WORST_CASE(?,O(n^1))
proof of input_LGULMv8iDU.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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^1).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxWeightedTrs
(3) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxWeightedTrs
(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxTypedWeightedTrs
(7) CompletionProof [UPPER BOUND(ID), 0 ms]
(8) CpxTypedWeightedCompleteTrs
(9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxTypedWeightedCompleteTrs
(11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(12) CpxRNTS
(13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxRNTS
(15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxRNTS
(17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(18) CpxRNTS
(19) IntTrsBoundProof [UPPER BOUND(ID), 216 ms]
(20) CpxRNTS
(21) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
(22) CpxRNTS
(23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 284 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 106 ms]
(28) CpxRNTS
(29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(30) CpxRNTS
(31) IntTrsBoundProof [UPPER BOUND(ID), 244 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 107 ms]
(34) CpxRNTS
(35) FinalProof [FINISHED, 0 ms]
(36) BOUNDS(1, n^1)


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


The TRS R consists of the following rules:

   +(X, 0) -> X
   +(X, s(Y)) -> s(+(X, Y))
   f(0, s(0), X) -> f(X, +(X, X), X)
   g(X, Y) -> X
   g(X, Y) -> 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^1).


The TRS R consists of the following rules:

   +(X, 0) -> X [1]
   +(X, s(Y)) -> s(+(X, Y)) [1]
   f(0, s(0), X) -> f(X, +(X, X), X) [1]
   g(X, Y) -> X [1]
   g(X, Y) -> Y [1]

Rewrite Strategy: INNERMOST
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(3) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

+ => plus

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(4)
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:

   plus(X, 0) -> X [1]
   plus(X, s(Y)) -> s(plus(X, Y)) [1]
   f(0, s(0), X) -> f(X, plus(X, X), X) [1]
   g(X, Y) -> X [1]
   g(X, Y) -> Y [1]

Rewrite Strategy: INNERMOST
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(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   plus(X, 0) -> X [1]
   plus(X, s(Y)) -> s(plus(X, Y)) [1]
   f(0, s(0), X) -> f(X, plus(X, X), X) [1]
   g(X, Y) -> X [1]
   g(X, Y) -> Y [1]

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

Rewrite Strategy: INNERMOST
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(7) 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:

   f_3
   g_2

(c) The following functions are completely defined:

   plus_2

Due to the following rules being added:
none

And the following fresh constants:    const, const1

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

(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:

   plus(X, 0) -> X [1]
   plus(X, s(Y)) -> s(plus(X, Y)) [1]
   f(0, s(0), X) -> f(X, plus(X, X), X) [1]
   g(X, Y) -> X [1]
   g(X, Y) -> Y [1]

The TRS has the following type information:
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
f :: 0:s -> 0:s -> 0:s -> f
g :: g -> g -> g
const :: f
const1 :: g

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

(10)
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:

   plus(X, 0) -> X [1]
   plus(X, s(Y)) -> s(plus(X, Y)) [1]
   f(0, s(0), 0) -> f(0, 0, 0) [2]
   f(0, s(0), s(Y')) -> f(s(Y'), s(plus(s(Y'), Y')), s(Y')) [2]
   g(X, Y) -> X [1]
   g(X, Y) -> Y [1]

The TRS has the following type information:
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
f :: 0:s -> 0:s -> 0:s -> f
g :: g -> g -> g
const :: f
const1 :: g

Rewrite Strategy: INNERMOST
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(11) 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
   const => 0
   const1 => 0

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

   f(z, z', z'') -{ 2 }-> f(0, 0, 0) :|: z'' = 0, z' = 1 + 0, z = 0
   f(z, z', z'') -{ 2 }-> f(1 + Y', 1 + plus(1 + Y', Y'), 1 + Y') :|: z'' = 1 + Y', Y' >= 0, z' = 1 + 0, z = 0
   g(z, z') -{ 1 }-> X :|: z' = Y, Y >= 0, X >= 0, z = X
   g(z, z') -{ 1 }-> Y :|: z' = Y, Y >= 0, X >= 0, z = X
   plus(z, z') -{ 1 }-> X :|: X >= 0, z = X, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(X, Y) :|: Y >= 0, z' = 1 + Y, X >= 0, z = X


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

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

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


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

   { g }
   { plus }
   { f }

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

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

Function symbols to be analyzed: {g}, {plus}, {f}

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

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

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

Function symbols to be analyzed: {g}, {plus}, {f}

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(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

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

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(25) 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'

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

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

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

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

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

(27) 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'

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

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

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

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

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

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

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

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

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

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

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

Function symbols to be analyzed: {f}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z + z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  f: runtime: ?, size: O(1) [0]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z''

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

(34)
Obligation:
Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: 
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [z + z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  f: runtime: O(n^1) [2 + z''], size: O(1) [0]

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(35) FinalProof (FINISHED)
Computed overall runtime complexity
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(36)
BOUNDS(1, n^1)