MAYBE
proof of input_16bzADiqRw.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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 [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
        (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 6 ms]
        (12) typed CpxTrs
        (13) OrderProof [LOWER BOUND(ID), 0 ms]
        (14) typed CpxTrs
    (15) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
    (16) TRS for Loop Detection
(17) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(18) CdtProblem
    (19) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CdtProblem
    (21) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CdtProblem
    (23) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CdtProblem
    (25) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CpxTRS
    (27) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (28) CpxTRS
    (29) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) CpxWeightedTrs
        (31) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
        (32) CpxTypedWeightedTrs
        (33) CompletionProof [UPPER BOUND(ID), 0 ms]
        (34) CpxTypedWeightedCompleteTrs
            (35) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
            (36) CpxTypedWeightedCompleteTrs
            (37) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
            (38) CpxRNTS
            (39) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
            (40) CpxRNTS
            (41) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
            (42) CpxRNTS
            (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
            (44) CpxRNTS
            (45) IntTrsBoundProof [UPPER BOUND(ID), 98 ms]
            (46) CpxRNTS
            (47) IntTrsBoundProof [UPPER BOUND(ID), 32 ms]
            (48) CpxRNTS
        (49) CompletionProof [UPPER BOUND(ID), 0 ms]
        (50) CpxTypedWeightedCompleteTrs
            (51) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
            (52) CpxRNTS
(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) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (60) CpxRNTS
    (61) CompletionProof [UPPER BOUND(ID), 0 ms]
    (62) CpxTypedWeightedCompleteTrs
        (63) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (64) CpxTypedWeightedCompleteTrs
        (65) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (66) CpxRNTS
        (67) InliningProof [UPPER BOUND(ID), 0 ms]
        (68) CpxRNTS
        (69) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (70) CpxRNTS
        (71) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (72) CpxRNTS
        (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (74) CpxRNTS
        (75) IntTrsBoundProof [UPPER BOUND(ID), 118 ms]
        (76) CpxRNTS
        (77) IntTrsBoundProof [UPPER BOUND(ID), 8 ms]
        (78) CpxRNTS
        (79) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (80) CpxRNTS
        (81) IntTrsBoundProof [UPPER BOUND(ID), 172 ms]
        (82) CpxRNTS
        (83) IntTrsBoundProof [UPPER BOUND(ID), 43 ms]
        (84) 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:

   c -> f(g(c))
   f(g(X)) -> g(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:

   c -> f(g(c))
   f(g(X)) -> g(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:

   c -> f(g(c))
   f(g(X)) -> g(X)

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   C, F_1

Compound Symbols:   c1_2, c2


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

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

(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:

   C -> c1(F(g(c)), C)
   F(g(z0)) -> c2

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

   c -> f(g(c))
   f(g(z0)) -> g(z0)

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

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

(10)
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:

   C -> c1(F(g(c)), C)
   F(g(z0)) -> c2

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

   c -> f(g(c))
   f(g(z0)) -> g(z0)

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

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

(12)
Obligation:
Innermost TRS:
Rules:
C -> c1(F(g(c)), C)
F(g(z0)) -> c2
c -> f(g(c))
f(g(z0)) -> g(z0)

Types:
C :: c1
c1 :: c2 -> c1 -> c1
F :: g -> c2
g :: g -> g
c :: g
c2 :: c2
f :: g -> g
hole_c11_3 :: c1
hole_c22_3 :: c2
hole_g3_3 :: g
gen_c14_3 :: Nat -> c1
gen_g5_3 :: Nat -> g

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

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
C, c

They will be analysed ascendingly in the following order:
c < C

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

(14)
Obligation:
Innermost TRS:
Rules:
C -> c1(F(g(c)), C)
F(g(z0)) -> c2
c -> f(g(c))
f(g(z0)) -> g(z0)

Types:
C :: c1
c1 :: c2 -> c1 -> c1
F :: g -> c2
g :: g -> g
c :: g
c2 :: c2
f :: g -> g
hole_c11_3 :: c1
hole_c22_3 :: c2
hole_g3_3 :: g
gen_c14_3 :: Nat -> c1
gen_g5_3 :: Nat -> g


Generator Equations:
gen_c14_3(0) <=> hole_c11_3
gen_c14_3(+(x, 1)) <=> c1(c2, gen_c14_3(x))
gen_g5_3(0) <=> hole_g3_3
gen_g5_3(+(x, 1)) <=> g(gen_g5_3(x))


The following defined symbols remain to be analysed:
c, C

They will be analysed ascendingly in the following order:
c < C

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

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

(16)
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:

   C -> c1(F(g(c)), C)
   F(g(z0)) -> c2

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

   c -> f(g(c))
   f(g(z0)) -> g(z0)

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

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

(18)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   C, F_1

Compound Symbols:   c1_2, c2


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

(19) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(g(z0)) -> c2

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

(20)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   C

Compound Symbols:   c1_2


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

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

(22)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   C

Compound Symbols:   c1_1


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

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

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

(24)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   C

Compound Symbols:   c1_1


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

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

(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:

   C -> c1(C)

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

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

(28)
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:

   C -> c1(C)

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

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

(30)
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:

   C -> c1(C) [1]

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

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

(32)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   C -> c1(C) [1]

The TRS has the following type information:
C :: c1
c1 :: c1 -> c1

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

(33) 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:

   C

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const

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

(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:

   C -> c1(C) [1]

The TRS has the following type information:
C :: c1
c1 :: c1 -> c1
const :: c1

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

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

(36)
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:

   C -> c1(C) [1]

The TRS has the following type information:
C :: c1
c1 :: c1 -> c1
const :: c1

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

(37) 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

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

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

   C -{ 1 }-> 1 + C :|: 


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

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

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

   C -{ 1 }-> 1 + C :|: 


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

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

   { C }

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

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

   C -{ 1 }-> 1 + C :|: 

Function symbols to be analyzed: {C}

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

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

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

   C -{ 1 }-> 1 + C :|: 

Function symbols to be analyzed: {C}

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   C -{ 1 }-> 1 + C :|: 

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

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   C -{ 1 }-> 1 + C :|: 

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

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

(49) 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

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

(50)
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:

   C -> c1(C) [1]

The TRS has the following type information:
C :: c1
c1 :: c1 -> c1
const :: c1

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

(51) 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

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

(52)
Obligation:
Complexity RNTS consisting of the following rules:

   C -{ 1 }-> 1 + C :|: 

Only complete derivations are relevant for the runtime complexity.

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

(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:

   c -> f(g(c)) [1]
   f(g(X)) -> g(X) [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:

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

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

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

(57) 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]

And the following fresh constants:    null_f

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

(58)
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:

   c -> f(g(c)) [1]
   f(g(X)) -> g(X) [1]
   f(v0) -> null_f [0]

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

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

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

   null_f => 0

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

(60)
Obligation:
Complexity RNTS consisting of the following rules:

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

Only complete derivations are relevant for the runtime complexity.

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

(61) 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:

   c
   f_1

Due to the following rules being added:

   f(v0) -> const [0]

And the following fresh constants:    const

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

(62)
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:

   c -> f(g(c)) [1]
   f(g(X)) -> g(X) [1]
   f(v0) -> const [0]

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

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

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

(64)
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:

   c -> f(g(f(g(c)))) [2]
   f(g(X)) -> g(X) [1]
   f(v0) -> const [0]

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

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

(65) 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

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

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

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


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

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

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

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

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

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


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

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

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

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


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

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

   { f }
   { c }

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

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

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

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

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

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

(74)
Obligation:
Complexity RNTS consisting of the following rules:

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

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

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

(75) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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(77) IntTrsBoundProof (UPPER BOUND(ID))

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

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(78)
Obligation:
Complexity RNTS consisting of the following rules:

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

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

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(79) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(80)
Obligation:
Complexity RNTS consisting of the following rules:

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

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

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(81) IntTrsBoundProof (UPPER BOUND(ID))

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

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(82)
Obligation:
Complexity RNTS consisting of the following rules:

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

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

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(83) IntTrsBoundProof (UPPER BOUND(ID))

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

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(84)
Obligation:
Complexity RNTS consisting of the following rules:

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

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