MAYBE
proof of input_4DX32nrcGy.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 7 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), 1639 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 204 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) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CdtProblem
    (43) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem


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

(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:

   *(X, +(Y, 1)) -> +(*(X, +(Y, *(1, 0))), X)
   *(X, 1) -> X
   *(X, 0) -> X
   *(X, 0) -> 0

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(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:

   *'(X, +'(Y, 1')) -> +'(*'(X, +'(Y, *'(1', 0'))), X)
   *'(X, 1') -> X
   *'(X, 0') -> X
   *'(X, 0') -> 0'

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(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:

   *(X, +(Y, 1)) -> +(*(X, +(Y, *(1, 0))), X)
   *(X, 1) -> X
   *(X, 0) -> X
   *(X, 0) -> 0

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(5) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(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:

   *(X, +(Y, 1)) -> +(*(X, +(Y, *(1, 0))), X) [1]
   *(X, 1) -> X [1]
   *(X, 0) -> X [1]
   *(X, 0) -> 0 [1]

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

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

* => times

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

(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:

   times(X, +(Y, 1)) -> +(times(X, +(Y, times(1, 0))), X) [1]
   times(X, 1) -> X [1]
   times(X, 0) -> X [1]
   times(X, 0) -> 0 [1]

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

(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(10)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   times(X, +(Y, 1)) -> +(times(X, +(Y, times(1, 0))), X) [1]
   times(X, 1) -> X [1]
   times(X, 0) -> X [1]
   times(X, 0) -> 0 [1]

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

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

(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:

   times(v0, v1) -> null_times [0]

And the following fresh constants:    null_times

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

(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:

   times(X, +(Y, 1)) -> +(times(X, +(Y, times(1, 0))), X) [1]
   times(X, 1) -> X [1]
   times(X, 0) -> X [1]
   times(X, 0) -> 0 [1]
   times(v0, v1) -> null_times [0]

The TRS has the following type information:
times :: 1:+:0:null_times -> 1:+:0:null_times -> 1:+:0:null_times
+ :: 1:+:0:null_times -> 1:+:0:null_times -> 1:+:0:null_times
1 :: 1:+:0:null_times
0 :: 1:+:0:null_times
null_times :: 1:+:0:null_times

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:

   1 => 1
   0 => 0
   null_times => 0

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

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

   times(z, z') -{ 1 }-> X :|: X >= 0, z = X, z' = 1
   times(z, z') -{ 1 }-> X :|: X >= 0, z = X, z' = 0
   times(z, z') -{ 1 }-> 0 :|: X >= 0, z = X, z' = 0
   times(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   times(z, z') -{ 1 }-> 1 + times(X, 1 + Y + times(1, 0)) + X :|: Y >= 0, X >= 0, z = X, z' = 1 + Y + 1

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:

   times_2

Due to the following rules being added:

   times(v0, v1) -> null_times [0]

And the following fresh constants:    null_times

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

(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:

   times(X, +(Y, 1)) -> +(times(X, +(Y, times(1, 0))), X) [1]
   times(X, 1) -> X [1]
   times(X, 0) -> X [1]
   times(X, 0) -> 0 [1]
   times(v0, v1) -> null_times [0]

The TRS has the following type information:
times :: 1:+:0:null_times -> 1:+:0:null_times -> 1:+:0:null_times
+ :: 1:+:0:null_times -> 1:+:0:null_times -> 1:+:0:null_times
1 :: 1:+:0:null_times
0 :: 1:+:0:null_times
null_times :: 1:+:0:null_times

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.
----------------------------------------

(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:

   times(X, +(Y, 1)) -> +(times(X, +(Y, 1)), X) [2]
   times(X, +(Y, 1)) -> +(times(X, +(Y, 0)), X) [2]
   times(X, +(Y, 1)) -> +(times(X, +(Y, null_times)), X) [1]
   times(X, 1) -> X [1]
   times(X, 0) -> X [1]
   times(X, 0) -> 0 [1]
   times(v0, v1) -> null_times [0]

The TRS has the following type information:
times :: 1:+:0:null_times -> 1:+:0:null_times -> 1:+:0:null_times
+ :: 1:+:0:null_times -> 1:+:0:null_times -> 1:+:0:null_times
1 :: 1:+:0:null_times
0 :: 1:+:0:null_times
null_times :: 1:+:0:null_times

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
   null_times => 0

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

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

   times(z, z') -{ 1 }-> X :|: X >= 0, z = X, z' = 1
   times(z, z') -{ 1 }-> X :|: X >= 0, z = X, z' = 0
   times(z, z') -{ 1 }-> 0 :|: X >= 0, z = X, z' = 0
   times(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   times(z, z') -{ 2 }-> 1 + times(X, 1 + Y + 1) + X :|: Y >= 0, X >= 0, z = X, z' = 1 + Y + 1
   times(z, z') -{ 2 }-> 1 + times(X, 1 + Y + 0) + X :|: Y >= 0, X >= 0, z = X, z' = 1 + Y + 1
   times(z, z') -{ 1 }-> 1 + times(X, 1 + Y + 0) + X :|: Y >= 0, X >= 0, z = X, z' = 1 + Y + 1


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

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

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

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


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

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

   { times }

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

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

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

Function symbols to be analyzed: {times}

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

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

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

Function symbols to be analyzed: {times}

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(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

Function symbols to be analyzed: {times}
Previous analysis results are:
  times: runtime: ?, size: INF

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(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

Function symbols to be analyzed: {times}
Previous analysis results are:
  times: runtime: INF, size: INF

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

(31) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, +(z1, 1)) -> +(*(z0, +(z1, *(1, 0))), z0)
   *(z0, 1) -> z0
   *(z0, 0) -> z0
   *(z0, 0) -> 0
Tuples:
   *'(z0, +(z1, 1)) -> c(*'(z0, +(z1, *(1, 0))), *'(1, 0))
   *'(z0, 1) -> c1
   *'(z0, 0) -> c2
   *'(z0, 0) -> c3
S tuples:
   *'(z0, +(z1, 1)) -> c(*'(z0, +(z1, *(1, 0))), *'(1, 0))
   *'(z0, 1) -> c1
   *'(z0, 0) -> c2
   *'(z0, 0) -> c3
K tuples:none
Defined Rule Symbols:   *_2

Defined Pair Symbols:   *'_2

Compound Symbols:   c_2, c1, c2, c3


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

(33) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   *'(z0, 0) -> c2
   *'(z0, 1) -> c1
   *'(z0, 0) -> c3

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, +(z1, 1)) -> +(*(z0, +(z1, *(1, 0))), z0)
   *(z0, 1) -> z0
   *(z0, 0) -> z0
   *(z0, 0) -> 0
Tuples:
   *'(z0, +(z1, 1)) -> c(*'(z0, +(z1, *(1, 0))), *'(1, 0))
S tuples:
   *'(z0, +(z1, 1)) -> c(*'(z0, +(z1, *(1, 0))), *'(1, 0))
K tuples:none
Defined Rule Symbols:   *_2

Defined Pair Symbols:   *'_2

Compound Symbols:   c_2


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

(35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, +(z1, 1)) -> +(*(z0, +(z1, *(1, 0))), z0)
   *(z0, 1) -> z0
   *(z0, 0) -> z0
   *(z0, 0) -> 0
Tuples:
   *'(z0, +(z1, 1)) -> c(*'(z0, +(z1, *(1, 0))))
S tuples:
   *'(z0, +(z1, 1)) -> c(*'(z0, +(z1, *(1, 0))))
K tuples:none
Defined Rule Symbols:   *_2

Defined Pair Symbols:   *'_2

Compound Symbols:   c_1


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

(37) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   *(z0, +(z1, 1)) -> +(*(z0, +(z1, *(1, 0))), z0)
   *(z0, 1) -> z0

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> z0
   *(z0, 0) -> 0
Tuples:
   *'(z0, +(z1, 1)) -> c(*'(z0, +(z1, *(1, 0))))
S tuples:
   *'(z0, +(z1, 1)) -> c(*'(z0, +(z1, *(1, 0))))
K tuples:none
Defined Rule Symbols:   *_2

Defined Pair Symbols:   *'_2

Compound Symbols:   c_1


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

(39) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace *'(z0, +(z1, 1)) -> c(*'(z0, +(z1, *(1, 0)))) by 
   *'(x0, +(x1, 1)) -> c(*'(x0, +(x1, 1)))
   *'(x0, +(x1, 1)) -> c(*'(x0, +(x1, 0)))

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> z0
   *(z0, 0) -> 0
Tuples:
   *'(x0, +(x1, 1)) -> c(*'(x0, +(x1, 1)))
   *'(x0, +(x1, 1)) -> c(*'(x0, +(x1, 0)))
S tuples:
   *'(x0, +(x1, 1)) -> c(*'(x0, +(x1, 1)))
   *'(x0, +(x1, 1)) -> c(*'(x0, +(x1, 0)))
K tuples:none
Defined Rule Symbols:   *_2

Defined Pair Symbols:   *'_2

Compound Symbols:   c_1


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

(41) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   *'(x0, +(x1, 1)) -> c(*'(x0, +(x1, 0)))

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> z0
   *(z0, 0) -> 0
Tuples:
   *'(x0, +(x1, 1)) -> c(*'(x0, +(x1, 1)))
S tuples:
   *'(x0, +(x1, 1)) -> c(*'(x0, +(x1, 1)))
K tuples:none
Defined Rule Symbols:   *_2

Defined Pair Symbols:   *'_2

Compound Symbols:   c_1


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

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

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   *'(x0, +(x1, 1)) -> c(*'(x0, +(x1, 1)))
S tuples:
   *'(x0, +(x1, 1)) -> c(*'(x0, +(x1, 1)))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   *'_2

Compound Symbols:   c_1