MAYBE
proof of input_HH8dwZlDWH.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) InliningProof [UPPER BOUND(ID), 0 ms]
        (16) CpxRNTS
        (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
        (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 220 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 68 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 163 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
        (32) CpxRNTS
    (33) CompletionProof [UPPER BOUND(ID), 0 ms]
    (34) CpxTypedWeightedCompleteTrs
        (35) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (36) CpxRNTS
(37) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(38) CdtProblem
    (39) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CdtProblem
    (43) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) 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(X) -> if(X, c, f(true))
   if(true, X, Y) -> X
   if(false, X, Y) -> Y

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

(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) -> if(X, c, f(true))
   if(true, X, Y) -> X
   if(false, X, Y) -> Y

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

(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(X) -> if(X, c, f(true))
   if(true, X, Y) -> X
   if(false, X, Y) -> Y

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

(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(X) -> if(X, c, f(true)) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [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(X) -> if(X, c, f(true)) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]

The TRS has the following type information:
f :: true:false -> c
if :: true:false -> c -> c -> c
c :: c
true :: true:false
false :: true:false

Rewrite Strategy: INNERMOST
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(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:
none

(c) The following functions are completely defined:

   f_1
   if_3

Due to the following rules being added:
none

And the following fresh constants: none

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

(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:

   f(X) -> if(X, c, f(true)) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]

The TRS has the following type information:
f :: true:false -> c
if :: true:false -> c -> c -> c
c :: c
true :: true:false
false :: true:false

Rewrite Strategy: INNERMOST
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(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:

   f(X) -> if(X, c, if(true, c, f(true))) [2]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]

The TRS has the following type information:
f :: true:false -> c
if :: true:false -> c -> c -> c
c :: c
true :: true:false
false :: true:false

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:

   c => 0
   true => 1
   false => 0

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

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


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(15) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
   if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0

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

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


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(17) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
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(18)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z) -{ 2 }-> if(z, 0, if(1, 0, f(1))) :|: z >= 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0


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

   { if }
   { f }

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

   f(z) -{ 2 }-> if(z, 0, if(1, 0, f(1))) :|: z >= 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0

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

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

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

   f(z) -{ 2 }-> if(z, 0, if(1, 0, f(1))) :|: z >= 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0

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

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(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   f(z) -{ 2 }-> if(z, 0, if(1, 0, f(1))) :|: z >= 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0

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

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(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z) -{ 2 }-> if(z, 0, if(1, 0, f(1))) :|: z >= 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0

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

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

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

   f(z) -{ 2 }-> if(z, 0, if(1, 0, f(1))) :|: z >= 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0

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

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(29) 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

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

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

   f(z) -{ 2 }-> if(z, 0, if(1, 0, f(1))) :|: z >= 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  f: runtime: ?, size: O(1) [0]

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(31) 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: ?

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

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

   f(z) -{ 2 }-> if(z, 0, if(1, 0, f(1))) :|: z >= 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  f: runtime: INF, size: O(1) [0]

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(33) 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: none

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(34)
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) -> if(X, c, f(true)) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]

The TRS has the following type information:
f :: true:false -> c
if :: true:false -> c -> c -> c
c :: c
true :: true:false
false :: true:false

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

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

   c => 0
   true => 1
   false => 0

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

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

   f(z) -{ 1 }-> if(X, 0, f(1)) :|: X >= 0, z = X
   if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
   if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0

Only complete derivations are relevant for the runtime complexity.

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0) -> if(z0, c, f(true))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
Tuples:
   F(z0) -> c1(IF(z0, c, f(true)), F(true))
   IF(true, z0, z1) -> c2
   IF(false, z0, z1) -> c3
S tuples:
   F(z0) -> c1(IF(z0, c, f(true)), F(true))
   IF(true, z0, z1) -> c2
   IF(false, z0, z1) -> c3
K tuples:none
Defined Rule Symbols:   f_1, if_3

Defined Pair Symbols:   F_1, IF_3

Compound Symbols:   c1_2, c2, c3


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(39) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   IF(false, z0, z1) -> c3
   IF(true, z0, z1) -> c2

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0) -> if(z0, c, f(true))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
Tuples:
   F(z0) -> c1(IF(z0, c, f(true)), F(true))
S tuples:
   F(z0) -> c1(IF(z0, c, f(true)), F(true))
K tuples:none
Defined Rule Symbols:   f_1, if_3

Defined Pair Symbols:   F_1

Compound Symbols:   c1_2


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

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0) -> if(z0, c, f(true))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1
Tuples:
   F(z0) -> c1(F(true))
S tuples:
   F(z0) -> c1(F(true))
K tuples:none
Defined Rule Symbols:   f_1, if_3

Defined Pair Symbols:   F_1

Compound Symbols:   c1_1


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(43) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(z0) -> if(z0, c, f(true))
   if(true, z0, z1) -> z0
   if(false, z0, z1) -> z1

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

(44)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_1

Compound Symbols:   c1_1


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(45) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace F(z0) -> c1(F(true)) by 
   F(true) -> c1(F(true))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_1

Compound Symbols:   c1_1