KILLED
proof of input_ULrs3dyRu3.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 1461 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 374 ms]
        (24) CpxRNTS
    (25) CompletionProof [UPPER BOUND(ID), 0 ms]
    (26) CpxTypedWeightedCompleteTrs
        (27) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
(29) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(30) CdtProblem
    (31) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) CdtProblem
    (33) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) CdtProblem
    (35) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (36) CdtProblem
    (37) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CdtProblem


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

(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, empty) -> x
   f(empty, cons(a, k)) -> f(cons(a, k), k)
   f(cons(a, k), y) -> f(y, k)

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, empty) -> x
   f(empty, cons(a, k)) -> f(cons(a, k), k)
   f(cons(a, k), y) -> f(y, k)

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, empty) -> x
   f(empty, cons(a, k)) -> f(cons(a, k), k)
   f(cons(a, k), y) -> f(y, k)

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

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

(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, empty) -> x [1]
   f(empty, cons(a, k)) -> f(cons(a, k), k) [1]
   f(cons(a, k), y) -> f(y, k) [1]

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

(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, empty) -> x [1]
   f(empty, cons(a, k)) -> f(cons(a, k), k) [1]
   f(cons(a, k), y) -> f(y, k) [1]

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

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

(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:

   f_2

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const

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

(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, empty) -> x [1]
   f(empty, cons(a, k)) -> f(cons(a, k), k) [1]
   f(cons(a, k), y) -> f(y, k) [1]

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

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

(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, empty) -> x [1]
   f(empty, cons(a, k)) -> f(cons(a, k), k) [1]
   f(cons(a, k), y) -> f(y, k) [1]

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

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:

   empty => 0
   const => 0

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

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

   f(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   f(z, z') -{ 1 }-> f(y, k) :|: z = 1 + a + k, a >= 0, y >= 0, k >= 0, z' = y
   f(z, z') -{ 1 }-> f(1 + a + k, k) :|: z' = 1 + a + k, a >= 0, k >= 0, z = 0


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

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

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

   f(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   f(z, z') -{ 1 }-> f(z', k) :|: z = 1 + a + k, a >= 0, z' >= 0, k >= 0
   f(z, z') -{ 1 }-> f(1 + a + k, k) :|: z' = 1 + a + k, a >= 0, k >= 0, z = 0


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

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

   { f }

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

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   f(z, z') -{ 1 }-> f(z', k) :|: z = 1 + a + k, a >= 0, z' >= 0, k >= 0
   f(z, z') -{ 1 }-> f(1 + a + k, k) :|: z' = 1 + a + k, a >= 0, k >= 0, z = 0

Function symbols to be analyzed: {f}

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

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

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

   f(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   f(z, z') -{ 1 }-> f(z', k) :|: z = 1 + a + k, a >= 0, z' >= 0, k >= 0
   f(z, z') -{ 1 }-> f(1 + a + k, k) :|: z' = 1 + a + k, a >= 0, k >= 0, z = 0

Function symbols to be analyzed: {f}

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

(21) 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 + z'

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

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

   f(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   f(z, z') -{ 1 }-> f(z', k) :|: z = 1 + a + k, a >= 0, z' >= 0, k >= 0
   f(z, z') -{ 1 }-> f(1 + a + k, k) :|: z' = 1 + a + k, a >= 0, k >= 0, z = 0

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

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

(23) 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: ?

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

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

   f(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   f(z, z') -{ 1 }-> f(z', k) :|: z = 1 + a + k, a >= 0, z' >= 0, k >= 0
   f(z, z') -{ 1 }-> f(1 + a + k, k) :|: z' = 1 + a + k, a >= 0, k >= 0, z = 0

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

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

(25) 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

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

(26)
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, empty) -> x [1]
   f(empty, cons(a, k)) -> f(cons(a, k), k) [1]
   f(cons(a, k), y) -> f(y, k) [1]

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

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:

   empty => 0
   const => 0

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

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

   f(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   f(z, z') -{ 1 }-> f(y, k) :|: z = 1 + a + k, a >= 0, y >= 0, k >= 0, z' = y
   f(z, z') -{ 1 }-> f(1 + a + k, k) :|: z' = 1 + a + k, a >= 0, k >= 0, z = 0

Only complete derivations are relevant for the runtime complexity.

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

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

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0, empty) -> z0
   f(empty, cons(z0, z1)) -> f(cons(z0, z1), z1)
   f(cons(z0, z1), z2) -> f(z2, z1)
Tuples:
   F(z0, empty) -> c
   F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1))
   F(cons(z0, z1), z2) -> c2(F(z2, z1))
S tuples:
   F(z0, empty) -> c
   F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1))
   F(cons(z0, z1), z2) -> c2(F(z2, z1))
K tuples:none
Defined Rule Symbols:   f_2

Defined Pair Symbols:   F_2

Compound Symbols:   c, c1_1, c2_1


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

(31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(z0, empty) -> c

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0, empty) -> z0
   f(empty, cons(z0, z1)) -> f(cons(z0, z1), z1)
   f(cons(z0, z1), z2) -> f(z2, z1)
Tuples:
   F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1))
   F(cons(z0, z1), z2) -> c2(F(z2, z1))
S tuples:
   F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1))
   F(cons(z0, z1), z2) -> c2(F(z2, z1))
K tuples:none
Defined Rule Symbols:   f_2

Defined Pair Symbols:   F_2

Compound Symbols:   c1_1, c2_1


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

(33) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(z0, empty) -> z0
   f(empty, cons(z0, z1)) -> f(cons(z0, z1), z1)
   f(cons(z0, z1), z2) -> f(z2, z1)

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1))
   F(cons(z0, z1), z2) -> c2(F(z2, z1))
S tuples:
   F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1))
   F(cons(z0, z1), z2) -> c2(F(z2, z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_2

Compound Symbols:   c1_1, c2_1


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

(35) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(cons(z0, z1), z2) -> c2(F(z2, z1)) by 
   F(cons(z0, cons(y0, y1)), empty) -> c2(F(empty, cons(y0, y1)))
   F(cons(z0, z1), cons(y0, y1)) -> c2(F(cons(y0, y1), z1))

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1))
   F(cons(z0, cons(y0, y1)), empty) -> c2(F(empty, cons(y0, y1)))
   F(cons(z0, z1), cons(y0, y1)) -> c2(F(cons(y0, y1), z1))
S tuples:
   F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1))
   F(cons(z0, cons(y0, y1)), empty) -> c2(F(empty, cons(y0, y1)))
   F(cons(z0, z1), cons(y0, y1)) -> c2(F(cons(y0, y1), z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_2

Compound Symbols:   c1_1, c2_1


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

(37) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1)) by 
   F(empty, cons(z0, cons(y2, y3))) -> c1(F(cons(z0, cons(y2, y3)), cons(y2, y3)))

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(cons(z0, cons(y0, y1)), empty) -> c2(F(empty, cons(y0, y1)))
   F(cons(z0, z1), cons(y0, y1)) -> c2(F(cons(y0, y1), z1))
   F(empty, cons(z0, cons(y2, y3))) -> c1(F(cons(z0, cons(y2, y3)), cons(y2, y3)))
S tuples:
   F(cons(z0, cons(y0, y1)), empty) -> c2(F(empty, cons(y0, y1)))
   F(cons(z0, z1), cons(y0, y1)) -> c2(F(cons(y0, y1), z1))
   F(empty, cons(z0, cons(y2, y3))) -> c1(F(cons(z0, cons(y2, y3)), cons(y2, y3)))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_2

Compound Symbols:   c2_1, c1_1


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

(39) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(cons(z0, cons(y0, y1)), empty) -> c2(F(empty, cons(y0, y1))) by 
   F(cons(z0, cons(z1, cons(y1, y2))), empty) -> c2(F(empty, cons(z1, cons(y1, y2))))

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(cons(z0, z1), cons(y0, y1)) -> c2(F(cons(y0, y1), z1))
   F(empty, cons(z0, cons(y2, y3))) -> c1(F(cons(z0, cons(y2, y3)), cons(y2, y3)))
   F(cons(z0, cons(z1, cons(y1, y2))), empty) -> c2(F(empty, cons(z1, cons(y1, y2))))
S tuples:
   F(cons(z0, z1), cons(y0, y1)) -> c2(F(cons(y0, y1), z1))
   F(empty, cons(z0, cons(y2, y3))) -> c1(F(cons(z0, cons(y2, y3)), cons(y2, y3)))
   F(cons(z0, cons(z1, cons(y1, y2))), empty) -> c2(F(empty, cons(z1, cons(y1, y2))))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_2

Compound Symbols:   c2_1, c1_1


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

(41) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(cons(z0, z1), cons(y0, y1)) -> c2(F(cons(y0, y1), z1)) by 
   F(cons(z0, cons(y2, y3)), cons(z2, z3)) -> c2(F(cons(z2, z3), cons(y2, y3)))
   F(cons(z0, empty), cons(z2, cons(y1, cons(y2, y3)))) -> c2(F(cons(z2, cons(y1, cons(y2, y3))), empty))

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(empty, cons(z0, cons(y2, y3))) -> c1(F(cons(z0, cons(y2, y3)), cons(y2, y3)))
   F(cons(z0, cons(z1, cons(y1, y2))), empty) -> c2(F(empty, cons(z1, cons(y1, y2))))
   F(cons(z0, cons(y2, y3)), cons(z2, z3)) -> c2(F(cons(z2, z3), cons(y2, y3)))
   F(cons(z0, empty), cons(z2, cons(y1, cons(y2, y3)))) -> c2(F(cons(z2, cons(y1, cons(y2, y3))), empty))
S tuples:
   F(empty, cons(z0, cons(y2, y3))) -> c1(F(cons(z0, cons(y2, y3)), cons(y2, y3)))
   F(cons(z0, cons(z1, cons(y1, y2))), empty) -> c2(F(empty, cons(z1, cons(y1, y2))))
   F(cons(z0, cons(y2, y3)), cons(z2, z3)) -> c2(F(cons(z2, z3), cons(y2, y3)))
   F(cons(z0, empty), cons(z2, cons(y1, cons(y2, y3)))) -> c2(F(cons(z2, cons(y1, cons(y2, y3))), empty))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_2

Compound Symbols:   c1_1, c2_1