MAYBE
proof of input_1tdIpNIHqd.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (12) CpxRNTS
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
        (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxTypedWeightedCompleteTrs
        (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (18) CpxRNTS
        (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
        (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (22) CpxRNTS
        (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 502 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 78 ms]
        (28) CpxRNTS
(29) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(30) CdtProblem
    (31) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) CdtProblem
    (33) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) CdtProblem
    (35) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (36) CdtProblem
    (37) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) 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(0) -> cons(0, f(s(0)))
   f(s(0)) -> f(p(s(0)))
   p(s(0)) -> 0

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(0') -> cons(0', f(s(0')))
   f(s(0')) -> f(p(s(0')))
   p(s(0')) -> 0'

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(0) -> cons(0, f(s(0)))
   f(s(0)) -> f(p(s(0)))
   p(s(0)) -> 0

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(0) -> cons(0, f(s(0))) [1]
   f(s(0)) -> f(p(s(0))) [1]
   p(s(0)) -> 0 [1]

Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

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

   f(0) -> cons(0, f(s(0))) [1]
   f(s(0)) -> f(p(s(0))) [1]
   p(s(0)) -> 0 [1]

The TRS has the following type information:
f :: 0:s -> cons
0 :: 0:s
cons :: 0:s -> cons -> cons
s :: 0:s -> 0:s
p :: 0:s -> 0:s

Rewrite Strategy: INNERMOST
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(9) 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:

   f(v0) -> null_f [0]
   p(v0) -> null_p [0]

And the following fresh constants:    null_f, null_p

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(10)
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(0) -> cons(0, f(s(0))) [1]
   f(s(0)) -> f(p(s(0))) [1]
   p(s(0)) -> 0 [1]
   f(v0) -> null_f [0]
   p(v0) -> null_p [0]

The TRS has the following type information:
f :: 0:s:null_p -> cons:null_f
0 :: 0:s:null_p
cons :: 0:s:null_p -> cons:null_f -> cons:null_f
s :: 0:s:null_p -> 0:s:null_p
p :: 0:s:null_p -> 0:s:null_p
null_f :: cons:null_f
null_p :: 0:s:null_p

Rewrite Strategy: INNERMOST
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(11) 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
   null_f => 0
   null_p => 0

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(12)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z) -{ 1 }-> f(p(1 + 0)) :|: z = 1 + 0
   f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   f(z) -{ 1 }-> 1 + 0 + f(1 + 0) :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.

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(13) 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

(c) The following functions are completely defined:

   p_1

Due to the following rules being added:

   p(v0) -> 0 [0]

And the following fresh constants:    const

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(14)
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(0) -> cons(0, f(s(0))) [1]
   f(s(0)) -> f(p(s(0))) [1]
   p(s(0)) -> 0 [1]
   p(v0) -> 0 [0]

The TRS has the following type information:
f :: 0:s -> cons
0 :: 0:s
cons :: 0:s -> cons -> cons
s :: 0:s -> 0:s
p :: 0:s -> 0:s
const :: cons

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

   f(0) -> cons(0, f(s(0))) [1]
   f(s(0)) -> f(0) [2]
   f(s(0)) -> f(0) [1]
   p(s(0)) -> 0 [1]
   p(v0) -> 0 [0]

The TRS has the following type information:
f :: 0:s -> cons
0 :: 0:s
cons :: 0:s -> cons -> cons
s :: 0:s -> 0:s
p :: 0:s -> 0:s
const :: cons

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

(17) 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

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

   f(z) -{ 2 }-> f(0) :|: z = 1 + 0
   f(z) -{ 1 }-> f(0) :|: z = 1 + 0
   f(z) -{ 1 }-> 1 + 0 + f(1 + 0) :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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(19) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
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(20)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z) -{ 2 }-> f(0) :|: z = 1 + 0
   f(z) -{ 1 }-> f(0) :|: z = 1 + 0
   f(z) -{ 1 }-> 1 + 0 + f(1 + 0) :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 0 }-> 0 :|: z >= 0


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

   { f }
   { p }

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

   f(z) -{ 2 }-> f(0) :|: z = 1 + 0
   f(z) -{ 1 }-> f(0) :|: z = 1 + 0
   f(z) -{ 1 }-> 1 + 0 + f(1 + 0) :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 0 }-> 0 :|: z >= 0

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

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(23) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(24)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z) -{ 2 }-> f(0) :|: z = 1 + 0
   f(z) -{ 1 }-> f(0) :|: z = 1 + 0
   f(z) -{ 1 }-> 1 + 0 + f(1 + 0) :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 0 }-> 0 :|: z >= 0

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

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(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   f(z) -{ 2 }-> f(0) :|: z = 1 + 0
   f(z) -{ 1 }-> f(0) :|: z = 1 + 0
   f(z) -{ 1 }-> 1 + 0 + f(1 + 0) :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 0 }-> 0 :|: z >= 0

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

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(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   f(z) -{ 2 }-> f(0) :|: z = 1 + 0
   f(z) -{ 1 }-> f(0) :|: z = 1 + 0
   f(z) -{ 1 }-> 1 + 0 + f(1 + 0) :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 0 }-> 0 :|: z >= 0

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

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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(0) -> cons(0, f(s(0)))
   f(s(0)) -> f(p(s(0)))
   p(s(0)) -> 0
Tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(p(s(0))), P(s(0)))
   P(s(0)) -> c2
S tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(p(s(0))), P(s(0)))
   P(s(0)) -> c2
K tuples:none
Defined Rule Symbols:   f_1, p_1

Defined Pair Symbols:   F_1, P_1

Compound Symbols:   c_1, c1_2, c2


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(31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   P(s(0)) -> c2

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(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> cons(0, f(s(0)))
   f(s(0)) -> f(p(s(0)))
   p(s(0)) -> 0
Tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(p(s(0))), P(s(0)))
S tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(p(s(0))), P(s(0)))
K tuples:none
Defined Rule Symbols:   f_1, p_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_1, c1_2


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(33) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
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(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> cons(0, f(s(0)))
   f(s(0)) -> f(p(s(0)))
   p(s(0)) -> 0
Tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(p(s(0))))
S tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(p(s(0))))
K tuples:none
Defined Rule Symbols:   f_1, p_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_1, c1_1


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(35) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(0) -> cons(0, f(s(0)))
   f(s(0)) -> f(p(s(0)))

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(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
Tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(p(s(0))))
S tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(p(s(0))))
K tuples:none
Defined Rule Symbols:   p_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_1, c1_1


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(37) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0)) -> c1(F(p(s(0)))) by 
   F(s(0)) -> c1(F(0))

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(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(0)) -> 0
Tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(0))
S tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(0))
K tuples:none
Defined Rule Symbols:   p_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_1, c1_1


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(39) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   p(s(0)) -> 0

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(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(0))
S tuples:
   F(0) -> c(F(s(0)))
   F(s(0)) -> c1(F(0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_1

Compound Symbols:   c_1, c1_1