MAYBE
proof of input_y9B4STMns3.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CdtProblem
    (13) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CdtProblem
    (15) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CdtProblem
    (17) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CdtProblem
(19) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(20) CpxWeightedTrs
    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxTypedWeightedTrs
    (23) CompletionProof [UPPER BOUND(ID), 0 ms]
    (24) CpxTypedWeightedCompleteTrs
        (25) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (26) CpxTypedWeightedCompleteTrs
        (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (30) CpxRNTS
        (31) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (32) CpxRNTS
        (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 124 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
        (38) CpxRNTS
    (39) CompletionProof [UPPER BOUND(ID), 0 ms]
    (40) CpxTypedWeightedCompleteTrs
        (41) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (42) 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:

   f(X) -> f(c)
   c -> b

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:

   f(X) -> f(c)
   c -> b

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:

   f(X) -> f(c)
   c -> b

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:
   f(z0) -> f(c)
   c -> b
Tuples:
   F(z0) -> c1(F(c), C)
   C -> c2
S tuples:
   F(z0) -> c1(F(c), C)
   C -> c2
K tuples:none
Defined Rule Symbols:   f_1, c

Defined Pair Symbols:   F_1, C

Compound Symbols:   c1_2, c2


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

(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   C -> c2

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0) -> f(c)
   c -> b
Tuples:
   F(z0) -> c1(F(c), C)
S tuples:
   F(z0) -> c1(F(c), C)
K tuples:none
Defined Rule Symbols:   f_1, c

Defined Pair Symbols:   F_1

Compound Symbols:   c1_2


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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0) -> f(c)
   c -> b
Tuples:
   F(z0) -> c1(F(c))
S tuples:
   F(z0) -> c1(F(c))
K tuples:none
Defined Rule Symbols:   f_1, c

Defined Pair Symbols:   F_1

Compound Symbols:   c1_1


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

(11) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(z0) -> f(c)

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   c -> b
Tuples:
   F(z0) -> c1(F(c))
S tuples:
   F(z0) -> c1(F(c))
K tuples:none
Defined Rule Symbols:   c

Defined Pair Symbols:   F_1

Compound Symbols:   c1_1


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

(13) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(z0) -> c1(F(c)) by 
   F(x0) -> c1(F(b))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   c -> b
Tuples:
   F(x0) -> c1(F(b))
S tuples:
   F(x0) -> c1(F(b))
K tuples:none
Defined Rule Symbols:   c

Defined Pair Symbols:   F_1

Compound Symbols:   c1_1


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

(15) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   c -> b

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(x0) -> c1(F(b))
S tuples:
   F(x0) -> c1(F(b))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_1

Compound Symbols:   c1_1


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

(17) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace F(x0) -> c1(F(b)) by 
   F(b) -> c1(F(b))

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

(18)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_1

Compound Symbols:   c1_1


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

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

(20)
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(X) -> f(c) [1]
   c -> b [1]

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

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

(22)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(X) -> f(c) [1]
   c -> b [1]

The TRS has the following type information:
f :: b -> f
c :: b
b :: b

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

(23) 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:

   c

Due to the following rules being added:
none

And the following fresh constants:    const

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

(24)
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(X) -> f(c) [1]
   c -> b [1]

The TRS has the following type information:
f :: b -> f
c :: b
b :: b
const :: f

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

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

(26)
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(X) -> f(b) [2]
   c -> b [1]

The TRS has the following type information:
f :: b -> f
c :: b
b :: b
const :: f

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:

   b => 0
   const => 0

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

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

   c -{ 1 }-> 0 :|: 
   f(z) -{ 2 }-> f(0) :|: X >= 0, z = X


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

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

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

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


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

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

   { f }
   { c }

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

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

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

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

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

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

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

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

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

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

(35) 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

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

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

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

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

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

(37) 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: ?

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

(38)
Obligation:
Complexity RNTS consisting of the following rules:

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

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

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

(39) 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

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

(40)
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(X) -> f(c) [1]
   c -> b [1]

The TRS has the following type information:
f :: b -> f
c :: b
b :: b
const :: f

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

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

   b => 0
   const => 0

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

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   c -{ 1 }-> 0 :|: 
   f(z) -{ 1 }-> f(c) :|: X >= 0, z = X

Only complete derivations are relevant for the runtime complexity.