WORST_CASE(?,O(n^3))
proof of input_yLzoglcDlh.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxWeightedTrs
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxTypedWeightedTrs
(5) CompletionProof [UPPER BOUND(ID), 0 ms]
(6) CpxTypedWeightedCompleteTrs
(7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTypedWeightedCompleteTrs
(9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(10) CpxRNTS
(11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxRNTS
(13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxRNTS
(15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(16) CpxRNTS
(17) IntTrsBoundProof [UPPER BOUND(ID), 391 ms]
(18) CpxRNTS
(19) IntTrsBoundProof [UPPER BOUND(ID), 98 ms]
(20) CpxRNTS
(21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(22) CpxRNTS
(23) IntTrsBoundProof [UPPER BOUND(ID), 420 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 16 ms]
(26) CpxRNTS
(27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(28) CpxRNTS
(29) IntTrsBoundProof [UPPER BOUND(ID), 614 ms]
(30) CpxRNTS
(31) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
(32) CpxRNTS
(33) FinalProof [FINISHED, 0 ms]
(34) BOUNDS(1, n^3)


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   mul0(Cons(x, xs), y) -> add0(mul0(xs, y), y)
   add0(Cons(x, xs), y) -> add0(xs, Cons(S, y))
   mul0(Nil, y) -> Nil
   add0(Nil, y) -> y
   goal(xs, ys) -> mul0(xs, ys)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   mul0(Cons(x, xs), y) -> add0(mul0(xs, y), y) [1]
   add0(Cons(x, xs), y) -> add0(xs, Cons(S, y)) [1]
   mul0(Nil, y) -> Nil [1]
   add0(Nil, y) -> y [1]
   goal(xs, ys) -> mul0(xs, ys) [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   mul0(Cons(x, xs), y) -> add0(mul0(xs, y), y) [1]
   add0(Cons(x, xs), y) -> add0(xs, Cons(S, y)) [1]
   mul0(Nil, y) -> Nil [1]
   add0(Nil, y) -> y [1]
   goal(xs, ys) -> mul0(xs, ys) [1]

The TRS has the following type information:
mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S -> Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
S :: S
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

Rewrite Strategy: INNERMOST
----------------------------------------

(5) CompletionProof (UPPER BOUND(ID))
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:

   goal_2

(c) The following functions are completely defined:

   mul0_2
   add0_2

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:

   mul0(Cons(x, xs), y) -> add0(mul0(xs, y), y) [1]
   add0(Cons(x, xs), y) -> add0(xs, Cons(S, y)) [1]
   mul0(Nil, y) -> Nil [1]
   add0(Nil, y) -> y [1]
   goal(xs, ys) -> mul0(xs, ys) [1]

The TRS has the following type information:
mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S -> Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
S :: S
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

Rewrite Strategy: INNERMOST
----------------------------------------

(7) NarrowingProof (BOTH BOUNDS(ID, ID))
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:

   mul0(Cons(x, Cons(x', xs')), y) -> add0(add0(mul0(xs', y), y), y) [2]
   mul0(Cons(x, Nil), y) -> add0(Nil, y) [2]
   add0(Cons(x, xs), y) -> add0(xs, Cons(S, y)) [1]
   mul0(Nil, y) -> Nil [1]
   add0(Nil, y) -> y [1]
   goal(xs, ys) -> mul0(xs, ys) [1]

The TRS has the following type information:
mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: S -> Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
S :: S
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

Rewrite Strategy: INNERMOST
----------------------------------------

(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   S => 0
   Nil => 0

----------------------------------------

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   add0(z, z') -{ 1 }-> add0(xs, 1 + 0 + y) :|: z = 1 + x + xs, xs >= 0, x >= 0, y >= 0, z' = y
   goal(z, z') -{ 1 }-> mul0(xs, ys) :|: xs >= 0, z = xs, z' = ys, ys >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(xs', y), y), y) :|: x >= 0, x' >= 0, xs' >= 0, y >= 0, z = 1 + x + (1 + x' + xs'), z' = y
   mul0(z, z') -{ 2 }-> add0(0, y) :|: x >= 0, y >= 0, z' = y, z = 1 + x + 0
   mul0(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y


----------------------------------------

(11) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add0(z, z') -{ 1 }-> add0(xs, 1 + 0 + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs')
   mul0(z, z') -{ 2 }-> add0(0, z') :|: z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0


----------------------------------------

(13) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { add0 }
   { mul0 }
   { goal }

----------------------------------------

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add0(z, z') -{ 1 }-> add0(xs, 1 + 0 + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs')
   mul0(z, z') -{ 2 }-> add0(0, z') :|: z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {add0}, {mul0}, {goal}

----------------------------------------

(15) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add0(z, z') -{ 1 }-> add0(xs, 1 + 0 + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs')
   mul0(z, z') -{ 2 }-> add0(0, z') :|: z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {add0}, {mul0}, {goal}

----------------------------------------

(17) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: add0
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z + z'

----------------------------------------

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add0(z, z') -{ 1 }-> add0(xs, 1 + 0 + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs')
   mul0(z, z') -{ 2 }-> add0(0, z') :|: z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {add0}, {mul0}, {goal}
Previous analysis results are:
  add0: runtime: ?, size: O(n^1) [z + z']

----------------------------------------

(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: add0
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z

----------------------------------------

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add0(z, z') -{ 1 }-> add0(xs, 1 + 0 + z') :|: z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs')
   mul0(z, z') -{ 2 }-> add0(0, z') :|: z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {mul0}, {goal}
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

----------------------------------------

(21) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= xs + (1 + 0 + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0
   mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs')
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {mul0}, {goal}
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

----------------------------------------

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: mul0
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 2*z*z' + z'

----------------------------------------

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= xs + (1 + 0 + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0
   mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs')
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {mul0}, {goal}
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  mul0: runtime: ?, size: O(n^2) [2*z*z' + z']

----------------------------------------

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: mul0
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 4 + 4*z + 3*z*z' + 4*z^2*z'

----------------------------------------

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= xs + (1 + 0 + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   goal(z, z') -{ 1 }-> mul0(z, z') :|: z >= 0, z' >= 0
   mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 2 }-> add0(add0(mul0(xs', z'), z'), z') :|: x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs')
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {goal}
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  mul0: runtime: O(n^3) [4 + 4*z + 3*z*z' + 4*z^2*z'], size: O(n^2) [2*z*z' + z']

----------------------------------------

(27) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= xs + (1 + 0 + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   goal(z, z') -{ 5 + 4*z + 3*z*z' + 4*z^2*z' }-> s3 :|: s3 >= 0, s3 <= 2 * (z' * z) + z', z >= 0, z' >= 0
   mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 8 + s'' + s1 + 4*xs' + 3*xs'*z' + 4*xs'^2*z' }-> s2 :|: s'' >= 0, s'' <= 2 * (z' * xs') + z', s1 >= 0, s1 <= s'' + z', s2 >= 0, s2 <= s1 + z', x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs')
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {goal}
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  mul0: runtime: O(n^3) [4 + 4*z + 3*z*z' + 4*z^2*z'], size: O(n^2) [2*z*z' + z']

----------------------------------------

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: goal
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 2*z*z' + z'

----------------------------------------

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= xs + (1 + 0 + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   goal(z, z') -{ 5 + 4*z + 3*z*z' + 4*z^2*z' }-> s3 :|: s3 >= 0, s3 <= 2 * (z' * z) + z', z >= 0, z' >= 0
   mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 8 + s'' + s1 + 4*xs' + 3*xs'*z' + 4*xs'^2*z' }-> s2 :|: s'' >= 0, s'' <= 2 * (z' * xs') + z', s1 >= 0, s1 <= s'' + z', s2 >= 0, s2 <= s1 + z', x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs')
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {goal}
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  mul0: runtime: O(n^3) [4 + 4*z + 3*z*z' + 4*z^2*z'], size: O(n^2) [2*z*z' + z']
  goal: runtime: ?, size: O(n^2) [2*z*z' + z']

----------------------------------------

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: goal
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 5 + 4*z + 3*z*z' + 4*z^2*z'

----------------------------------------

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 2 + xs }-> s' :|: s' >= 0, s' <= xs + (1 + 0 + z'), z = 1 + x + xs, xs >= 0, x >= 0, z' >= 0
   add0(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   goal(z, z') -{ 5 + 4*z + 3*z*z' + 4*z^2*z' }-> s3 :|: s3 >= 0, s3 <= 2 * (z' * z) + z', z >= 0, z' >= 0
   mul0(z, z') -{ 3 }-> s :|: s >= 0, s <= 0 + z', z - 1 >= 0, z' >= 0
   mul0(z, z') -{ 8 + s'' + s1 + 4*xs' + 3*xs'*z' + 4*xs'^2*z' }-> s2 :|: s'' >= 0, s'' <= 2 * (z' * xs') + z', s1 >= 0, s1 <= s'' + z', s2 >= 0, s2 <= s1 + z', x >= 0, x' >= 0, xs' >= 0, z' >= 0, z = 1 + x + (1 + x' + xs')
   mul0(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: 
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  mul0: runtime: O(n^3) [4 + 4*z + 3*z*z' + 4*z^2*z'], size: O(n^2) [2*z*z' + z']
  goal: runtime: O(n^3) [5 + 4*z + 3*z*z' + 4*z^2*z'], size: O(n^2) [2*z*z' + z']

----------------------------------------

(33) FinalProof (FINISHED)
Computed overall runtime complexity
----------------------------------------

(34)
BOUNDS(1, n^3)