MAYBE
proof of input_VmQDz8KrjM.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 858 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 137 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
    (35) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (36) CdtProblem
    (37) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) 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:

   2nd(cons1(X, cons(Y, Z))) -> Y
   2nd(cons(X, X1)) -> 2nd(cons1(X, X1))
   from(X) -> cons(X, from(s(X)))

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:

   2nd(cons1(X, cons(Y, Z))) -> Y
   2nd(cons(X, X1)) -> 2nd(cons1(X, X1))
   from(X) -> cons(X, from(s(X)))

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:

   2nd(cons1(X, cons(Y, Z))) -> Y
   2nd(cons(X, X1)) -> 2nd(cons1(X, X1))
   from(X) -> cons(X, from(s(X)))

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:

   2nd(cons1(X, cons(Y, Z))) -> Y [1]
   2nd(cons(X, X1)) -> 2nd(cons1(X, X1)) [1]
   from(X) -> cons(X, from(s(X))) [1]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   2nd(cons1(X, cons(Y, Z))) -> Y [1]
   2nd(cons(X, X1)) -> 2nd(cons1(X, X1)) [1]
   from(X) -> cons(X, from(s(X))) [1]

The TRS has the following type information:
2nd :: cons:cons1 -> s
cons1 :: s -> cons:cons1 -> cons:cons1
cons :: s -> cons:cons1 -> cons:cons1
from :: s -> cons:cons1
s :: s -> s

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

(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:

   2nd_1
   from_1

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const, const1

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

(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:

   2nd(cons1(X, cons(Y, Z))) -> Y [1]
   2nd(cons(X, X1)) -> 2nd(cons1(X, X1)) [1]
   from(X) -> cons(X, from(s(X))) [1]

The TRS has the following type information:
2nd :: cons:cons1 -> s
cons1 :: s -> cons:cons1 -> cons:cons1
cons :: s -> cons:cons1 -> cons:cons1
from :: s -> cons:cons1
s :: s -> s
const :: s
const1 :: cons:cons1

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

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

(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:

   2nd(cons1(X, cons(Y, Z))) -> Y [1]
   2nd(cons(X, X1)) -> 2nd(cons1(X, X1)) [1]
   from(X) -> cons(X, from(s(X))) [1]

The TRS has the following type information:
2nd :: cons:cons1 -> s
cons1 :: s -> cons:cons1 -> cons:cons1
cons :: s -> cons:cons1 -> cons:cons1
from :: s -> cons:cons1
s :: s -> s
const :: s
const1 :: cons:cons1

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:

   const => 0
   const1 => 0

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

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

   2nd(z) -{ 1 }-> Y :|: z = 1 + X + (1 + Y + Z), Z >= 0, Y >= 0, X >= 0
   2nd(z) -{ 1 }-> 2nd(1 + X + X1) :|: X1 >= 0, X >= 0, z = 1 + X + X1
   from(z) -{ 1 }-> 1 + X + from(1 + X) :|: X >= 0, z = X


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

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

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

   2nd(z) -{ 1 }-> Y :|: z = 1 + X + (1 + Y + Z), Z >= 0, Y >= 0, X >= 0
   2nd(z) -{ 1 }-> 2nd(1 + X + X1) :|: X1 >= 0, X >= 0, z = 1 + X + X1
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0


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

   { from }
   { 2nd }

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

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

   2nd(z) -{ 1 }-> Y :|: z = 1 + X + (1 + Y + Z), Z >= 0, Y >= 0, X >= 0
   2nd(z) -{ 1 }-> 2nd(1 + X + X1) :|: X1 >= 0, X >= 0, z = 1 + X + X1
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0

Function symbols to be analyzed: {from}, {2nd}

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

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

   2nd(z) -{ 1 }-> Y :|: z = 1 + X + (1 + Y + Z), Z >= 0, Y >= 0, X >= 0
   2nd(z) -{ 1 }-> 2nd(1 + X + X1) :|: X1 >= 0, X >= 0, z = 1 + X + X1
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0

Function symbols to be analyzed: {from}, {2nd}

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(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   2nd(z) -{ 1 }-> Y :|: z = 1 + X + (1 + Y + Z), Z >= 0, Y >= 0, X >= 0
   2nd(z) -{ 1 }-> 2nd(1 + X + X1) :|: X1 >= 0, X >= 0, z = 1 + X + X1
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0

Function symbols to be analyzed: {from}, {2nd}
Previous analysis results are:
  from: runtime: ?, size: O(1) [0]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   2nd(z) -{ 1 }-> Y :|: z = 1 + X + (1 + Y + Z), Z >= 0, Y >= 0, X >= 0
   2nd(z) -{ 1 }-> 2nd(1 + X + X1) :|: X1 >= 0, X >= 0, z = 1 + X + X1
   from(z) -{ 1 }-> 1 + z + from(1 + z) :|: z >= 0

Function symbols to be analyzed: {from}, {2nd}
Previous analysis results are:
  from: runtime: INF, size: O(1) [0]

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

(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:

   2nd(v0) -> null_2nd [0]

And the following fresh constants:    null_2nd, const

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

(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:

   2nd(cons1(X, cons(Y, Z))) -> Y [1]
   2nd(cons(X, X1)) -> 2nd(cons1(X, X1)) [1]
   from(X) -> cons(X, from(s(X))) [1]
   2nd(v0) -> null_2nd [0]

The TRS has the following type information:
2nd :: cons:cons1 -> s:null_2nd
cons1 :: s:null_2nd -> cons:cons1 -> cons:cons1
cons :: s:null_2nd -> cons:cons1 -> cons:cons1
from :: s:null_2nd -> cons:cons1
s :: s:null_2nd -> s:null_2nd
null_2nd :: s:null_2nd
const :: cons:cons1

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:

   null_2nd => 0
   const => 0

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

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

   2nd(z) -{ 1 }-> Y :|: z = 1 + X + (1 + Y + Z), Z >= 0, Y >= 0, X >= 0
   2nd(z) -{ 1 }-> 2nd(1 + X + X1) :|: X1 >= 0, X >= 0, z = 1 + X + X1
   2nd(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   from(z) -{ 1 }-> 1 + X + from(1 + 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
----------------------------------------

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   2ND(cons1(z0, cons(z1, z2))) -> c
   2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, z1)))
   FROM(z0) -> c2(FROM(s(z0)))
S tuples:
   2ND(cons1(z0, cons(z1, z2))) -> c
   2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, z1)))
   FROM(z0) -> c2(FROM(s(z0)))
K tuples:none
Defined Rule Symbols:   2nd_1, from_1

Defined Pair Symbols:   2ND_1, FROM_1

Compound Symbols:   c, c1_1, c2_1


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

(31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   2ND(cons(z0, z1)) -> c1(2ND(cons1(z0, z1)))
   2ND(cons1(z0, cons(z1, z2))) -> c

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))
Tuples:
   FROM(z0) -> c2(FROM(s(z0)))
S tuples:
   FROM(z0) -> c2(FROM(s(z0)))
K tuples:none
Defined Rule Symbols:   2nd_1, from_1

Defined Pair Symbols:   FROM_1

Compound Symbols:   c2_1


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

(33) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   2nd(cons1(z0, cons(z1, z2))) -> z1
   2nd(cons(z0, z1)) -> 2nd(cons1(z0, z1))
   from(z0) -> cons(z0, from(s(z0)))

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

(34)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   FROM_1

Compound Symbols:   c2_1


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

(35) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace FROM(z0) -> c2(FROM(s(z0))) by 
   FROM(s(x0)) -> c2(FROM(s(s(x0))))

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   FROM(s(x0)) -> c2(FROM(s(s(x0))))
S tuples:
   FROM(s(x0)) -> c2(FROM(s(s(x0))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   FROM_1

Compound Symbols:   c2_1


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

(37) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace FROM(s(x0)) -> c2(FROM(s(s(x0)))) by 
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
S tuples:
   FROM(s(s(x0))) -> c2(FROM(s(s(s(x0)))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   FROM_1

Compound Symbols:   c2_1