WORST_CASE(?,O(n^3))
proof of input_tSnVUcEQ6m.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 553 ms]
(18) CpxRNTS
(19) IntTrsBoundProof [UPPER BOUND(ID), 151 ms]
(20) CpxRNTS
(21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(22) CpxRNTS
(23) IntTrsBoundProof [UPPER BOUND(ID), 251 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
(26) CpxRNTS
(27) FinalProof [FINISHED, 0 ms]
(28) BOUNDS(1, n^3)


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(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:

   times(x, 0) -> 0
   times(x, s(y)) -> plus(times(x, y), x)
   plus(x, 0) -> x
   plus(0, x) -> x
   plus(x, s(y)) -> s(plus(x, y))
   plus(s(x), y) -> s(plus(x, y))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
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(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:

   times(x, 0) -> 0 [1]
   times(x, s(y)) -> plus(times(x, y), x) [1]
   plus(x, 0) -> x [1]
   plus(0, x) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   plus(s(x), y) -> s(plus(x, y)) [1]

Rewrite Strategy: INNERMOST
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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   times(x, 0) -> 0 [1]
   times(x, s(y)) -> plus(times(x, y), x) [1]
   plus(x, 0) -> x [1]
   plus(0, x) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   plus(s(x), y) -> s(plus(x, y)) [1]

The TRS has the following type information:
times :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s

Rewrite Strategy: INNERMOST
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(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:
none

(c) The following functions are completely defined:

   times_2
   plus_2

Due to the following rules being added:
none

And the following fresh constants: none

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(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:

   times(x, 0) -> 0 [1]
   times(x, s(y)) -> plus(times(x, y), x) [1]
   plus(x, 0) -> x [1]
   plus(0, x) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   plus(s(x), y) -> s(plus(x, y)) [1]

The TRS has the following type information:
times :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s

Rewrite Strategy: INNERMOST
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(7) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
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(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:

   times(x, 0) -> 0 [1]
   times(x, s(0)) -> plus(0, x) [2]
   times(x, s(s(y'))) -> plus(plus(times(x, y'), x), x) [2]
   plus(x, 0) -> x [1]
   plus(0, x) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   plus(s(x), y) -> s(plus(x, y)) [1]

The TRS has the following type information:
times :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s

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

   0 => 0

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

   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 1 }-> x :|: z' = x, x >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   times(z, z') -{ 2 }-> plus(plus(times(x, y'), x), x) :|: z' = 1 + (1 + y'), x >= 0, y' >= 0, z = x
   times(z, z') -{ 2 }-> plus(0, x) :|: x >= 0, z' = 1 + 0, z = x
   times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0


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(11) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
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(12)
Obligation:
Complexity RNTS consisting of the following rules:

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0


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

   { plus }
   { times }

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {plus}, {times}

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(15) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(16)
Obligation:
Complexity RNTS consisting of the following rules:

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {plus}, {times}

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(17) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

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

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(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

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

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(21) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(22)
Obligation:
Complexity RNTS consisting of the following rules:

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 + z + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z }-> s :|: s >= 0, s <= 0 + z, z >= 0, z' = 1 + 0
   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

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

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(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 + z + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z }-> s :|: s >= 0, s <= 0 + z, z >= 0, z' = 1 + 0
   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

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

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(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 + z + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z }-> s :|: s >= 0, s <= 0 + z, z >= 0, z' = 1 + 0
   times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

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

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(27) FinalProof (FINISHED)
Computed overall runtime complexity
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(28)
BOUNDS(1, n^3)