MAYBE
proof of input_nroLuiqqel.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 103 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
        (24) CpxRNTS
    (25) CompletionProof [UPPER BOUND(ID), 0 ms]
    (26) CpxTypedWeightedCompleteTrs
        (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
(29) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(30) CdtProblem
    (31) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) CdtProblem
    (33) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) CdtProblem


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


The TRS R consists of the following rules:

   f(S(x)) -> x
   f(0) -> 0
   g(x) -> g(x)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(S(x)) -> x
   f(0') -> 0'
   g(x) -> g(x)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(4)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(S(x)) -> x
   f(0) -> 0
   g(x) -> g(x)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(5) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
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(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   f(S(x)) -> x [1]
   f(0) -> 0 [1]
   g(x) -> g(x) [1]

Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(S(x)) -> x [1]
   f(0) -> 0 [1]
   g(x) -> g(x) [1]

The TRS has the following type information:
f :: S:0 -> S:0
S :: S:0 -> S:0
0 :: S:0
g :: a -> g

Rewrite Strategy: INNERMOST
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(9) 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_1
   g_1

(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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(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:

   f(S(x)) -> x [1]
   f(0) -> 0 [1]
   g(x) -> g(x) [1]

The TRS has the following type information:
f :: S:0 -> S:0
S :: S:0 -> S:0
0 :: S:0
g :: a -> g
const :: g
const1 :: a

Rewrite Strategy: INNERMOST
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(11) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
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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(S(x)) -> x [1]
   f(0) -> 0 [1]
   g(x) -> g(x) [1]

The TRS has the following type information:
f :: S:0 -> S:0
S :: S:0 -> S:0
0 :: S:0
g :: a -> g
const :: g
const1 :: a

Rewrite Strategy: INNERMOST
----------------------------------------

(13) 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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(14)
Obligation:
Complexity RNTS consisting of the following rules:

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


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(15) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
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(16)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   g(z) -{ 1 }-> g(z) :|: z >= 0


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

   { g }
   { f }

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

   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   g(z) -{ 1 }-> g(z) :|: z >= 0

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

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(19) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(20)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   g(z) -{ 1 }-> g(z) :|: z >= 0

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

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(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   g(z) -{ 1 }-> g(z) :|: z >= 0

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

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(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: g
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   g(z) -{ 1 }-> g(z) :|: z >= 0

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

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(25) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
none

And the following fresh constants:    const, const1

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(26)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(S(x)) -> x [1]
   f(0) -> 0 [1]
   g(x) -> g(x) [1]

The TRS has the following type information:
f :: S:0 -> S:0
S :: S:0 -> S:0
0 :: S:0
g :: a -> g
const :: g
const1 :: a

Rewrite Strategy: INNERMOST
----------------------------------------

(27) 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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(28)
Obligation:
Complexity RNTS consisting of the following rules:

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

Only complete derivations are relevant for the runtime complexity.

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(29) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
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(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(S(z0)) -> z0
   f(0) -> 0
   g(z0) -> g(z0)
Tuples:
   F(S(z0)) -> c
   F(0) -> c1
   G(z0) -> c2(G(z0))
S tuples:
   F(S(z0)) -> c
   F(0) -> c1
   G(z0) -> c2(G(z0))
K tuples:none
Defined Rule Symbols:   f_1, g_1

Defined Pair Symbols:   F_1, G_1

Compound Symbols:   c, c1, c2_1


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(31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   F(S(z0)) -> c
   F(0) -> c1

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(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(S(z0)) -> z0
   f(0) -> 0
   g(z0) -> g(z0)
Tuples:
   G(z0) -> c2(G(z0))
S tuples:
   G(z0) -> c2(G(z0))
K tuples:none
Defined Rule Symbols:   f_1, g_1

Defined Pair Symbols:   G_1

Compound Symbols:   c2_1


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(33) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(S(z0)) -> z0
   f(0) -> 0
   g(z0) -> g(z0)

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(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   G(z0) -> c2(G(z0))
S tuples:
   G(z0) -> c2(G(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   G_1

Compound Symbols:   c2_1