double(x2))) :|: z = 1 + (1 + x2), x2 >= 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:

double(z) -{ 1 }→ 0 :|: z = 0
double(z) -{ 1 }→ 1 + (1 + double(z - 1)) :|: z - 1 >= 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
sqr(z) -{ 3 }→ plus(plus(sqr(z - 2), 1 + double(z - 2)), 1 + (1 + (1 + double(z - 2)))) :|: z - 2 >= 0
sqr(z) -{ 3 }→ plus(0, 1 + 0) :|: z = 1 + 0
sqr(z) -{ 3 }→ plus(1 + plus(sqr(z - 2), double(z - 2)), 1 + (1 + (1 + double(z - 2)))) :|: z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 3 }→ 1 + plus(plus(sqr(z - 2), 1 + double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0
sqr(z) -{ 3 }→ 1 + plus(0, 0) :|: z = 1 + 0
sqr(z) -{ 3 }→ 1 + plus(1 + plus(sqr(z - 2), double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0

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

Found the following analysis order by SCC decomposition:

{ double } { plus } { sqr }

(16) Obligation:

Complexity RNTS consisting of the following rules:

double(z) -{ 1 }→ 0 :|: z = 0
double(z) -{ 1 }→ 1 + (1 + double(z - 1)) :|: z - 1 >= 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
sqr(z) -{ 3 }→ plus(plus(sqr(z - 2), 1 + double(z - 2)), 1 + (1 + (1 + double(z - 2)))) :|: z - 2 >= 0
sqr(z) -{ 3 }→ plus(0, 1 + 0) :|: z = 1 + 0
sqr(z) -{ 3 }→ plus(1 + plus(sqr(z - 2), double(z - 2)), 1 + (1 + (1 + double(z - 2)))) :|: z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 3 }→ 1 + plus(plus(sqr(z - 2), 1 + double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0
sqr(z) -{ 3 }→ 1 + plus(0, 0) :|: z = 1 + 0
sqr(z) -{ 3 }→ 1 + plus(1 + plus(sqr(z - 2), double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0

Function symbols to be analyzed: {double}, {plus}, {sqr}

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

Computed SIZE bound using CoFloCo for: double
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 2·z

(18) Obligation:

Complexity RNTS consisting of the following rules:

double(z) -{ 1 }→ 0 :|: z = 0
double(z) -{ 1 }→ 1 + (1 + double(z - 1)) :|: z - 1 >= 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
sqr(z) -{ 3 }→ plus(plus(sqr(z - 2), 1 + double(z - 2)), 1 + (1 + (1 + double(z - 2)))) :|: z - 2 >= 0
sqr(z) -{ 3 }→ plus(0, 1 + 0) :|: z = 1 + 0
sqr(z) -{ 3 }→ plus(1 + plus(sqr(z - 2), double(z - 2)), 1 + (1 + (1 + double(z - 2)))) :|: z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 3 }→ 1 + plus(plus(sqr(z - 2), 1 + double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0
sqr(z) -{ 3 }→ 1 + plus(0, 0) :|: z = 1 + 0
sqr(z) -{ 3 }→ 1 + plus(1 + plus(sqr(z - 2), double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0

Function symbols to be analyzed: {double}, {plus}, {sqr}
Previous analysis results are:

double: runtime: ?, size: O(n1) [2·z]

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

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

(20) Obligation:

Complexity RNTS consisting of the following rules:

double(z) -{ 1 }→ 0 :|: z = 0
double(z) -{ 1 }→ 1 + (1 + double(z - 1)) :|: z - 1 >= 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
sqr(z) -{ 3 }→ plus(plus(sqr(z - 2), 1 + double(z - 2)), 1 + (1 + (1 + double(z - 2)))) :|: z - 2 >= 0
sqr(z) -{ 3 }→ plus(0, 1 + 0) :|: z = 1 + 0
sqr(z) -{ 3 }→ plus(1 + plus(sqr(z - 2), double(z - 2)), 1 + (1 + (1 + double(z - 2)))) :|: z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 3 }→ 1 + plus(plus(sqr(z - 2), 1 + double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0
sqr(z) -{ 3 }→ 1 + plus(0, 0) :|: z = 1 + 0
sqr(z) -{ 3 }→ 1 + plus(1 + plus(sqr(z - 2), double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0

Function symbols to be analyzed: {plus}, {sqr}
Previous analysis results are:

double: runtime: O(n1) [1 + z], size: O(n1) [2·z]

(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:

double(z) -{ 1 }→ 0 :|: z = 0
double(z) -{ 1 + z }→ 1 + (1 + s2) :|: s2 >= 0, s2 <= 2 * (z - 1), z - 1 >= 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
sqr(z) -{ 1 + 2·z }→ plus(plus(sqr(z - 2), 1 + s), 1 + (1 + (1 + s'))) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
sqr(z) -{ 3 }→ plus(0, 1 + 0) :|: z = 1 + 0
sqr(z) -{ 1 + 2·z }→ plus(1 + plus(sqr(z - 2), s''), 1 + (1 + (1 + s1))) :|: s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 1 + 2·z }→ 1 + plus(plus(sqr(z - 2), 1 + s3), 1 + (1 + s4)) :|: s3 >= 0, s3 <= 2 * (z - 2), s4 >= 0, s4 <= 2 * (z - 2), z - 2 >= 0
sqr(z) -{ 3 }→ 1 + plus(0, 0) :|: z = 1 + 0
sqr(z) -{ 1 + 2·z }→ 1 + plus(1 + plus(sqr(z - 2), s5), 1 + (1 + s6)) :|: s5 >= 0, s5 <= 2 * (z - 2), s6 >= 0, s6 <= 2 * (z - 2), z - 2 >= 0

Function symbols to be analyzed: {plus}, {sqr}
Previous analysis results are:

double: runtime: O(n1) [1 + z], size: O(n1) [2·z]

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

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

(24) Obligation:

Complexity RNTS consisting of the following rules:

double(z) -{ 1 }→ 0 :|: z = 0
double(z) -{ 1 + z }→ 1 + (1 + s2) :|: s2 >= 0, s2 <= 2 * (z - 1), z - 1 >= 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
sqr(z) -{ 1 + 2·z }→ plus(plus(sqr(z - 2), 1 + s), 1 + (1 + (1 + s'))) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
sqr(z) -{ 3 }→ plus(0, 1 + 0) :|: z = 1 + 0
sqr(z) -{ 1 + 2·z }→ plus(1 + plus(sqr(z - 2), s''), 1 + (1 + (1 + s1))) :|: s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 1 + 2·z }→ 1 + plus(plus(sqr(z - 2), 1 + s3), 1 + (1 + s4)) :|: s3 >= 0, s3 <= 2 * (z - 2), s4 >= 0, s4 <= 2 * (z - 2), z - 2 >= 0
sqr(z) -{ 3 }→ 1 + plus(0, 0) :|: z = 1 + 0
sqr(z) -{ 1 + 2·z }→ 1 + plus(1 + plus(sqr(z - 2), s5), 1 + (1 + s6)) :|: s5 >= 0, s5 <= 2 * (z - 2), s6 >= 0, s6 <= 2 * (z - 2), z - 2 >= 0

Function symbols to be analyzed: {plus}, {sqr}
Previous analysis results are:

double: runtime: O(n1) [1 + z], size: O(n1) [2·z] plus: runtime: ?, size: O(n1) [z + z']

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

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

(26) Obligation:

Complexity RNTS consisting of the following rules:

double(z) -{ 1 }→ 0 :|: z = 0
double(z) -{ 1 + z }→ 1 + (1 + s2) :|: s2 >= 0, s2 <= 2 * (z - 1), z - 1 >= 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 1 }→ 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
sqr(z) -{ 1 + 2·z }→ plus(plus(sqr(z - 2), 1 + s), 1 + (1 + (1 + s'))) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
sqr(z) -{ 3 }→ plus(0, 1 + 0) :|: z = 1 + 0
sqr(z) -{ 1 + 2·z }→ plus(1 + plus(sqr(z - 2), s''), 1 + (1 + (1 + s1))) :|: s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 1 + 2·z }→ 1 + plus(plus(sqr(z - 2), 1 + s3), 1 + (1 + s4)) :|: s3 >= 0, s3 <= 2 * (z - 2), s4 >= 0, s4 <= 2 * (z - 2), z - 2 >= 0
sqr(z) -{ 3 }→ 1 + plus(0, 0) :|: z = 1 + 0
sqr(z) -{ 1 + 2·z }→ 1 + plus(1 + plus(sqr(z - 2), s5), 1 + (1 + s6)) :|: s5 >= 0, s5 <= 2 * (z - 2), s6 >= 0, s6 <= 2 * (z - 2), z - 2 >= 0

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

double: runtime: O(n1) [1 + z], size: O(n1) [2·z] plus: runtime: O(n1) [1 + z'], size: O(n1) [z + z']

(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:

double(z) -{ 1 }→ 0 :|: z = 0
double(z) -{ 1 + z }→ 1 + (1 + s2) :|: s2 >= 0, s2 <= 2 * (z - 1), z - 1 >= 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 1 + z' }→ 1 + s8 :|: s8 >= 0, s8 <= 1 * z + 1 * (z' - 1), z >= 0, z' - 1 >= 0
sqr(z) -{ 5 }→ s7 :|: s7 >= 0, s7 <= 1 * 0 + 1 * (1 + 0), z = 1 + 0
sqr(z) -{ 1 + 2·z }→ plus(plus(sqr(z - 2), 1 + s), 1 + (1 + (1 + s'))) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
sqr(z) -{ 1 + 2·z }→ plus(1 + plus(sqr(z - 2), s''), 1 + (1 + (1 + s1))) :|: s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 4 }→ 1 + s9 :|: s9 >= 0, s9 <= 1 * 0 + 1 * 0, z = 1 + 0
sqr(z) -{ 1 + 2·z }→ 1 + plus(plus(sqr(z - 2), 1 + s3), 1 + (1 + s4)) :|: s3 >= 0, s3 <= 2 * (z - 2), s4 >= 0, s4 <= 2 * (z - 2), z - 2 >= 0
sqr(z) -{ 1 + 2·z }→ 1 + plus(1 + plus(sqr(z - 2), s5), 1 + (1 + s6)) :|: s5 >= 0, s5 <= 2 * (z - 2), s6 >= 0, s6 <= 2 * (z - 2), z - 2 >= 0

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

double: runtime: O(n1) [1 + z], size: O(n1) [2·z] plus: runtime: O(n1) [1 + z'], size: O(n1) [z + z']

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

Computed SIZE bound using CoFloCo for: sqr
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 5 + 8·z + 4·z²

(30) Obligation:

Complexity RNTS consisting of the following rules:

double(z) -{ 1 }→ 0 :|: z = 0
double(z) -{ 1 + z }→ 1 + (1 + s2) :|: s2 >= 0, s2 <= 2 * (z - 1), z - 1 >= 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 1 + z' }→ 1 + s8 :|: s8 >= 0, s8 <= 1 * z + 1 * (z' - 1), z >= 0, z' - 1 >= 0
sqr(z) -{ 5 }→ s7 :|: s7 >= 0, s7 <= 1 * 0 + 1 * (1 + 0), z = 1 + 0
sqr(z) -{ 1 + 2·z }→ plus(plus(sqr(z - 2), 1 + s), 1 + (1 + (1 + s'))) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
sqr(z) -{ 1 + 2·z }→ plus(1 + plus(sqr(z - 2), s''), 1 + (1 + (1 + s1))) :|: s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 4 }→ 1 + s9 :|: s9 >= 0, s9 <= 1 * 0 + 1 * 0, z = 1 + 0
sqr(z) -{ 1 + 2·z }→ 1 + plus(plus(sqr(z - 2), 1 + s3), 1 + (1 + s4)) :|: s3 >= 0, s3 <= 2 * (z - 2), s4 >= 0, s4 <= 2 * (z - 2), z - 2 >= 0
sqr(z) -{ 1 + 2·z }→ 1 + plus(1 + plus(sqr(z - 2), s5), 1 + (1 + s6)) :|: s5 >= 0, s5 <= 2 * (z - 2), s6 >= 0, s6 <= 2 * (z - 2), z - 2 >= 0

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

double: runtime: O(n1) [1 + z], size: O(n1) [2·z] plus: runtime: O(n1) [1 + z'], size: O(n1) [z + z'] sqr: runtime: ?, size: O(n2) [5 + 8·z + 4·z2]

(31) IntTrsBoundProof (UPPER BOUND(ID) transformation)

Computed RUNTIME bound using KoAT for: sqr
after applying outer abstraction to obtain an ITS,
resulting in: O(n²) with polynomial bound: 10 + 24·z²

(32) Obligation:

Complexity RNTS consisting of the following rules:

double(z) -{ 1 }→ 0 :|: z = 0
double(z) -{ 1 + z }→ 1 + (1 + s2) :|: s2 >= 0, s2 <= 2 * (z - 1), z - 1 >= 0
plus(z, z') -{ 1 }→ z :|: z >= 0, z' = 0
plus(z, z') -{ 1 + z' }→ 1 + s8 :|: s8 >= 0, s8 <= 1 * z + 1 * (z' - 1), z >= 0, z' - 1 >= 0
sqr(z) -{ 5 }→ s7 :|: s7 >= 0, s7 <= 1 * 0 + 1 * (1 + 0), z = 1 + 0
sqr(z) -{ 1 + 2·z }→ plus(plus(sqr(z - 2), 1 + s), 1 + (1 + (1 + s'))) :|: s >= 0, s <= 2 * (z - 2), s' >= 0, s' <= 2 * (z - 2), z - 2 >= 0
sqr(z) -{ 1 + 2·z }→ plus(1 + plus(sqr(z - 2), s''), 1 + (1 + (1 + s1))) :|: s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0
sqr(z) -{ 1 }→ 0 :|: z = 0
sqr(z) -{ 4 }→ 1 + s9 :|: s9 >= 0, s9 <= 1 * 0 + 1 * 0, z = 1 + 0
sqr(z) -{ 1 + 2·z }→ 1 + plus(plus(sqr(z - 2), 1 + s3), 1 + (1 + s4)) :|: s3 >= 0, s3 <= 2 * (z - 2), s4 >= 0, s4 <= 2 * (z - 2), z - 2 >= 0
sqr(z) -{ 1 + 2·z }→ 1 + plus(1 + plus(sqr(z - 2), s5), 1 + (1 + s6)) :|: s5 >= 0, s5 <= 2 * (z - 2), s6 >= 0, s6 <= 2 * (z - 2), z - 2 >= 0

Function symbols to be analyzed:
Previous analysis results are:

double: runtime: O(n1) [1 + z], size: O(n1) [2·z] plus: runtime: O(n1) [1 + z'], size: O(n1) [z + z'] sqr: runtime: O(n2) [10 + 24·z2], size: O(n2) [5 + 8·z + 4·z2]

(33) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(34) BOUNDS(1, n^2)