WORST_CASE(?,O(n^1))
proof of input_enISBPhRR8.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
(3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CdtProblem
(5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
(7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTRS
(9) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxWeightedTrs
(11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTypedWeightedTrs
(13) CompletionProof [UPPER BOUND(ID), 0 ms]
(14) CpxTypedWeightedCompleteTrs
(15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxTypedWeightedCompleteTrs
(17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(18) CpxRNTS
(19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CpxRNTS
(21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
(22) CpxRNTS
(23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 212 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 125 ms]
(28) CpxRNTS
(29) FinalProof [FINISHED, 0 ms]
(30) 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:

   average(s(x), y) -> average(x, s(y))
   average(x, s(s(s(y)))) -> s(average(s(x), y))
   average(0, 0) -> 0
   average(0, s(0)) -> 0
   average(0, s(s(0))) -> s(0)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
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(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   average(s(z0), z1) -> average(z0, s(z1))
   average(z0, s(s(s(z1)))) -> s(average(s(z0), z1))
   average(0, 0) -> 0
   average(0, s(0)) -> 0
   average(0, s(s(0))) -> s(0)
Tuples:
   AVERAGE(s(z0), z1) -> c(AVERAGE(z0, s(z1)))
   AVERAGE(z0, s(s(s(z1)))) -> c1(AVERAGE(s(z0), z1))
   AVERAGE(0, 0) -> c2
   AVERAGE(0, s(0)) -> c3
   AVERAGE(0, s(s(0))) -> c4
S tuples:
   AVERAGE(s(z0), z1) -> c(AVERAGE(z0, s(z1)))
   AVERAGE(z0, s(s(s(z1)))) -> c1(AVERAGE(s(z0), z1))
   AVERAGE(0, 0) -> c2
   AVERAGE(0, s(0)) -> c3
   AVERAGE(0, s(s(0))) -> c4
K tuples:none
Defined Rule Symbols:   average_2

Defined Pair Symbols:   AVERAGE_2

Compound Symbols:   c_1, c1_1, c2, c3, c4


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(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   AVERAGE(0, s(s(0))) -> c4
   AVERAGE(0, s(0)) -> c3
   AVERAGE(0, 0) -> c2

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   average(s(z0), z1) -> average(z0, s(z1))
   average(z0, s(s(s(z1)))) -> s(average(s(z0), z1))
   average(0, 0) -> 0
   average(0, s(0)) -> 0
   average(0, s(s(0))) -> s(0)
Tuples:
   AVERAGE(s(z0), z1) -> c(AVERAGE(z0, s(z1)))
   AVERAGE(z0, s(s(s(z1)))) -> c1(AVERAGE(s(z0), z1))
S tuples:
   AVERAGE(s(z0), z1) -> c(AVERAGE(z0, s(z1)))
   AVERAGE(z0, s(s(s(z1)))) -> c1(AVERAGE(s(z0), z1))
K tuples:none
Defined Rule Symbols:   average_2

Defined Pair Symbols:   AVERAGE_2

Compound Symbols:   c_1, c1_1


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(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   average(s(z0), z1) -> average(z0, s(z1))
   average(z0, s(s(s(z1)))) -> s(average(s(z0), z1))
   average(0, 0) -> 0
   average(0, s(0)) -> 0
   average(0, s(s(0))) -> s(0)

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   AVERAGE(s(z0), z1) -> c(AVERAGE(z0, s(z1)))
   AVERAGE(z0, s(s(s(z1)))) -> c1(AVERAGE(s(z0), z1))
S tuples:
   AVERAGE(s(z0), z1) -> c(AVERAGE(z0, s(z1)))
   AVERAGE(z0, s(s(s(z1)))) -> c1(AVERAGE(s(z0), z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   AVERAGE_2

Compound Symbols:   c_1, c1_1


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(7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(8)
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:

   AVERAGE(s(z0), z1) -> c(AVERAGE(z0, s(z1)))
   AVERAGE(z0, s(s(s(z1)))) -> c1(AVERAGE(s(z0), z1))

S is empty.
Rewrite Strategy: INNERMOST
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(9) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(10)
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:

   AVERAGE(s(z0), z1) -> c(AVERAGE(z0, s(z1))) [1]
   AVERAGE(z0, s(s(s(z1)))) -> c1(AVERAGE(s(z0), z1)) [1]

Rewrite Strategy: INNERMOST
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(11) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(12)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   AVERAGE(s(z0), z1) -> c(AVERAGE(z0, s(z1))) [1]
   AVERAGE(z0, s(s(s(z1)))) -> c1(AVERAGE(s(z0), z1)) [1]

The TRS has the following type information:
AVERAGE :: s -> s -> c:c1
s :: s -> s
c :: c:c1 -> c:c1
c1 :: c:c1 -> c:c1

Rewrite Strategy: INNERMOST
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(13) 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:

   AVERAGE_2

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const, const1

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(14)
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:

   AVERAGE(s(z0), z1) -> c(AVERAGE(z0, s(z1))) [1]
   AVERAGE(z0, s(s(s(z1)))) -> c1(AVERAGE(s(z0), z1)) [1]

The TRS has the following type information:
AVERAGE :: s -> s -> c:c1
s :: s -> s
c :: c:c1 -> c:c1
c1 :: c:c1 -> c:c1
const :: c:c1
const1 :: s

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

(16)
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:

   AVERAGE(s(z0), z1) -> c(AVERAGE(z0, s(z1))) [1]
   AVERAGE(z0, s(s(s(z1)))) -> c1(AVERAGE(s(z0), z1)) [1]

The TRS has the following type information:
AVERAGE :: s -> s -> c:c1
s :: s -> s
c :: c:c1 -> c:c1
c1 :: c:c1 -> c:c1
const :: c:c1
const1 :: s

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

   const => 0
   const1 => 0

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

   AVERAGE(z, z') -{ 1 }-> 1 + AVERAGE(z0, 1 + z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + AVERAGE(1 + z0, z1) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + (1 + (1 + z1))


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

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

   AVERAGE(z, z') -{ 1 }-> 1 + AVERAGE(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + AVERAGE(1 + z, z' - 3) :|: z' - 3 >= 0, z >= 0


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

   { AVERAGE }

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

   AVERAGE(z, z') -{ 1 }-> 1 + AVERAGE(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + AVERAGE(1 + z, z' - 3) :|: z' - 3 >= 0, z >= 0

Function symbols to be analyzed: {AVERAGE}

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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:

   AVERAGE(z, z') -{ 1 }-> 1 + AVERAGE(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + AVERAGE(1 + z, z' - 3) :|: z' - 3 >= 0, z >= 0

Function symbols to be analyzed: {AVERAGE}

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(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   AVERAGE(z, z') -{ 1 }-> 1 + AVERAGE(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + AVERAGE(1 + z, z' - 3) :|: z' - 3 >= 0, z >= 0

Function symbols to be analyzed: {AVERAGE}
Previous analysis results are:
  AVERAGE: runtime: ?, size: O(1) [0]

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(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   AVERAGE(z, z') -{ 1 }-> 1 + AVERAGE(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   AVERAGE(z, z') -{ 1 }-> 1 + AVERAGE(1 + z, z' - 3) :|: z' - 3 >= 0, z >= 0

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

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(29) FinalProof (FINISHED)
Computed overall runtime complexity
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(30)
BOUNDS(1, n^1)