WORST_CASE(?,O(n^1))
proof of input_MOte4gaImL.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CdtProblem
(5) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxTRS
(7) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxWeightedTrs
(9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxTypedWeightedTrs
(11) CompletionProof [UPPER BOUND(ID), 0 ms]
(12) CpxTypedWeightedCompleteTrs
(13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTypedWeightedCompleteTrs
(15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(16) CpxRNTS
(17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) CpxRNTS
(19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CpxRNTS
(21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(22) CpxRNTS
(23) IntTrsBoundProof [UPPER BOUND(ID), 286 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 114 ms]
(26) CpxRNTS
(27) FinalProof [FINISHED, 0 ms]
(28) 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:

   f(0, 1, x) -> f(s(x), x, x)
   f(x, y, s(z)) -> s(f(0, 1, z))

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:
   f(0, 1, z0) -> f(s(z0), z0, z0)
   f(z0, z1, s(z2)) -> s(f(0, 1, z2))
Tuples:
   F(0, 1, z0) -> c(F(s(z0), z0, z0))
   F(z0, z1, s(z2)) -> c1(F(0, 1, z2))
S tuples:
   F(0, 1, z0) -> c(F(s(z0), z0, z0))
   F(z0, z1, s(z2)) -> c1(F(0, 1, z2))
K tuples:none
Defined Rule Symbols:   f_3

Defined Pair Symbols:   F_3

Compound Symbols:   c_1, c1_1


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(3) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(0, 1, z0) -> f(s(z0), z0, z0)
   f(z0, z1, s(z2)) -> s(f(0, 1, z2))

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(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(0, 1, z0) -> c(F(s(z0), z0, z0))
   F(z0, z1, s(z2)) -> c1(F(0, 1, z2))
S tuples:
   F(0, 1, z0) -> c(F(s(z0), z0, z0))
   F(z0, z1, s(z2)) -> c1(F(0, 1, z2))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_3

Compound Symbols:   c_1, c1_1


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(5) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
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(6)
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:

   F(0, 1, z0) -> c(F(s(z0), z0, z0))
   F(z0, z1, s(z2)) -> c1(F(0, 1, z2))

S is empty.
Rewrite Strategy: INNERMOST
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(7) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
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(8)
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:

   F(0, 1, z0) -> c(F(s(z0), z0, z0)) [1]
   F(z0, z1, s(z2)) -> c1(F(0, 1, z2)) [1]

Rewrite Strategy: INNERMOST
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(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(10)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   F(0, 1, z0) -> c(F(s(z0), z0, z0)) [1]
   F(z0, z1, s(z2)) -> c1(F(0, 1, z2)) [1]

The TRS has the following type information:
F :: 0:1:s -> 0:1:s -> 0:1:s -> c:c1
0 :: 0:1:s
1 :: 0:1:s
c :: c:c1 -> c:c1
s :: 0:1:s -> 0:1:s
c1 :: c:c1 -> c:c1

Rewrite Strategy: INNERMOST
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(11) 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

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const

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(12)
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:

   F(0, 1, z0) -> c(F(s(z0), z0, z0)) [1]
   F(z0, z1, s(z2)) -> c1(F(0, 1, z2)) [1]

The TRS has the following type information:
F :: 0:1:s -> 0:1:s -> 0:1:s -> c:c1
0 :: 0:1:s
1 :: 0:1:s
c :: c:c1 -> c:c1
s :: 0:1:s -> 0:1:s
c1 :: c:c1 -> c:c1
const :: c:c1

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

   F(0, 1, z0) -> c(F(s(z0), z0, z0)) [1]
   F(z0, z1, s(z2)) -> c1(F(0, 1, z2)) [1]

The TRS has the following type information:
F :: 0:1:s -> 0:1:s -> 0:1:s -> c:c1
0 :: 0:1:s
1 :: 0:1:s
c :: c:c1 -> c:c1
s :: 0:1:s -> 0:1:s
c1 :: c:c1 -> c:c1
const :: c:c1

Rewrite Strategy: INNERMOST
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(15) 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
   1 => 1
   const => 0

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

   F(z, z', z'') -{ 1 }-> 1 + F(0, 1, z2) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0, z'' = 1 + z2, z2 >= 0
   F(z, z', z'') -{ 1 }-> 1 + F(1 + z0, z0, z0) :|: z'' = z0, z0 >= 0, z' = 1, z = 0


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(17) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
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(18)
Obligation:
Complexity RNTS consisting of the following rules:

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


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

   { F }

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

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

Function symbols to be analyzed: {F}

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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:

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

Function symbols to be analyzed: {F}

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(23) 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

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

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

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

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(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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