KILLED
proof of input_SxiVyKMk6L.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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 [UPPER BOUND(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), 17 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), 15.2 s]
                (44) CpxRNTS
                (45) IntTrsBoundProof [UPPER BOUND(ID), 5386 ms]
                (46) CpxRNTS
            (47) CompletionProof [UPPER BOUND(ID), 0 ms]
            (48) CpxTypedWeightedCompleteTrs
                (49) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (50) CpxRNTS
    (51) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
        (53) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (54) CdtProblem
        (55) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (56) CdtProblem
        (57) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (58) CdtProblem
(59) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(60) CpxWeightedTrs
    (61) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CpxTypedWeightedTrs
    (63) CompletionProof [UPPER BOUND(ID), 0 ms]
    (64) CpxTypedWeightedCompleteTrs
        (65) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (66) CpxTypedWeightedCompleteTrs
        (67) CpxTypedWeightedTrsToRntsProof [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), 1624 ms]
        (76) CpxRNTS
        (77) IntTrsBoundProof [UPPER BOUND(ID), 401 ms]
        (78) CpxRNTS
    (79) CompletionProof [UPPER BOUND(ID), 0 ms]
    (80) CpxTypedWeightedCompleteTrs
        (81) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (82) 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, 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) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(4)
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


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

(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, empty) -> c
   F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1))
   F(cons(z0, z1), z2) -> c2(F(z2, z1))

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

   f(z0, empty) -> z0
   f(empty, cons(z0, z1)) -> f(cons(z0, z1), z1)
   f(cons(z0, z1), z2) -> f(z2, z1)

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, empty) -> c
   F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1))
   F(cons(z0, z1), z2) -> c2(F(z2, z1))

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

   f(z0, empty) -> z0
   f(empty, cons(z0, z1)) -> f(cons(z0, z1), z1)
   f(cons(z0, z1), z2) -> f(z2, z1)

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

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

(10)
Obligation:
Innermost TRS:
Rules:
F(z0, empty) -> c
F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1))
F(cons(z0, z1), z2) -> c2(F(z2, z1))
f(z0, empty) -> z0
f(empty, cons(z0, z1)) -> f(cons(z0, z1), z1)
f(cons(z0, z1), z2) -> f(z2, z1)

Types:
F :: empty:cons -> empty:cons -> c:c1:c2
empty :: empty:cons
c :: c:c1:c2
cons :: a -> empty:cons -> empty:cons
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
f :: empty:cons -> empty:cons -> empty:cons
hole_c:c1:c21_3 :: c:c1:c2
hole_empty:cons2_3 :: empty:cons
hole_a3_3 :: a
gen_c:c1:c24_3 :: Nat -> c:c1:c2
gen_empty:cons5_3 :: Nat -> empty:cons

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

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

(12)
Obligation:
Innermost TRS:
Rules:
F(z0, empty) -> c
F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1))
F(cons(z0, z1), z2) -> c2(F(z2, z1))
f(z0, empty) -> z0
f(empty, cons(z0, z1)) -> f(cons(z0, z1), z1)
f(cons(z0, z1), z2) -> f(z2, z1)

Types:
F :: empty:cons -> empty:cons -> c:c1:c2
empty :: empty:cons
c :: c:c1:c2
cons :: a -> empty:cons -> empty:cons
c1 :: c:c1:c2 -> c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
f :: empty:cons -> empty:cons -> empty:cons
hole_c:c1:c21_3 :: c:c1:c2
hole_empty:cons2_3 :: empty:cons
hole_a3_3 :: a
gen_c:c1:c24_3 :: Nat -> c:c1:c2
gen_empty:cons5_3 :: Nat -> empty:cons


Generator Equations:
gen_c:c1:c24_3(0) <=> c
gen_c:c1:c24_3(+(x, 1)) <=> c1(gen_c:c1:c24_3(x))
gen_empty:cons5_3(0) <=> empty
gen_empty:cons5_3(+(x, 1)) <=> cons(hole_a3_3, gen_empty:cons5_3(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, empty) -> c
   F(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1))
   F(cons(z0, z1), z2) -> c2(F(z2, z1))

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

   f(z0, empty) -> z0
   f(empty, cons(z0, z1)) -> f(cons(z0, z1), z1)
   f(cons(z0, z1), z2) -> f(z2, z1)

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, 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
----------------------------------------

(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, 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


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

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

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

(20)
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


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

(21) 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)

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

(22)
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


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

(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(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1))
   F(cons(z0, z1), z2) -> c2(F(z2, z1))

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(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1))
   F(cons(z0, z1), z2) -> c2(F(z2, z1))

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(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1)) [1]
   F(cons(z0, z1), z2) -> c2(F(z2, z1)) [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(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1)) [1]
   F(cons(z0, z1), z2) -> c2(F(z2, z1)) [1]

The TRS has the following type information:
F :: empty:cons -> empty:cons -> c1:c2
empty :: empty:cons
cons :: a -> empty:cons -> empty:cons
c1 :: c1:c2 -> c1:c2
c2 :: c1:c2 -> c1:c2

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_2

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const, const1

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

(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(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1)) [1]
   F(cons(z0, z1), z2) -> c2(F(z2, z1)) [1]

The TRS has the following type information:
F :: empty:cons -> empty:cons -> c1:c2
empty :: empty:cons
cons :: a -> empty:cons -> empty:cons
c1 :: c1:c2 -> c1:c2
c2 :: c1:c2 -> c1:c2
const :: c1:c2
const1 :: a

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(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1)) [1]
   F(cons(z0, z1), z2) -> c2(F(z2, z1)) [1]

The TRS has the following type information:
F :: empty:cons -> empty:cons -> c1:c2
empty :: empty:cons
cons :: a -> empty:cons -> empty:cons
c1 :: c1:c2 -> c1:c2
c2 :: c1:c2 -> c1:c2
const :: c1:c2
const1 :: a

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:

   empty => 0
   const => 0
   const1 => 0

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

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

   F(z, z') -{ 1 }-> 1 + F(z2, z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   F(z, z') -{ 1 }-> 1 + F(1 + z0 + z1, z1) :|: z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 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, z') -{ 1 }-> 1 + F(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   F(z, z') -{ 1 }-> 1 + F(1 + z0 + z1, z1) :|: z' = 1 + z0 + z1, z1 >= 0, z0 >= 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, z') -{ 1 }-> 1 + F(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   F(z, z') -{ 1 }-> 1 + F(1 + z0 + z1, z1) :|: z' = 1 + z0 + z1, z1 >= 0, z0 >= 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, z') -{ 1 }-> 1 + F(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   F(z, z') -{ 1 }-> 1 + F(1 + z0 + z1, z1) :|: z' = 1 + z0 + z1, z1 >= 0, z0 >= 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, z') -{ 1 }-> 1 + F(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   F(z, z') -{ 1 }-> 1 + F(1 + z0 + z1, z1) :|: z' = 1 + z0 + z1, z1 >= 0, z0 >= 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, z') -{ 1 }-> 1 + F(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   F(z, z') -{ 1 }-> 1 + F(1 + z0 + z1, z1) :|: z' = 1 + z0 + z1, z1 >= 0, z0 >= 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:

   F(v0, v1) -> null_F [0]

And the following fresh constants:    null_F, 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(empty, cons(z0, z1)) -> c1(F(cons(z0, z1), z1)) [1]
   F(cons(z0, z1), z2) -> c2(F(z2, z1)) [1]
   F(v0, v1) -> null_F [0]

The TRS has the following type information:
F :: empty:cons -> empty:cons -> c1:c2:null_F
empty :: empty:cons
cons :: a -> empty:cons -> empty:cons
c1 :: c1:c2:null_F -> c1:c2:null_F
c2 :: c1:c2:null_F -> c1:c2:null_F
null_F :: c1:c2:null_F
const :: a

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:

   empty => 0
   null_F => 0
   const => 0

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

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

   F(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   F(z, z') -{ 1 }-> 1 + F(z2, z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   F(z, z') -{ 1 }-> 1 + F(1 + z0 + z1, z1) :|: z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 0

Only complete derivations are relevant for the runtime complexity.

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

(51) 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))

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

(52)
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


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

(53) 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)))

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

(54)
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


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

(55) 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))))

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

(56)
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


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

(57) 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))

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

(58)
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


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

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

(60)
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
----------------------------------------

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

(62)
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
----------------------------------------

(63) 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

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

(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:

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

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

(66)
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
----------------------------------------

(67) 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

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

(68)
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


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

(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:

   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


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

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

   { f }

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

(72)
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}

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

(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:

   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}

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

(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 + z'

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

(76)
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']

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

(77) 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: ?

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

(78)
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']

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

(79) 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

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

(80)
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
----------------------------------------

(81) 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

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

(82)
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.