MAYBE
proof of input_h6ZrPHToo2.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
    (7) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
(11) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(12) CpxWeightedTrs
    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxTypedWeightedTrs
    (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), 163 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 45 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 803 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 43 ms]
        (36) CpxRNTS
    (37) CompletionProof [UPPER BOUND(ID), 0 ms]
    (38) CpxTypedWeightedCompleteTrs
        (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS


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

(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:

   from(X) -> cons(X)
   length -> 0
   length -> s(length1)
   length1 -> length

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:

   from(X) -> cons(X)
   length -> 0'
   length -> s(length1)
   length1 -> length

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:

   from(X) -> cons(X)
   length -> 0
   length -> s(length1)
   length1 -> length

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0)
   length -> 0
   length -> s(length1)
   length1 -> length
Tuples:
   FROM(z0) -> c
   LENGTH -> c1
   LENGTH -> c2(LENGTH1)
   LENGTH1 -> c3(LENGTH)
S tuples:
   FROM(z0) -> c
   LENGTH -> c1
   LENGTH -> c2(LENGTH1)
   LENGTH1 -> c3(LENGTH)
K tuples:none
Defined Rule Symbols:   from_1, length, length1

Defined Pair Symbols:   FROM_1, LENGTH, LENGTH1

Compound Symbols:   c, c1, c2_1, c3_1


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

(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   LENGTH -> c1
   FROM(z0) -> c

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0)
   length -> 0
   length -> s(length1)
   length1 -> length
Tuples:
   LENGTH -> c2(LENGTH1)
   LENGTH1 -> c3(LENGTH)
S tuples:
   LENGTH -> c2(LENGTH1)
   LENGTH1 -> c3(LENGTH)
K tuples:none
Defined Rule Symbols:   from_1, length, length1

Defined Pair Symbols:   LENGTH, LENGTH1

Compound Symbols:   c2_1, c3_1


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

(9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   from(z0) -> cons(z0)
   length -> 0
   length -> s(length1)
   length1 -> length

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   LENGTH -> c2(LENGTH1)
   LENGTH1 -> c3(LENGTH)
S tuples:
   LENGTH -> c2(LENGTH1)
   LENGTH1 -> c3(LENGTH)
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   LENGTH, LENGTH1

Compound Symbols:   c2_1, c3_1


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

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

(12)
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:

   from(X) -> cons(X) [1]
   length -> 0 [1]
   length -> s(length1) [1]
   length1 -> length [1]

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

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

(14)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   from(X) -> cons(X) [1]
   length -> 0 [1]
   length -> s(length1) [1]
   length1 -> length [1]

The TRS has the following type information:
from :: a -> cons
cons :: a -> cons
length :: 0:s
0 :: 0:s
s :: 0:s -> 0:s
length1 :: 0:s

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

(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:

   from_1
   length
   length1

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const, const1

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

(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:

   from(X) -> cons(X) [1]
   length -> 0 [1]
   length -> s(length1) [1]
   length1 -> length [1]

The TRS has the following type information:
from :: a -> cons
cons :: a -> cons
length :: 0:s
0 :: 0:s
s :: 0:s -> 0:s
length1 :: 0:s
const :: cons
const1 :: a

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

(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:

   from(X) -> cons(X) [1]
   length -> 0 [1]
   length -> s(length1) [1]
   length1 -> length [1]

The TRS has the following type information:
from :: a -> cons
cons :: a -> cons
length :: 0:s
0 :: 0:s
s :: 0:s -> 0:s
length1 :: 0:s
const :: cons
const1 :: a

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:

   0 => 0
   const => 0
   const1 => 0

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

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

   from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   length -{ 1 }-> 0 :|: 
   length -{ 1 }-> 1 + length1 :|: 
   length1 -{ 1 }-> length :|: 


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

(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:

   from(z) -{ 1 }-> 1 + z :|: z >= 0
   length -{ 1 }-> 0 :|: 
   length -{ 1 }-> 1 + length1 :|: 
   length1 -{ 1 }-> length :|: 


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

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

   { from }
   { length1, length }

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

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

   from(z) -{ 1 }-> 1 + z :|: z >= 0
   length -{ 1 }-> 0 :|: 
   length -{ 1 }-> 1 + length1 :|: 
   length1 -{ 1 }-> length :|: 

Function symbols to be analyzed: {from}, {length1,length}

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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:

   from(z) -{ 1 }-> 1 + z :|: z >= 0
   length -{ 1 }-> 0 :|: 
   length -{ 1 }-> 1 + length1 :|: 
   length1 -{ 1 }-> length :|: 

Function symbols to be analyzed: {from}, {length1,length}

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: from
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z

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

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

   from(z) -{ 1 }-> 1 + z :|: z >= 0
   length -{ 1 }-> 0 :|: 
   length -{ 1 }-> 1 + length1 :|: 
   length1 -{ 1 }-> length :|: 

Function symbols to be analyzed: {from}, {length1,length}
Previous analysis results are:
  from: runtime: ?, size: O(n^1) [1 + z]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   from(z) -{ 1 }-> 1 + z :|: z >= 0
   length -{ 1 }-> 0 :|: 
   length -{ 1 }-> 1 + length1 :|: 
   length1 -{ 1 }-> length :|: 

Function symbols to be analyzed: {length1,length}
Previous analysis results are:
  from: runtime: O(1) [1], size: O(n^1) [1 + z]

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

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

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   from(z) -{ 1 }-> 1 + z :|: z >= 0
   length -{ 1 }-> 0 :|: 
   length -{ 1 }-> 1 + length1 :|: 
   length1 -{ 1 }-> length :|: 

Function symbols to be analyzed: {length1,length}
Previous analysis results are:
  from: runtime: O(1) [1], size: O(n^1) [1 + z]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: length1
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   from(z) -{ 1 }-> 1 + z :|: z >= 0
   length -{ 1 }-> 0 :|: 
   length -{ 1 }-> 1 + length1 :|: 
   length1 -{ 1 }-> length :|: 

Function symbols to be analyzed: {length1,length}
Previous analysis results are:
  from: runtime: O(1) [1], size: O(n^1) [1 + z]
  length1: runtime: ?, size: INF
  length: runtime: ?, size: INF

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: length1
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   from(z) -{ 1 }-> 1 + z :|: z >= 0
   length -{ 1 }-> 0 :|: 
   length -{ 1 }-> 1 + length1 :|: 
   length1 -{ 1 }-> length :|: 

Function symbols to be analyzed: {length1,length}
Previous analysis results are:
  from: runtime: O(1) [1], size: O(n^1) [1 + z]
  length1: runtime: INF, size: INF
  length: runtime: ?, size: INF

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

(37) 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

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

(38)
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:

   from(X) -> cons(X) [1]
   length -> 0 [1]
   length -> s(length1) [1]
   length1 -> length [1]

The TRS has the following type information:
from :: a -> cons
cons :: a -> cons
length :: 0:s
0 :: 0:s
s :: 0:s -> 0:s
length1 :: 0:s
const :: cons
const1 :: a

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

(39) 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

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

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X
   length -{ 1 }-> 0 :|: 
   length -{ 1 }-> 1 + length1 :|: 
   length1 -{ 1 }-> length :|: 

Only complete derivations are relevant for the runtime complexity.