WORST_CASE(Omega(n^3),O(n^3))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, 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), 289 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 119 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 501 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 611 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) FinalProof [FINISHED, 0 ms]
    (34) BOUNDS(1, n^3)
(35) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(36) CpxTRS
    (37) SlicingProof [LOWER BOUND(ID), 0 ms]
    (38) CpxTRS
    (39) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) typed CpxTrs
    (41) OrderProof [LOWER BOUND(ID), 0 ms]
    (42) typed CpxTrs
    (43) RewriteLemmaProof [LOWER BOUND(ID), 301 ms]
    (44) BEST
        (45) proven lower bound
            (46) LowerBoundPropagationProof [FINISHED, 0 ms]
            (47) BOUNDS(n^1, INF)
        (48) typed CpxTrs
            (49) RewriteLemmaProof [LOWER BOUND(ID), 73 ms]
            (50) proven lower bound
            (51) LowerBoundPropagationProof [FINISHED, 0 ms]
            (52) BOUNDS(n^3, INF)


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

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

(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, 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)

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

(35) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(36)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).


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: INNERMOST
----------------------------------------

(37) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
Cons/0

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

(38)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

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

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

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

(40)
Obligation:
Innermost TRS:
Rules:
mul0(Cons(xs), y) -> add0(mul0(xs, y), y)
add0(Cons(xs), y) -> add0(xs, Cons(y))
mul0(Nil, y) -> Nil
add0(Nil, y) -> y
goal(xs, ys) -> mul0(xs, ys)

Types:
mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat -> Cons:Nil

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(41) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
mul0, add0

They will be analysed ascendingly in the following order:
add0 < mul0

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(42)
Obligation:
Innermost TRS:
Rules:
mul0(Cons(xs), y) -> add0(mul0(xs, y), y)
add0(Cons(xs), y) -> add0(xs, Cons(y))
mul0(Nil, y) -> Nil
add0(Nil, y) -> y
goal(xs, ys) -> mul0(xs, ys)

Types:
mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil2_1(0) <=> Nil
gen_Cons:Nil2_1(+(x, 1)) <=> Cons(gen_Cons:Nil2_1(x))


The following defined symbols remain to be analysed:
add0, mul0

They will be analysed ascendingly in the following order:
add0 < mul0

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(43) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
add0(gen_Cons:Nil2_1(n4_1), gen_Cons:Nil2_1(b)) -> gen_Cons:Nil2_1(+(n4_1, b)), rt in Omega(1 + n4_1)

Induction Base:
add0(gen_Cons:Nil2_1(0), gen_Cons:Nil2_1(b)) ->_R^Omega(1)
gen_Cons:Nil2_1(b)

Induction Step:
add0(gen_Cons:Nil2_1(+(n4_1, 1)), gen_Cons:Nil2_1(b)) ->_R^Omega(1)
add0(gen_Cons:Nil2_1(n4_1), Cons(gen_Cons:Nil2_1(b))) ->_IH
gen_Cons:Nil2_1(+(+(b, 1), c5_1))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(44)
Complex Obligation (BEST)

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(45)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
mul0(Cons(xs), y) -> add0(mul0(xs, y), y)
add0(Cons(xs), y) -> add0(xs, Cons(y))
mul0(Nil, y) -> Nil
add0(Nil, y) -> y
goal(xs, ys) -> mul0(xs, ys)

Types:
mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil2_1(0) <=> Nil
gen_Cons:Nil2_1(+(x, 1)) <=> Cons(gen_Cons:Nil2_1(x))


The following defined symbols remain to be analysed:
add0, mul0

They will be analysed ascendingly in the following order:
add0 < mul0

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(46) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(47)
BOUNDS(n^1, INF)

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(48)
Obligation:
Innermost TRS:
Rules:
mul0(Cons(xs), y) -> add0(mul0(xs, y), y)
add0(Cons(xs), y) -> add0(xs, Cons(y))
mul0(Nil, y) -> Nil
add0(Nil, y) -> y
goal(xs, ys) -> mul0(xs, ys)

Types:
mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat -> Cons:Nil


Lemmas:
add0(gen_Cons:Nil2_1(n4_1), gen_Cons:Nil2_1(b)) -> gen_Cons:Nil2_1(+(n4_1, b)), rt in Omega(1 + n4_1)


Generator Equations:
gen_Cons:Nil2_1(0) <=> Nil
gen_Cons:Nil2_1(+(x, 1)) <=> Cons(gen_Cons:Nil2_1(x))


The following defined symbols remain to be analysed:
mul0
----------------------------------------

(49) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mul0(gen_Cons:Nil2_1(n457_1), gen_Cons:Nil2_1(b)) -> gen_Cons:Nil2_1(*(n457_1, b)), rt in Omega(1 + b*n457_1^2 + n457_1)

Induction Base:
mul0(gen_Cons:Nil2_1(0), gen_Cons:Nil2_1(b)) ->_R^Omega(1)
Nil

Induction Step:
mul0(gen_Cons:Nil2_1(+(n457_1, 1)), gen_Cons:Nil2_1(b)) ->_R^Omega(1)
add0(mul0(gen_Cons:Nil2_1(n457_1), gen_Cons:Nil2_1(b)), gen_Cons:Nil2_1(b)) ->_IH
add0(gen_Cons:Nil2_1(*(c458_1, b)), gen_Cons:Nil2_1(b)) ->_L^Omega(1 + b*n457_1)
gen_Cons:Nil2_1(+(*(n457_1, b), b))

We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3).
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(50)
Obligation:
Proved the lower bound n^3 for the following obligation:

Innermost TRS:
Rules:
mul0(Cons(xs), y) -> add0(mul0(xs, y), y)
add0(Cons(xs), y) -> add0(xs, Cons(y))
mul0(Nil, y) -> Nil
add0(Nil, y) -> y
goal(xs, ys) -> mul0(xs, ys)

Types:
mul0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil
hole_Cons:Nil1_1 :: Cons:Nil
gen_Cons:Nil2_1 :: Nat -> Cons:Nil


Lemmas:
add0(gen_Cons:Nil2_1(n4_1), gen_Cons:Nil2_1(b)) -> gen_Cons:Nil2_1(+(n4_1, b)), rt in Omega(1 + n4_1)


Generator Equations:
gen_Cons:Nil2_1(0) <=> Nil
gen_Cons:Nil2_1(+(x, 1)) <=> Cons(gen_Cons:Nil2_1(x))


The following defined symbols remain to be analysed:
mul0
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(51) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(52)
BOUNDS(n^3, INF)