MAYBE
proof of input_sBim049yQJ.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxWeightedTrsRenamingProof [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) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
    (15) CompletionProof [UPPER BOUND(ID), 0 ms]
    (16) CpxTypedWeightedCompleteTrs
        (17) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxTypedWeightedCompleteTrs
        (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (22) CpxRNTS
        (23) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 851 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 79 ms]
        (30) CpxRNTS
(31) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(32) CdtProblem
    (33) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) CdtProblem
    (35) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (36) CdtProblem
    (37) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) 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:

   +(1, x) -> +(+(0, 1), x)
   +(0, x) -> 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:

   +'(1', x) -> +'(+'(0', 1'), x)
   +'(0', x) -> 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:

   +(1, x) -> +(+(0, 1), x)
   +(0, x) -> 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:

   +(1, x) -> +(+(0, 1), x) [1]
   +(0, x) -> x [1]

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

+ => plus

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(8)
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:

   plus(1, x) -> plus(plus(0, 1), x) [1]
   plus(0, x) -> x [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:

   plus(1, x) -> plus(plus(0, 1), x) [1]
   plus(0, x) -> x [1]

The TRS has the following type information:
plus :: 1:0 -> 1:0 -> 1:0
1 :: 1:0
0 :: 1:0

Rewrite Strategy: INNERMOST
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(11) 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: none

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(12)
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:

   plus(1, x) -> plus(plus(0, 1), x) [1]
   plus(0, x) -> x [1]

The TRS has the following type information:
plus :: 1:0 -> 1:0 -> 1:0
1 :: 1:0
0 :: 1:0

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

   1 => 1
   0 => 0

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

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

Only complete derivations are relevant for the runtime complexity.

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(15) 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:

   plus_2

Due to the following rules being added:
none

And the following fresh constants: none

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(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:

   plus(1, x) -> plus(plus(0, 1), x) [1]
   plus(0, x) -> x [1]

The TRS has the following type information:
plus :: 1:0 -> 1:0 -> 1:0
1 :: 1:0
0 :: 1:0

Rewrite Strategy: INNERMOST
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(17) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
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(18)
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(1, x) -> plus(1, x) [2]
   plus(0, x) -> x [1]

The TRS has the following type information:
plus :: 1:0 -> 1:0 -> 1:0
1 :: 1:0
0 :: 1:0

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

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

   1 => 1
   0 => 0

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

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

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


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(21) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
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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') -{ 2 }-> plus(1, z') :|: z = 1, z' >= 0


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

   { plus }

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

Function symbols to be analyzed: {plus}

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(25) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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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') -{ 2 }-> plus(1, z') :|: z = 1, z' >= 0

Function symbols to be analyzed: {plus}

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(27) 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'

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

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

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

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(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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(31) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
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(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(1, z0) -> +(+(0, 1), z0)
   +(0, z0) -> z0
Tuples:
   +'(1, z0) -> c(+'(+(0, 1), z0), +'(0, 1))
   +'(0, z0) -> c1
S tuples:
   +'(1, z0) -> c(+'(+(0, 1), z0), +'(0, 1))
   +'(0, z0) -> c1
K tuples:none
Defined Rule Symbols:   +_2

Defined Pair Symbols:   +'_2

Compound Symbols:   c_2, c1


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(33) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   +'(0, z0) -> c1

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(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(1, z0) -> +(+(0, 1), z0)
   +(0, z0) -> z0
Tuples:
   +'(1, z0) -> c(+'(+(0, 1), z0), +'(0, 1))
S tuples:
   +'(1, z0) -> c(+'(+(0, 1), z0), +'(0, 1))
K tuples:none
Defined Rule Symbols:   +_2

Defined Pair Symbols:   +'_2

Compound Symbols:   c_2


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(35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
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(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(1, z0) -> +(+(0, 1), z0)
   +(0, z0) -> z0
Tuples:
   +'(1, z0) -> c(+'(+(0, 1), z0))
S tuples:
   +'(1, z0) -> c(+'(+(0, 1), z0))
K tuples:none
Defined Rule Symbols:   +_2

Defined Pair Symbols:   +'_2

Compound Symbols:   c_1


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(37) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   +(1, z0) -> +(+(0, 1), z0)

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(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(0, z0) -> z0
Tuples:
   +'(1, z0) -> c(+'(+(0, 1), z0))
S tuples:
   +'(1, z0) -> c(+'(+(0, 1), z0))
K tuples:none
Defined Rule Symbols:   +_2

Defined Pair Symbols:   +'_2

Compound Symbols:   c_1


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(39) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace +'(1, z0) -> c(+'(+(0, 1), z0)) by 
   +'(1, x0) -> c(+'(1, x0))

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(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(0, z0) -> z0
Tuples:
   +'(1, x0) -> c(+'(1, x0))
S tuples:
   +'(1, x0) -> c(+'(1, x0))
K tuples:none
Defined Rule Symbols:   +_2

Defined Pair Symbols:   +'_2

Compound Symbols:   c_1


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(41) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   +(0, z0) -> z0

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(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   +'(1, x0) -> c(+'(1, x0))
S tuples:
   +'(1, x0) -> c(+'(1, x0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   +'_2

Compound Symbols:   c_1