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:

compS_f#1(compS_f(x2), x1) → compS_f#1(x2, S(x1))
compS_f#1(id, x3) → S(x3)
iter#3(0) → id
iter#3(S(x6)) → compS_f(iter#3(x6))
main(0) → 0
main(S(x9)) → compS_f#1(iter#3(x9), 0)

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:

compS_f#1(compS_f(x2), x1) → compS_f#1(x2, S(x1)) [1]
compS_f#1(id, x3) → S(x3) [1]
iter#3(0) → id [1]
iter#3(S(x6)) → compS_f(iter#3(x6)) [1]
main(0) → 0 [1]
main(S(x9)) → compS_f#1(iter#3(x9), 0) [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:

compS_f#1(compS_f(x2), x1) → compS_f#1(x2, S(x1)) [1]
compS_f#1(id, x3) → S(x3) [1]
iter#3(0) → id [1]
iter#3(S(x6)) → compS_f(iter#3(x6)) [1]
main(0) → 0 [1]
main(S(x9)) → compS_f#1(iter#3(x9), 0) [1]

The TRS has the following type information:

compS_f#1 :: compS_f:id → S:0 → S:0 compS_f :: compS_f:id → compS_f:id S :: S:0 → S:0 id :: compS_f:id iter#3 :: S:0 → compS_f:id 0 :: S:0 main :: S:0 → S:0

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:

compS_f#1
main

(c) The following functions are completely defined:

iter#3

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: compS_f#1(compS_f(x2), x1) → compS_f#1(x2, S(x1)) [1] compS_f#1(id, x3) → S(x3) [1] iter#3(0) → id [1] iter#3(S(x6)) → compS_f(iter#3(x6)) [1] main(0) → 0 [1] main(S(x9)) → compS_f#1(iter#3(x9), 0) [1] The TRS has the following type information: compS_f#1 :: compS_f:id → S:0 → S:0 compS_f :: compS_f:id → compS_f:id S :: S:0 → S:0 id :: compS_f:id iter#3 :: S:0 → compS_f:id 0 :: S:0 main :: S:0 → S:0 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: compS_f#1(compS_f(x2), x1) → compS_f#1(x2, S(x1)) [1] compS_f#1(id, x3) → S(x3) [1] iter#3(0) → id [1] iter#3(S(x6)) → compS_f(iter#3(x6)) [1] main(0) → 0 [1] main(S(0)) → compS_f#1(id, 0) [2] main(S(S(x6'))) → compS_f#1(compS_f(iter#3(x6')), 0) [2] The TRS has the following type information: compS_f#1 :: compS_f:id → S:0 → S:0 compS_f :: compS_f:id → compS_f:id S :: S:0 → S:0 id :: compS_f:id iter#3 :: S:0 → compS_f:id 0 :: S:0 main :: S:0 → S:0 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:

id => 0
0 => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

compS_f#1(z, z') -{ 1 }→ compS_f#1(x2, 1 + x1) :|: x1 >= 0, z = 1 + x2, z' = x1, x2 >= 0
compS_f#1(z, z') -{ 1 }→ 1 + x3 :|: z' = x3, z = 0, x3 >= 0
iter#3(z) -{ 1 }→ 0 :|: z = 0
iter#3(z) -{ 1 }→ 1 + iter#3(x6) :|: z = 1 + x6, x6 >= 0
main(z) -{ 2 }→ compS_f#1(0, 0) :|: z = 1 + 0
main(z) -{ 2 }→ compS_f#1(1 + iter#3(x6'), 0) :|: x6' >= 0, z = 1 + (1 + x6')
main(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:

compS_f#1(z, z') -{ 1 }→ compS_f#1(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
compS_f#1(z, z') -{ 1 }→ 1 + z' :|: z = 0, z' >= 0
iter#3(z) -{ 1 }→ 0 :|: z = 0
iter#3(z) -{ 1 }→ 1 + iter#3(z - 1) :|: z - 1 >= 0
main(z) -{ 2 }→ compS_f#1(0, 0) :|: z = 1 + 0
main(z) -{ 2 }→ compS_f#1(1 + iter#3(z - 2), 0) :|: z - 2 >= 0
main(z) -{ 1 }→ 0 :|: z = 0

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

Found the following analysis order by SCC decomposition:

{ compS_f#1 } { iter#3 } { main }

(14) Obligation:

Complexity RNTS consisting of the following rules:

compS_f#1(z, z') -{ 1 }→ compS_f#1(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
compS_f#1(z, z') -{ 1 }→ 1 + z' :|: z = 0, z' >= 0
iter#3(z) -{ 1 }→ 0 :|: z = 0
iter#3(z) -{ 1 }→ 1 + iter#3(z - 1) :|: z - 1 >= 0
main(z) -{ 2 }→ compS_f#1(0, 0) :|: z = 1 + 0
main(z) -{ 2 }→ compS_f#1(1 + iter#3(z - 2), 0) :|: z - 2 >= 0
main(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {compS_f#1}, {iter#3}, {main}

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

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

(16) Obligation:

Complexity RNTS consisting of the following rules:

compS_f#1(z, z') -{ 1 }→ compS_f#1(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
compS_f#1(z, z') -{ 1 }→ 1 + z' :|: z = 0, z' >= 0
iter#3(z) -{ 1 }→ 0 :|: z = 0
iter#3(z) -{ 1 }→ 1 + iter#3(z - 1) :|: z - 1 >= 0
main(z) -{ 2 }→ compS_f#1(0, 0) :|: z = 1 + 0
main(z) -{ 2 }→ compS_f#1(1 + iter#3(z - 2), 0) :|: z - 2 >= 0
main(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {compS_f#1}, {iter#3}, {main}
Previous analysis results are:

compS_f#1: runtime: ?, size: O(n1) [1 + z + z']

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

Computed RUNTIME bound using CoFloCo for: compS_f#1
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:

compS_f#1(z, z') -{ 1 }→ compS_f#1(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
compS_f#1(z, z') -{ 1 }→ 1 + z' :|: z = 0, z' >= 0
iter#3(z) -{ 1 }→ 0 :|: z = 0
iter#3(z) -{ 1 }→ 1 + iter#3(z - 1) :|: z - 1 >= 0
main(z) -{ 2 }→ compS_f#1(0, 0) :|: z = 1 + 0
main(z) -{ 2 }→ compS_f#1(1 + iter#3(z - 2), 0) :|: z - 2 >= 0
main(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {iter#3}, {main}
Previous analysis results are:

compS_f#1: runtime: O(n1) [1 + z], size: O(n1) [1 + z + 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:

compS_f#1(z, z') -{ 1 + z }→ s :|: s >= 0, s <= 1 * (z - 1) + 1 * (1 + z') + 1, z' >= 0, z - 1 >= 0
compS_f#1(z, z') -{ 1 }→ 1 + z' :|: z = 0, z' >= 0
iter#3(z) -{ 1 }→ 0 :|: z = 0
iter#3(z) -{ 1 }→ 1 + iter#3(z - 1) :|: z - 1 >= 0
main(z) -{ 3 }→ s' :|: s' >= 0, s' <= 1 * 0 + 1 * 0 + 1, z = 1 + 0
main(z) -{ 2 }→ compS_f#1(1 + iter#3(z - 2), 0) :|: z - 2 >= 0
main(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {iter#3}, {main}
Previous analysis results are:

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

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

Computed SIZE bound using CoFloCo for: iter#3
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:

compS_f#1(z, z') -{ 1 + z }→ s :|: s >= 0, s <= 1 * (z - 1) + 1 * (1 + z') + 1, z' >= 0, z - 1 >= 0
compS_f#1(z, z') -{ 1 }→ 1 + z' :|: z = 0, z' >= 0
iter#3(z) -{ 1 }→ 0 :|: z = 0
iter#3(z) -{ 1 }→ 1 + iter#3(z - 1) :|: z - 1 >= 0
main(z) -{ 3 }→ s' :|: s' >= 0, s' <= 1 * 0 + 1 * 0 + 1, z = 1 + 0
main(z) -{ 2 }→ compS_f#1(1 + iter#3(z - 2), 0) :|: z - 2 >= 0
main(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed: {iter#3}, {main}
Previous analysis results are:

compS_f#1: runtime: O(n1) [1 + z], size: O(n1) [1 + z + z'] iter#3: runtime: ?, size: O(n1) [z]

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

Computed RUNTIME bound using CoFloCo for: iter#3
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:

compS_f#1(z, z') -{ 1 + z }→ s :|: s >= 0, s <= 1 * (z - 1) + 1 * (1 + z') + 1, z' >= 0, z - 1 >= 0
compS_f#1(z, z') -{ 1 }→ 1 + z' :|: z = 0, z' >= 0
iter#3(z) -{ 1 }→ 0 :|: z = 0
iter#3(z) -{ 1 }→ 1 + iter#3(z - 1) :|: z - 1 >= 0
main(z) -{ 3 }→ s' :|: s' >= 0, s' <= 1 * 0 + 1 * 0 + 1, z = 1 + 0
main(z) -{ 2 }→ compS_f#1(1 + iter#3(z - 2), 0) :|: z - 2 >= 0
main(z) -{ 1 }→ 0 :|: z = 0

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

compS_f#1: runtime: O(n1) [1 + z], size: O(n1) [1 + z + z'] iter#3: runtime: O(n1) [1 + z], 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:

compS_f#1(z, z') -{ 1 + z }→ s :|: s >= 0, s <= 1 * (z - 1) + 1 * (1 + z') + 1, z' >= 0, z - 1 >= 0
compS_f#1(z, z') -{ 1 }→ 1 + z' :|: z = 0, z' >= 0
iter#3(z) -{ 1 }→ 0 :|: z = 0
iter#3(z) -{ 1 + z }→ 1 + s'' :|: s'' >= 0, s'' <= 1 * (z - 1), z - 1 >= 0
main(z) -{ 3 }→ s' :|: s' >= 0, s' <= 1 * 0 + 1 * 0 + 1, z = 1 + 0
main(z) -{ 3 + s1 + z }→ s2 :|: s1 >= 0, s1 <= 1 * (z - 2), s2 >= 0, s2 <= 1 * (1 + s1) + 1 * 0 + 1, z - 2 >= 0
main(z) -{ 1 }→ 0 :|: z = 0

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

compS_f#1: runtime: O(n1) [1 + z], size: O(n1) [1 + z + z'] iter#3: runtime: O(n1) [1 + z], size: O(n1) [z]

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

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

(28) Obligation:

Complexity RNTS consisting of the following rules:

compS_f#1(z, z') -{ 1 + z }→ s :|: s >= 0, s <= 1 * (z - 1) + 1 * (1 + z') + 1, z' >= 0, z - 1 >= 0
compS_f#1(z, z') -{ 1 }→ 1 + z' :|: z = 0, z' >= 0
iter#3(z) -{ 1 }→ 0 :|: z = 0
iter#3(z) -{ 1 + z }→ 1 + s'' :|: s'' >= 0, s'' <= 1 * (z - 1), z - 1 >= 0
main(z) -{ 3 }→ s' :|: s' >= 0, s' <= 1 * 0 + 1 * 0 + 1, z = 1 + 0
main(z) -{ 3 + s1 + z }→ s2 :|: s1 >= 0, s1 <= 1 * (z - 2), s2 >= 0, s2 <= 1 * (1 + s1) + 1 * 0 + 1, z - 2 >= 0
main(z) -{ 1 }→ 0 :|: z = 0

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

compS_f#1: runtime: O(n1) [1 + z], size: O(n1) [1 + z + z'] iter#3: runtime: O(n1) [1 + z], size: O(n1) [z] main: runtime: ?, size: O(n1) [z]

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

Computed RUNTIME bound using CoFloCo for: main
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 3 + 2·z

(30) Obligation:

Complexity RNTS consisting of the following rules:

compS_f#1(z, z') -{ 1 + z }→ s :|: s >= 0, s <= 1 * (z - 1) + 1 * (1 + z') + 1, z' >= 0, z - 1 >= 0
compS_f#1(z, z') -{ 1 }→ 1 + z' :|: z = 0, z' >= 0
iter#3(z) -{ 1 }→ 0 :|: z = 0
iter#3(z) -{ 1 + z }→ 1 + s'' :|: s'' >= 0, s'' <= 1 * (z - 1), z - 1 >= 0
main(z) -{ 3 }→ s' :|: s' >= 0, s' <= 1 * 0 + 1 * 0 + 1, z = 1 + 0
main(z) -{ 3 + s1 + z }→ s2 :|: s1 >= 0, s1 <= 1 * (z - 2), s2 >= 0, s2 <= 1 * (1 + s1) + 1 * 0 + 1, z - 2 >= 0
main(z) -{ 1 }→ 0 :|: z = 0

Function symbols to be analyzed:
Previous analysis results are:

compS_f#1: runtime: O(n1) [1 + z], size: O(n1) [1 + z + z'] iter#3: runtime: O(n1) [1 + z], size: O(n1) [z] main: runtime: O(n1) [3 + 2·z], size: O(n1) [z]

(31) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(32) BOUNDS(1, n^1)