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:

walk#1(Nil) → walk_xs
walk#1(Cons(x4, x3)) → comp_f_g(walk#1(x3), walk_xs_3(x4))
comp_f_g#1(comp_f_g(x7, x9), walk_xs_3(x8), x12) → comp_f_g#1(x7, x9, Cons(x8, x12))
comp_f_g#1(walk_xs, walk_xs_3(x8), x12) → Cons(x8, x12)
main(Nil) → Nil
main(Cons(x4, x5)) → comp_f_g#1(walk#1(x5), walk_xs_3(x4), Nil)

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:

walk#1(Nil) → walk_xs [1]
walk#1(Cons(x4, x3)) → comp_f_g(walk#1(x3), walk_xs_3(x4)) [1]
comp_f_g#1(comp_f_g(x7, x9), walk_xs_3(x8), x12) → comp_f_g#1(x7, x9, Cons(x8, x12)) [1]
comp_f_g#1(walk_xs, walk_xs_3(x8), x12) → Cons(x8, x12) [1]
main(Nil) → Nil [1]
main(Cons(x4, x5)) → comp_f_g#1(walk#1(x5), walk_xs_3(x4), Nil) [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:

walk#1(Nil) → walk_xs [1]
walk#1(Cons(x4, x3)) → comp_f_g(walk#1(x3), walk_xs_3(x4)) [1]
comp_f_g#1(comp_f_g(x7, x9), walk_xs_3(x8), x12) → comp_f_g#1(x7, x9, Cons(x8, x12)) [1]
comp_f_g#1(walk_xs, walk_xs_3(x8), x12) → Cons(x8, x12) [1]
main(Nil) → Nil [1]
main(Cons(x4, x5)) → comp_f_g#1(walk#1(x5), walk_xs_3(x4), Nil) [1]

The TRS has the following type information:

walk#1 :: Nil:Cons → walk_xs:comp_f_g Nil :: Nil:Cons walk_xs :: walk_xs:comp_f_g Cons :: a → Nil:Cons → Nil:Cons comp_f_g :: walk_xs:comp_f_g → walk_xs_3 → walk_xs:comp_f_g walk_xs_3 :: a → walk_xs_3 comp_f_g#1 :: walk_xs:comp_f_g → walk_xs_3 → Nil:Cons → Nil:Cons main :: Nil:Cons → Nil:Cons

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:

comp_f_g#1
main

(c) The following functions are completely defined:

walk#1

Due to the following rules being added:
none

And the following fresh constants:

const, const1

(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: walk#1(Nil) → walk_xs [1] walk#1(Cons(x4, x3)) → comp_f_g(walk#1(x3), walk_xs_3(x4)) [1] comp_f_g#1(comp_f_g(x7, x9), walk_xs_3(x8), x12) → comp_f_g#1(x7, x9, Cons(x8, x12)) [1] comp_f_g#1(walk_xs, walk_xs_3(x8), x12) → Cons(x8, x12) [1] main(Nil) → Nil [1] main(Cons(x4, x5)) → comp_f_g#1(walk#1(x5), walk_xs_3(x4), Nil) [1] The TRS has the following type information: walk#1 :: Nil:Cons → walk_xs:comp_f_g Nil :: Nil:Cons walk_xs :: walk_xs:comp_f_g Cons :: a → Nil:Cons → Nil:Cons comp_f_g :: walk_xs:comp_f_g → walk_xs_3 → walk_xs:comp_f_g walk_xs_3 :: a → walk_xs_3 comp_f_g#1 :: walk_xs:comp_f_g → walk_xs_3 → Nil:Cons → Nil:Cons main :: Nil:Cons → Nil:Cons const :: a const1 :: walk_xs_3 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: walk#1(Nil) → walk_xs [1] walk#1(Cons(x4, x3)) → comp_f_g(walk#1(x3), walk_xs_3(x4)) [1] comp_f_g#1(comp_f_g(x7, x9), walk_xs_3(x8), x12) → comp_f_g#1(x7, x9, Cons(x8, x12)) [1] comp_f_g#1(walk_xs, walk_xs_3(x8), x12) → Cons(x8, x12) [1] main(Nil) → Nil [1] main(Cons(x4, Nil)) → comp_f_g#1(walk_xs, walk_xs_3(x4), Nil) [2] main(Cons(x4, Cons(x4', x3'))) → comp_f_g#1(comp_f_g(walk#1(x3'), walk_xs_3(x4')), walk_xs_3(x4), Nil) [2] The TRS has the following type information: walk#1 :: Nil:Cons → walk_xs:comp_f_g Nil :: Nil:Cons walk_xs :: walk_xs:comp_f_g Cons :: a → Nil:Cons → Nil:Cons comp_f_g :: walk_xs:comp_f_g → walk_xs_3 → walk_xs:comp_f_g walk_xs_3 :: a → walk_xs_3 comp_f_g#1 :: walk_xs:comp_f_g → walk_xs_3 → Nil:Cons → Nil:Cons main :: Nil:Cons → Nil:Cons const :: a const1 :: walk_xs_3 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:

Nil => 0
walk_xs => 0
const => 0
const1 => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

comp_f_g#1(z, z', z'') -{ 1 }→ comp_f_g#1(x7, x9, 1 + x8 + x12) :|: z' = 1 + x8, x8 >= 0, z'' = x12, z = 1 + x7 + x9, x7 >= 0, x12 >= 0, x9 >= 0
comp_f_g#1(z, z', z'') -{ 1 }→ 1 + x8 + x12 :|: z' = 1 + x8, x8 >= 0, z'' = x12, x12 >= 0, z = 0
main(z) -{ 2 }→ comp_f_g#1(0, 1 + x4, 0) :|: z = 1 + x4 + 0, x4 >= 0
main(z) -{ 2 }→ comp_f_g#1(1 + walk#1(x3') + (1 + x4'), 1 + x4, 0) :|: z = 1 + x4 + (1 + x4' + x3'), x4 >= 0, x4' >= 0, x3' >= 0
main(z) -{ 1 }→ 0 :|: z = 0
walk#1(z) -{ 1 }→ 0 :|: z = 0
walk#1(z) -{ 1 }→ 1 + walk#1(x3) + (1 + x4) :|: x4 >= 0, z = 1 + x4 + x3, x3 >= 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:

comp_f_g#1(z, z', z'') -{ 1 }→ comp_f_g#1(x7, x9, 1 + (z' - 1) + z'') :|: z' - 1 >= 0, z = 1 + x7 + x9, x7 >= 0, z'' >= 0, x9 >= 0
comp_f_g#1(z, z', z'') -{ 1 }→ 1 + (z' - 1) + z'' :|: z' - 1 >= 0, z'' >= 0, z = 0
main(z) -{ 2 }→ comp_f_g#1(0, 1 + (z - 1), 0) :|: z - 1 >= 0
main(z) -{ 2 }→ comp_f_g#1(1 + walk#1(x3') + (1 + x4'), 1 + x4, 0) :|: z = 1 + x4 + (1 + x4' + x3'), x4 >= 0, x4' >= 0, x3' >= 0
main(z) -{ 1 }→ 0 :|: z = 0
walk#1(z) -{ 1 }→ 0 :|: z = 0
walk#1(z) -{ 1 }→ 1 + walk#1(x3) + (1 + x4) :|: x4 >= 0, z = 1 + x4 + x3, x3 >= 0

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

Found the following analysis order by SCC decomposition:

{ walk#1 } { comp_f_g#1 } { main }

(14) Obligation:

Complexity RNTS consisting of the following rules:

comp_f_g#1(z, z', z'') -{ 1 }→ comp_f_g#1(x7, x9, 1 + (z' - 1) + z'') :|: z' - 1 >= 0, z = 1 + x7 + x9, x7 >= 0, z'' >= 0, x9 >= 0
comp_f_g#1(z, z', z'') -{ 1 }→ 1 + (z' - 1) + z'' :|: z' - 1 >= 0, z'' >= 0, z = 0
main(z) -{ 2 }→ comp_f_g#1(0, 1 + (z - 1), 0) :|: z - 1 >= 0
main(z) -{ 2 }→ comp_f_g#1(1 + walk#1(x3') + (1 + x4'), 1 + x4, 0) :|: z = 1 + x4 + (1 + x4' + x3'), x4 >= 0, x4' >= 0, x3' >= 0
main(z) -{ 1 }→ 0 :|: z = 0
walk#1(z) -{ 1 }→ 0 :|: z = 0
walk#1(z) -{ 1 }→ 1 + walk#1(x3) + (1 + x4) :|: x4 >= 0, z = 1 + x4 + x3, x3 >= 0

Function symbols to be analyzed: {walk#1}, {comp_f_g#1}, {main}

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

Computed SIZE bound using CoFloCo for: walk#1
after applying outer abstraction to obtain an ITS,
resulting in: O(n¹) with polynomial bound: 2·z

(16) Obligation:

Complexity RNTS consisting of the following rules:

comp_f_g#1(z, z', z'') -{ 1 }→ comp_f_g#1(x7, x9, 1 + (z' - 1) + z'') :|: z' - 1 >= 0, z = 1 + x7 + x9, x7 >= 0, z'' >= 0, x9 >= 0
comp_f_g#1(z, z', z'') -{ 1 }→ 1 + (z' - 1) + z'' :|: z' - 1 >= 0, z'' >= 0, z = 0
main(z) -{ 2 }→ comp_f_g#1(0, 1 + (z - 1), 0) :|: z - 1 >= 0
main(z) -{ 2 }→ comp_f_g#1(1 + walk#1(x3') + (1 + x4'), 1 + x4, 0) :|: z = 1 + x4 + (1 + x4' + x3'), x4 >= 0, x4' >= 0, x3' >= 0
main(z) -{ 1 }→ 0 :|: z = 0
walk#1(z) -{ 1 }→ 0 :|: z = 0
walk#1(z) -{ 1 }→ 1 + walk#1(x3) + (1 + x4) :|: x4 >= 0, z = 1 + x4 + x3, x3 >= 0

Function symbols to be analyzed: {walk#1}, {comp_f_g#1}, {main}
Previous analysis results are:

walk#1: runtime: ?, size: O(n1) [2·z]

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

Computed RUNTIME bound using CoFloCo for: walk#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:

comp_f_g#1(z, z', z'') -{ 1 }→ comp_f_g#1(x7, x9, 1 + (z' - 1) + z'') :|: z' - 1 >= 0, z = 1 + x7 + x9, x7 >= 0, z'' >= 0, x9 >= 0
comp_f_g#1(z, z', z'') -{ 1 }→ 1 + (z' - 1) + z'' :|: z' - 1 >= 0, z'' >= 0, z = 0
main(z) -{ 2 }→ comp_f_g#1(0, 1 + (z - 1), 0) :|: z - 1 >= 0
main(z) -{ 2 }→ comp_f_g#1(1 + walk#1(x3') + (1 + x4'), 1 + x4, 0) :|: z = 1 + x4 + (1 + x4' + x3'), x4 >= 0, x4' >= 0, x3' >= 0
main(z) -{ 1 }→ 0 :|: z = 0
walk#1(z) -{ 1 }→ 0 :|: z = 0
walk#1(z) -{ 1 }→ 1 + walk#1(x3) + (1 + x4) :|: x4 >= 0, z = 1 + x4 + x3, x3 >= 0

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

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

comp_f_g#1(z, z', z'') -{ 1 }→ comp_f_g#1(x7, x9, 1 + (z' - 1) + z'') :|: z' - 1 >= 0, z = 1 + x7 + x9, x7 >= 0, z'' >= 0, x9 >= 0
comp_f_g#1(z, z', z'') -{ 1 }→ 1 + (z' - 1) + z'' :|: z' - 1 >= 0, z'' >= 0, z = 0
main(z) -{ 2 }→ comp_f_g#1(0, 1 + (z - 1), 0) :|: z - 1 >= 0
main(z) -{ 3 + x3' }→ comp_f_g#1(1 + s' + (1 + x4'), 1 + x4, 0) :|: s' >= 0, s' <= 2 * x3', z = 1 + x4 + (1 + x4' + x3'), x4 >= 0, x4' >= 0, x3' >= 0
main(z) -{ 1 }→ 0 :|: z = 0
walk#1(z) -{ 1 }→ 0 :|: z = 0
walk#1(z) -{ 2 + x3 }→ 1 + s + (1 + x4) :|: s >= 0, s <= 2 * x3, x4 >= 0, z = 1 + x4 + x3, x3 >= 0

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

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

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

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

(22) Obligation:

Complexity RNTS consisting of the following rules:

comp_f_g#1(z, z', z'') -{ 1 }→ comp_f_g#1(x7, x9, 1 + (z' - 1) + z'') :|: z' - 1 >= 0, z = 1 + x7 + x9, x7 >= 0, z'' >= 0, x9 >= 0
comp_f_g#1(z, z', z'') -{ 1 }→ 1 + (z' - 1) + z'' :|: z' - 1 >= 0, z'' >= 0, z = 0
main(z) -{ 2 }→ comp_f_g#1(0, 1 + (z - 1), 0) :|: z - 1 >= 0
main(z) -{ 3 + x3' }→ comp_f_g#1(1 + s' + (1 + x4'), 1 + x4, 0) :|: s' >= 0, s' <= 2 * x3', z = 1 + x4 + (1 + x4' + x3'), x4 >= 0, x4' >= 0, x3' >= 0
main(z) -{ 1 }→ 0 :|: z = 0
walk#1(z) -{ 1 }→ 0 :|: z = 0
walk#1(z) -{ 2 + x3 }→ 1 + s + (1 + x4) :|: s >= 0, s <= 2 * x3, x4 >= 0, z = 1 + x4 + x3, x3 >= 0

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

walk#1: runtime: O(n1) [1 + z], size: O(n1) [2·z] comp_f_g#1: runtime: ?, size: O(n1) [z + z' + z'']

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

Computed RUNTIME bound using PUBS for: comp_f_g#1
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:

comp_f_g#1(z, z', z'') -{ 1 }→ comp_f_g#1(x7, x9, 1 + (z' - 1) + z'') :|: z' - 1 >= 0, z = 1 + x7 + x9, x7 >= 0, z'' >= 0, x9 >= 0
comp_f_g#1(z, z', z'') -{ 1 }→ 1 + (z' - 1) + z'' :|: z' - 1 >= 0, z'' >= 0, z = 0
main(z) -{ 2 }→ comp_f_g#1(0, 1 + (z - 1), 0) :|: z - 1 >= 0
main(z) -{ 3 + x3' }→ comp_f_g#1(1 + s' + (1 + x4'), 1 + x4, 0) :|: s' >= 0, s' <= 2 * x3', z = 1 + x4 + (1 + x4' + x3'), x4 >= 0, x4' >= 0, x3' >= 0
main(z) -{ 1 }→ 0 :|: z = 0
walk#1(z) -{ 1 }→ 0 :|: z = 0
walk#1(z) -{ 2 + x3 }→ 1 + s + (1 + x4) :|: s >= 0, s <= 2 * x3, x4 >= 0, z = 1 + x4 + x3, x3 >= 0

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

walk#1: runtime: O(n1) [1 + z], size: O(n1) [2·z] comp_f_g#1: runtime: O(n1) [1 + z], size: O(n1) [z + z' + 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:

comp_f_g#1(z, z', z'') -{ 2 + x7 }→ s'' :|: s'' >= 0, s'' <= 1 * x7 + 1 * x9 + 1 * (1 + (z' - 1) + z''), z' - 1 >= 0, z = 1 + x7 + x9, x7 >= 0, z'' >= 0, x9 >= 0
comp_f_g#1(z, z', z'') -{ 1 }→ 1 + (z' - 1) + z'' :|: z' - 1 >= 0, z'' >= 0, z = 0
main(z) -{ 3 }→ s1 :|: s1 >= 0, s1 <= 1 * 0 + 1 * (1 + (z - 1)) + 1 * 0, z - 1 >= 0
main(z) -{ 6 + s' + x3' + x4' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + s' + (1 + x4')) + 1 * (1 + x4) + 1 * 0, s' >= 0, s' <= 2 * x3', z = 1 + x4 + (1 + x4' + x3'), x4 >= 0, x4' >= 0, x3' >= 0
main(z) -{ 1 }→ 0 :|: z = 0
walk#1(z) -{ 1 }→ 0 :|: z = 0
walk#1(z) -{ 2 + x3 }→ 1 + s + (1 + x4) :|: s >= 0, s <= 2 * x3, x4 >= 0, z = 1 + x4 + x3, x3 >= 0

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

walk#1: runtime: O(n1) [1 + z], size: O(n1) [2·z] comp_f_g#1: runtime: O(n1) [1 + z], size: O(n1) [z + z' + 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: 2·z

(28) Obligation:

Complexity RNTS consisting of the following rules:

comp_f_g#1(z, z', z'') -{ 2 + x7 }→ s'' :|: s'' >= 0, s'' <= 1 * x7 + 1 * x9 + 1 * (1 + (z' - 1) + z''), z' - 1 >= 0, z = 1 + x7 + x9, x7 >= 0, z'' >= 0, x9 >= 0
comp_f_g#1(z, z', z'') -{ 1 }→ 1 + (z' - 1) + z'' :|: z' - 1 >= 0, z'' >= 0, z = 0
main(z) -{ 3 }→ s1 :|: s1 >= 0, s1 <= 1 * 0 + 1 * (1 + (z - 1)) + 1 * 0, z - 1 >= 0
main(z) -{ 6 + s' + x3' + x4' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + s' + (1 + x4')) + 1 * (1 + x4) + 1 * 0, s' >= 0, s' <= 2 * x3', z = 1 + x4 + (1 + x4' + x3'), x4 >= 0, x4' >= 0, x3' >= 0
main(z) -{ 1 }→ 0 :|: z = 0
walk#1(z) -{ 1 }→ 0 :|: z = 0
walk#1(z) -{ 2 + x3 }→ 1 + s + (1 + x4) :|: s >= 0, s <= 2 * x3, x4 >= 0, z = 1 + x4 + x3, x3 >= 0

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

walk#1: runtime: O(n1) [1 + z], size: O(n1) [2·z] comp_f_g#1: runtime: O(n1) [1 + z], size: O(n1) [z + z' + z''] main: runtime: ?, size: O(n1) [2·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 + 3·z

(30) Obligation:

Complexity RNTS consisting of the following rules:

comp_f_g#1(z, z', z'') -{ 2 + x7 }→ s'' :|: s'' >= 0, s'' <= 1 * x7 + 1 * x9 + 1 * (1 + (z' - 1) + z''), z' - 1 >= 0, z = 1 + x7 + x9, x7 >= 0, z'' >= 0, x9 >= 0
comp_f_g#1(z, z', z'') -{ 1 }→ 1 + (z' - 1) + z'' :|: z' - 1 >= 0, z'' >= 0, z = 0
main(z) -{ 3 }→ s1 :|: s1 >= 0, s1 <= 1 * 0 + 1 * (1 + (z - 1)) + 1 * 0, z - 1 >= 0
main(z) -{ 6 + s' + x3' + x4' }→ s2 :|: s2 >= 0, s2 <= 1 * (1 + s' + (1 + x4')) + 1 * (1 + x4) + 1 * 0, s' >= 0, s' <= 2 * x3', z = 1 + x4 + (1 + x4' + x3'), x4 >= 0, x4' >= 0, x3' >= 0
main(z) -{ 1 }→ 0 :|: z = 0
walk#1(z) -{ 1 }→ 0 :|: z = 0
walk#1(z) -{ 2 + x3 }→ 1 + s + (1 + x4) :|: s >= 0, s <= 2 * x3, x4 >= 0, z = 1 + x4 + x3, x3 >= 0

Function symbols to be analyzed:
Previous analysis results are:

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

(31) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(32) BOUNDS(1, n^1)