MAYBE
proof of input_oFhqlNvuA6.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CdtProblem
    (5) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxRelTRS
    (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
        (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
        (10) typed CpxTrs
        (11) OrderProof [LOWER BOUND(ID), 1 ms]
        (12) typed CpxTrs
    (13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
    (14) TRS for Loop Detection
(15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(16) CpxTRS
(17) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(18) CdtProblem
    (19) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CdtProblem
    (21) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CdtProblem
    (23) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CpxTRS
        (25) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
        (26) CpxTRS
        (27) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (28) CpxWeightedTrs
            (29) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
            (30) CpxTypedWeightedTrs
            (31) CompletionProof [UPPER BOUND(ID), 0 ms]
            (32) CpxTypedWeightedCompleteTrs
                (33) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
                (34) CpxTypedWeightedCompleteTrs
                (35) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (36) CpxRNTS
                (37) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
                (38) CpxRNTS
                (39) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
                (40) CpxRNTS
                (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (42) CpxRNTS
                (43) IntTrsBoundProof [UPPER BOUND(ID), 231 ms]
                (44) CpxRNTS
                (45) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
                (46) CpxRNTS
            (47) CompletionProof [UPPER BOUND(ID), 0 ms]
            (48) CpxTypedWeightedCompleteTrs
                (49) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (50) CpxRNTS
    (51) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
(53) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(54) CpxWeightedTrs
    (55) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CpxTypedWeightedTrs
    (57) CompletionProof [UPPER BOUND(ID), 0 ms]
    (58) CpxTypedWeightedCompleteTrs
        (59) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (60) CpxTypedWeightedCompleteTrs
        (61) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (62) CpxRNTS
        (63) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (64) CpxRNTS
        (65) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (66) CpxRNTS
        (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (68) CpxRNTS
        (69) IntTrsBoundProof [UPPER BOUND(ID), 143 ms]
        (70) CpxRNTS
        (71) IntTrsBoundProof [UPPER BOUND(ID), 66 ms]
        (72) CpxRNTS
    (73) CompletionProof [UPPER BOUND(ID), 0 ms]
    (74) CpxTypedWeightedCompleteTrs
        (75) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (76) 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(a)
   b -> a

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(a)
   b -> a

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

(3) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(4)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_1, B

Compound Symbols:   c_1, c1


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

(5) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   F(z0) -> c(F(a))
   B -> c1

The (relative) TRS S consists of the following rules:

   f(z0) -> f(a)
   b -> a

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

(7) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   F(z0) -> c(F(a))
   B -> c1

The (relative) TRS S consists of the following rules:

   f(z0) -> f(a)
   b -> a

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

(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(10)
Obligation:
Innermost TRS:
Rules:
F(z0) -> c(F(a))
B -> c1
f(z0) -> f(a)
b -> a

Types:
F :: a -> c
c :: c -> c
a :: a
B :: c1
c1 :: c1
f :: a -> f
b :: a
hole_c1_2 :: c
hole_a2_2 :: a
hole_c13_2 :: c1
hole_f4_2 :: f
gen_c5_2 :: Nat -> c

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

(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
F, f
----------------------------------------

(12)
Obligation:
Innermost TRS:
Rules:
F(z0) -> c(F(a))
B -> c1
f(z0) -> f(a)
b -> a

Types:
F :: a -> c
c :: c -> c
a :: a
B :: c1
c1 :: c1
f :: a -> f
b :: a
hole_c1_2 :: c
hole_a2_2 :: a
hole_c13_2 :: c1
hole_f4_2 :: f
gen_c5_2 :: Nat -> c


Generator Equations:
gen_c5_2(0) <=> hole_c1_2
gen_c5_2(+(x, 1)) <=> c(gen_c5_2(x))


The following defined symbols remain to be analysed:
F, f
----------------------------------------

(13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(14)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   F(z0) -> c(F(a))
   B -> c1

The (relative) TRS S consists of the following rules:

   f(z0) -> f(a)
   b -> a

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

(15) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(16)
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(a)
   b -> a

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

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

(18)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_1, B

Compound Symbols:   c_1, c1


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

(19) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   B -> c1

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

(20)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_1

Compound Symbols:   c_1


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

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

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

(22)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_1

Compound Symbols:   c_1


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

(23) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(24)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   F(z0) -> c(F(a))

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

(25) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(26)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   F(z0) -> c(F(a))

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

(27) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(28)
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(z0) -> c(F(a)) [1]

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

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

(30)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   F(z0) -> c(F(a)) [1]

The TRS has the following type information:
F :: a -> c
c :: c -> c
a :: a

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

(31) 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:
none

Due to the following rules being added:
none

And the following fresh constants:    const

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

(32)
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(z0) -> c(F(a)) [1]

The TRS has the following type information:
F :: a -> c
c :: c -> c
a :: a
const :: c

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

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

(34)
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(z0) -> c(F(a)) [1]

The TRS has the following type information:
F :: a -> c
c :: c -> c
a :: a
const :: c

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:

   a => 0
   const => 0

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

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

   F(z) -{ 1 }-> 1 + F(0) :|: z = z0, z0 >= 0


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

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

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

   F(z) -{ 1 }-> 1 + F(0) :|: z >= 0


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

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

   { F }

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

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

   F(z) -{ 1 }-> 1 + F(0) :|: z >= 0

Function symbols to be analyzed: {F}

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

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

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

   F(z) -{ 1 }-> 1 + F(0) :|: z >= 0

Function symbols to be analyzed: {F}

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

(43) 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

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

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z) -{ 1 }-> 1 + F(0) :|: z >= 0

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

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

(45) 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: ?

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

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z) -{ 1 }-> 1 + F(0) :|: z >= 0

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

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

(47) 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

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

(48)
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(z0) -> c(F(a)) [1]

The TRS has the following type information:
F :: a -> c
c :: c -> c
a :: a
const :: c

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

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

   a => 0
   const => 0

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

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z) -{ 1 }-> 1 + F(0) :|: z = z0, z0 >= 0

Only complete derivations are relevant for the runtime complexity.

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

(51) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace F(z0) -> c(F(a)) by 
   F(a) -> c(F(a))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_1

Compound Symbols:   c_1


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

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

(54)
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(a) [1]
   b -> a [1]

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

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

(56)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

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

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

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

(57) 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
   b

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const

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

(58)
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(a) [1]
   b -> a [1]

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

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

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

(60)
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(a) [1]
   b -> a [1]

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

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

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

   a => 0
   const => 0

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

(62)
Obligation:
Complexity RNTS consisting of the following rules:

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


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

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

(64)
Obligation:
Complexity RNTS consisting of the following rules:

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


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

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

   { f }
   { b }

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

(66)
Obligation:
Complexity RNTS consisting of the following rules:

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

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

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

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

(68)
Obligation:
Complexity RNTS consisting of the following rules:

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

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

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

(69) 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

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

(70)
Obligation:
Complexity RNTS consisting of the following rules:

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

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

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

(71) 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: ?

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

(72)
Obligation:
Complexity RNTS consisting of the following rules:

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

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

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

(73) 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

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

(74)
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(a) [1]
   b -> a [1]

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

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

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

   a => 0
   const => 0

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

(76)
Obligation:
Complexity RNTS consisting of the following rules:

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

Only complete derivations are relevant for the runtime complexity.