KILLED
proof of input_vFq9GwE70K.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 0 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) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxRelTRS
        (21) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
        (22) CpxTRS
        (23) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (24) CpxWeightedTrs
            (25) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
            (26) CpxTypedWeightedTrs
            (27) CompletionProof [UPPER BOUND(ID), 0 ms]
            (28) CpxTypedWeightedCompleteTrs
                (29) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
                (30) CpxTypedWeightedCompleteTrs
                (31) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (32) CpxRNTS
                (33) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
                (34) CpxRNTS
            (35) CompletionProof [UPPER BOUND(ID), 0 ms]
            (36) CpxTypedWeightedCompleteTrs
                (37) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (38) CpxRNTS
    (39) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
        (41) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (42) CdtProblem
        (43) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (44) CdtProblem
        (45) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (46) CdtProblem
(47) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(48) CpxWeightedTrs
    (49) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CpxTypedWeightedTrs
    (51) CompletionProof [UPPER BOUND(ID), 0 ms]
    (52) CpxTypedWeightedCompleteTrs
        (53) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (54) CpxTypedWeightedCompleteTrs
        (55) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (56) CpxRNTS
        (57) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (58) CpxRNTS
    (59) CompletionProof [UPPER BOUND(ID), 0 ms]
    (60) CpxTypedWeightedCompleteTrs
        (61) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (62) 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(f(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:

   f(x) -> f(f(x))

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

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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

(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(f(z0)), F(z0))

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

   f(z0) -> f(f(z0))

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(f(z0)), F(z0))

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

   f(z0) -> f(f(z0))

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

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

(10)
Obligation:
Innermost TRS:
Rules:
F(z0) -> c(F(f(z0)), F(z0))
f(z0) -> f(f(z0))

Types:
F :: f -> c
c :: c -> c -> c
f :: f -> f
hole_c1_1 :: c
hole_f2_1 :: f
gen_c3_1 :: Nat -> c

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

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

They will be analysed ascendingly in the following order:
f < F

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

(12)
Obligation:
Innermost TRS:
Rules:
F(z0) -> c(F(f(z0)), F(z0))
f(z0) -> f(f(z0))

Types:
F :: f -> c
c :: c -> c -> c
f :: f -> f
hole_c1_1 :: c
hole_f2_1 :: f
gen_c3_1 :: Nat -> c


Generator Equations:
gen_c3_1(0) <=> hole_c1_1
gen_c3_1(+(x, 1)) <=> c(hole_c1_1, gen_c3_1(x))


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

They will be analysed ascendingly in the following order:
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(f(z0)), F(z0))

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

   f(z0) -> f(f(z0))

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(f(x))

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

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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

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

(20)
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(f(z0)), F(z0))

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

   f(z0) -> f(f(z0))

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

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

(22)
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(f(z0)), F(z0))
   f(z0) -> f(f(z0))

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

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

(24)
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(f(z0)), F(z0)) [1]
   f(z0) -> f(f(z0)) [0]

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

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

(26)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   F(z0) -> c(F(f(z0)), F(z0)) [1]
   f(z0) -> f(f(z0)) [0]

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

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

(27) 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:

   f_1

Due to the following rules being added:

   f(v0) -> const1 [0]

And the following fresh constants:    const1, const

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

(28)
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(f(z0)), F(z0)) [1]
   f(z0) -> f(f(z0)) [0]
   f(v0) -> const1 [0]

The TRS has the following type information:
F :: const1 -> c
c :: c -> c -> c
f :: const1 -> const1
const1 :: const1
const :: c

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

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

(30)
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(f(f(z0))), F(z0)) [1]
   F(z0) -> c(F(const1), F(z0)) [1]
   f(z0) -> f(f(f(z0))) [0]
   f(z0) -> f(const1) [0]
   f(v0) -> const1 [0]

The TRS has the following type information:
F :: const1 -> c
c :: c -> c -> c
f :: const1 -> const1
const1 :: const1
const :: c

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

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

   const1 => 0
   const => 0

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

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

   F(z) -{ 1 }-> 1 + F(f(f(z0))) + F(z0) :|: z = z0, z0 >= 0
   F(z) -{ 1 }-> 1 + F(0) + F(z0) :|: z = z0, z0 >= 0
   f(z) -{ 0 }-> f(f(f(z0))) :|: z = z0, z0 >= 0
   f(z) -{ 0 }-> f(0) :|: z = z0, z0 >= 0
   f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

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

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

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


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

(35) 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, const

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

(36)
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(f(z0)), F(z0)) [1]
   f(z0) -> f(f(z0)) [0]
   f(v0) -> null_f [0]

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

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:

   null_f => 0
   const => 0

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

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

   F(z) -{ 1 }-> 1 + F(f(z0)) + F(z0) :|: z = z0, z0 >= 0
   f(z) -{ 0 }-> f(f(z0)) :|: z = z0, z0 >= 0
   f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.

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

(39) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(z0) -> c(F(f(z0)), F(z0)) by 
   F(z0) -> c(F(f(f(z0))), F(z0))

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

(40)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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

(41) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(z0) -> c(F(f(f(z0))), F(z0)) by 
   F(x0) -> c(F(f(f(f(x0)))), F(x0))

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0) -> f(f(z0))
Tuples:
   F(x0) -> c(F(f(f(f(x0)))), F(x0))
S tuples:
   F(x0) -> c(F(f(f(f(x0)))), F(x0))
K tuples:none
Defined Rule Symbols:   f_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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

(43) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(x0) -> c(F(f(f(f(x0)))), F(x0)) by F(z0) -> c(F(f(f(f(f(z0))))), F(z0))
----------------------------------------

(44)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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

(45) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(z0) -> c(F(f(f(f(f(z0))))), F(z0)) by F(z0) -> c(F(f(f(f(f(f(z0)))))), F(z0))
----------------------------------------

(46)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_1

Compound Symbols:   c_2


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

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

(48)
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(f(x)) [1]

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

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

(50)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(x) -> f(f(x)) [1]

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

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

(51) 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

Due to the following rules being added:
none

And the following fresh constants:    const

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

(52)
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(f(x)) [1]

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

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

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

(54)
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(f(f(x))) [2]

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

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

(55) 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

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

(56)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z) -{ 2 }-> f(f(f(x))) :|: x >= 0, z = x


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

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

(58)
Obligation:
Complexity RNTS consisting of the following rules:

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


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

(59) 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

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

(60)
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(f(x)) [1]

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

   const => 0

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

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

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

Only complete derivations are relevant for the runtime complexity.