/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^2), O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, n^2).

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 4 ms]
    (16) CpxRNTS
    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 311 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 108 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 129 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 43 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 128 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 43 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 127 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 43 ms]
    (40) CpxRNTS
    (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 331 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 94 ms]
    (46) CpxRNTS
    (47) FinalProof [FINISHED, 0 ms]
    (48) BOUNDS(1, n^2)
(49) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(50) CpxTRS
    (51) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) typed CpxTrs
    (53) OrderProof [LOWER BOUND(ID), 0 ms]
    (54) typed CpxTrs
    (55) RewriteLemmaProof [LOWER BOUND(ID), 305 ms]
    (56) BEST
        (57) proven lower bound
            (58) LowerBoundPropagationProof [FINISHED, 0 ms]
            (59) BOUNDS(n^1, INF)
        (60) typed CpxTrs
            (61) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
            (62) proven lower bound
            (63) LowerBoundPropagationProof [FINISHED, 0 ms]
            (64) BOUNDS(n^2, INF)


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, n^2).


The TRS R consists of the following rules:

   rev(nil) -> nil
   rev(.(x, y)) -> ++(rev(y), .(x, nil))
   car(.(x, y)) -> x
   cdr(.(x, y)) -> y
   null(nil) -> true
   null(.(x, y)) -> false
   ++(nil, y) -> y
   ++(.(x, y), z) -> .(x, ++(y, z))

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

(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS is a non-duplicating overlay system, we have rc = irc.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   rev(nil) -> nil
   rev(.(x, y)) -> ++(rev(y), .(x, nil))
   car(.(x, y)) -> x
   cdr(.(x, y)) -> y
   null(nil) -> true
   null(.(x, y)) -> false
   ++(nil, y) -> y
   ++(.(x, y), z) -> .(x, ++(y, z))

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

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   rev(nil) -> nil [1]
   rev(.(x, y)) -> ++(rev(y), .(x, nil)) [1]
   car(.(x, y)) -> x [1]
   cdr(.(x, y)) -> y [1]
   null(nil) -> true [1]
   null(.(x, y)) -> false [1]
   ++(nil, y) -> y [1]
   ++(.(x, y), z) -> .(x, ++(y, z)) [1]

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

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

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   rev(nil) -> nil [1]
   rev(.(x, y)) -> ++(rev(y), .(x, nil)) [1]
   car(.(x, y)) -> x [1]
   cdr(.(x, y)) -> y [1]
   null(nil) -> true [1]
   null(.(x, y)) -> false [1]
   ++(nil, y) -> y [1]
   ++(.(x, y), z) -> .(x, ++(y, z)) [1]

The TRS has the following type information:
rev :: nil:. -> nil:.
nil :: nil:.
. :: car -> nil:. -> nil:.
++ :: nil:. -> nil:. -> nil:.
car :: nil:. -> car
cdr :: nil:. -> nil:.
null :: nil:. -> true:false
true :: true:false
false :: true:false

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

(7) 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:

   car_1
   cdr_1
   null_1

(c) The following functions are completely defined:

   rev_1
   ++_2

Due to the following rules being added:
none

And the following fresh constants:    const

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

(8)
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:

   rev(nil) -> nil [1]
   rev(.(x, y)) -> ++(rev(y), .(x, nil)) [1]
   car(.(x, y)) -> x [1]
   cdr(.(x, y)) -> y [1]
   null(nil) -> true [1]
   null(.(x, y)) -> false [1]
   ++(nil, y) -> y [1]
   ++(.(x, y), z) -> .(x, ++(y, z)) [1]

The TRS has the following type information:
rev :: nil:. -> nil:.
nil :: nil:.
. :: car -> nil:. -> nil:.
++ :: nil:. -> nil:. -> nil:.
car :: nil:. -> car
cdr :: nil:. -> nil:.
null :: nil:. -> true:false
true :: true:false
false :: true:false
const :: car

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

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

(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:

   rev(nil) -> nil [1]
   rev(.(x, nil)) -> ++(nil, .(x, nil)) [2]
   rev(.(x, .(x', y'))) -> ++(++(rev(y'), .(x', nil)), .(x, nil)) [2]
   car(.(x, y)) -> x [1]
   cdr(.(x, y)) -> y [1]
   null(nil) -> true [1]
   null(.(x, y)) -> false [1]
   ++(nil, y) -> y [1]
   ++(.(x, y), z) -> .(x, ++(y, z)) [1]

The TRS has the following type information:
rev :: nil:. -> nil:.
nil :: nil:.
. :: car -> nil:. -> nil:.
++ :: nil:. -> nil:. -> nil:.
car :: nil:. -> car
cdr :: nil:. -> nil:.
null :: nil:. -> true:false
true :: true:false
false :: true:false
const :: car

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

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

   nil => 0
   true => 1
   false => 0
   const => 0

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   ++(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   ++(z', z'') -{ 1 }-> 1 + x + ++(y, z) :|: z'' = z, z >= 0, z' = 1 + x + y, x >= 0, y >= 0
   car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0
   cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
   rev(z') -{ 1 }-> 0 :|: z' = 0
   rev(z') -{ 2 }-> ++(0, 1 + x + 0) :|: x >= 0, z' = 1 + x + 0
   rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0


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

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

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

   ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   ++(z', z'') -{ 1 }-> 1 + x + ++(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0
   cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
   rev(z') -{ 1 }-> 0 :|: z' = 0
   rev(z') -{ 2 }-> ++(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0


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

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

   { ++ }
   { null }
   { cdr }
   { car }
   { rev }

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

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

   ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   ++(z', z'') -{ 1 }-> 1 + x + ++(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0
   cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
   rev(z') -{ 1 }-> 0 :|: z' = 0
   rev(z') -{ 2 }-> ++(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0

Function symbols to be analyzed: {++}, {null}, {cdr}, {car}, {rev}

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

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

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

   ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   ++(z', z'') -{ 1 }-> 1 + x + ++(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0
   cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
   rev(z') -{ 1 }-> 0 :|: z' = 0
   rev(z') -{ 2 }-> ++(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0

Function symbols to be analyzed: {++}, {null}, {cdr}, {car}, {rev}

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   ++(z', z'') -{ 1 }-> 1 + x + ++(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0
   cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
   rev(z') -{ 1 }-> 0 :|: z' = 0
   rev(z') -{ 2 }-> ++(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0

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

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   ++(z', z'') -{ 1 }-> 1 + x + ++(y, z'') :|: z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0
   cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
   rev(z') -{ 1 }-> 0 :|: z' = 0
   rev(z') -{ 2 }-> ++(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0

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

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

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

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

   ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0
   cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
   rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   rev(z') -{ 1 }-> 0 :|: z' = 0
   rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0

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

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0
   cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
   rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   rev(z') -{ 1 }-> 0 :|: z' = 0
   rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0
   cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
   rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   rev(z') -{ 1 }-> 0 :|: z' = 0
   rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0

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

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

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

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0
   cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
   rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   rev(z') -{ 1 }-> 0 :|: z' = 0
   rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0

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

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0
   cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
   rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   rev(z') -{ 1 }-> 0 :|: z' = 0
   rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0

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

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0
   cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
   rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   rev(z') -{ 1 }-> 0 :|: z' = 0
   rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0

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

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

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

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

   ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0
   cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
   rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   rev(z') -{ 1 }-> 0 :|: z' = 0
   rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0

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

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0
   cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
   rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   rev(z') -{ 1 }-> 0 :|: z' = 0
   rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0

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

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0
   cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
   rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   rev(z') -{ 1 }-> 0 :|: z' = 0
   rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0

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

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

(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:

   ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0
   cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
   rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   rev(z') -{ 1 }-> 0 :|: z' = 0
   rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0

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

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0
   cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
   rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   rev(z') -{ 1 }-> 0 :|: z' = 0
   rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0

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

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: rev
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 4 + 3*z' + 2*z'^2

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

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

   ++(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   ++(z', z'') -{ 2 + y }-> 1 + x + s' :|: s' >= 0, s' <= y + z'', z'' >= 0, z' = 1 + x + y, x >= 0, y >= 0
   car(z') -{ 1 }-> x :|: z' = 1 + x + y, x >= 0, y >= 0
   cdr(z') -{ 1 }-> y :|: z' = 1 + x + y, x >= 0, y >= 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: z' = 1 + x + y, x >= 0, y >= 0
   rev(z') -{ 3 }-> s :|: s >= 0, s <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   rev(z') -{ 1 }-> 0 :|: z' = 0
   rev(z') -{ 2 }-> ++(++(rev(y'), 1 + x' + 0), 1 + x + 0) :|: z' = 1 + x + (1 + x' + y'), x >= 0, x' >= 0, y' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  ++: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  null: runtime: O(1) [1], size: O(1) [1]
  cdr: runtime: O(1) [1], size: O(n^1) [z']
  car: runtime: O(1) [1], size: O(n^1) [z']
  rev: runtime: O(n^2) [4 + 3*z' + 2*z'^2], size: O(n^1) [z']

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

(47) FinalProof (FINISHED)
Computed overall runtime complexity
----------------------------------------

(48)
BOUNDS(1, n^2)

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

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

(50)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   rev(nil) -> nil
   rev(.(x, y)) -> ++(rev(y), .(x, nil))
   car(.(x, y)) -> x
   cdr(.(x, y)) -> y
   null(nil) -> true
   null(.(x, y)) -> false
   ++(nil, y) -> y
   ++(.(x, y), z) -> .(x, ++(y, z))

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

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

(52)
Obligation:
TRS:
Rules:
rev(nil) -> nil
rev(.(x, y)) -> ++(rev(y), .(x, nil))
car(.(x, y)) -> x
cdr(.(x, y)) -> y
null(nil) -> true
null(.(x, y)) -> false
++(nil, y) -> y
++(.(x, y), z) -> .(x, ++(y, z))

Types:
rev :: nil:. -> nil:.
nil :: nil:.
. :: car -> nil:. -> nil:.
++ :: nil:. -> nil:. -> nil:.
car :: nil:. -> car
cdr :: nil:. -> nil:.
null :: nil:. -> true:false
true :: true:false
false :: true:false
hole_nil:.1_0 :: nil:.
hole_car2_0 :: car
hole_true:false3_0 :: true:false
gen_nil:.4_0 :: Nat -> nil:.

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

(53) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
rev, ++

They will be analysed ascendingly in the following order:
++ < rev

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

(54)
Obligation:
TRS:
Rules:
rev(nil) -> nil
rev(.(x, y)) -> ++(rev(y), .(x, nil))
car(.(x, y)) -> x
cdr(.(x, y)) -> y
null(nil) -> true
null(.(x, y)) -> false
++(nil, y) -> y
++(.(x, y), z) -> .(x, ++(y, z))

Types:
rev :: nil:. -> nil:.
nil :: nil:.
. :: car -> nil:. -> nil:.
++ :: nil:. -> nil:. -> nil:.
car :: nil:. -> car
cdr :: nil:. -> nil:.
null :: nil:. -> true:false
true :: true:false
false :: true:false
hole_nil:.1_0 :: nil:.
hole_car2_0 :: car
hole_true:false3_0 :: true:false
gen_nil:.4_0 :: Nat -> nil:.


Generator Equations:
gen_nil:.4_0(0) <=> nil
gen_nil:.4_0(+(x, 1)) <=> .(hole_car2_0, gen_nil:.4_0(x))


The following defined symbols remain to be analysed:
++, rev

They will be analysed ascendingly in the following order:
++ < rev

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

(55) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
++(gen_nil:.4_0(n6_0), gen_nil:.4_0(b)) -> gen_nil:.4_0(+(n6_0, b)), rt in Omega(1 + n6_0)

Induction Base:
++(gen_nil:.4_0(0), gen_nil:.4_0(b)) ->_R^Omega(1)
gen_nil:.4_0(b)

Induction Step:
++(gen_nil:.4_0(+(n6_0, 1)), gen_nil:.4_0(b)) ->_R^Omega(1)
.(hole_car2_0, ++(gen_nil:.4_0(n6_0), gen_nil:.4_0(b))) ->_IH
.(hole_car2_0, gen_nil:.4_0(+(b, c7_0)))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(56)
Complex Obligation (BEST)

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

(57)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
rev(nil) -> nil
rev(.(x, y)) -> ++(rev(y), .(x, nil))
car(.(x, y)) -> x
cdr(.(x, y)) -> y
null(nil) -> true
null(.(x, y)) -> false
++(nil, y) -> y
++(.(x, y), z) -> .(x, ++(y, z))

Types:
rev :: nil:. -> nil:.
nil :: nil:.
. :: car -> nil:. -> nil:.
++ :: nil:. -> nil:. -> nil:.
car :: nil:. -> car
cdr :: nil:. -> nil:.
null :: nil:. -> true:false
true :: true:false
false :: true:false
hole_nil:.1_0 :: nil:.
hole_car2_0 :: car
hole_true:false3_0 :: true:false
gen_nil:.4_0 :: Nat -> nil:.


Generator Equations:
gen_nil:.4_0(0) <=> nil
gen_nil:.4_0(+(x, 1)) <=> .(hole_car2_0, gen_nil:.4_0(x))


The following defined symbols remain to be analysed:
++, rev

They will be analysed ascendingly in the following order:
++ < rev

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

(58) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(59)
BOUNDS(n^1, INF)

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

(60)
Obligation:
TRS:
Rules:
rev(nil) -> nil
rev(.(x, y)) -> ++(rev(y), .(x, nil))
car(.(x, y)) -> x
cdr(.(x, y)) -> y
null(nil) -> true
null(.(x, y)) -> false
++(nil, y) -> y
++(.(x, y), z) -> .(x, ++(y, z))

Types:
rev :: nil:. -> nil:.
nil :: nil:.
. :: car -> nil:. -> nil:.
++ :: nil:. -> nil:. -> nil:.
car :: nil:. -> car
cdr :: nil:. -> nil:.
null :: nil:. -> true:false
true :: true:false
false :: true:false
hole_nil:.1_0 :: nil:.
hole_car2_0 :: car
hole_true:false3_0 :: true:false
gen_nil:.4_0 :: Nat -> nil:.


Lemmas:
++(gen_nil:.4_0(n6_0), gen_nil:.4_0(b)) -> gen_nil:.4_0(+(n6_0, b)), rt in Omega(1 + n6_0)


Generator Equations:
gen_nil:.4_0(0) <=> nil
gen_nil:.4_0(+(x, 1)) <=> .(hole_car2_0, gen_nil:.4_0(x))


The following defined symbols remain to be analysed:
rev
----------------------------------------

(61) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
rev(gen_nil:.4_0(n495_0)) -> gen_nil:.4_0(n495_0), rt in Omega(1 + n495_0 + n495_0^2)

Induction Base:
rev(gen_nil:.4_0(0)) ->_R^Omega(1)
nil

Induction Step:
rev(gen_nil:.4_0(+(n495_0, 1))) ->_R^Omega(1)
++(rev(gen_nil:.4_0(n495_0)), .(hole_car2_0, nil)) ->_IH
++(gen_nil:.4_0(c496_0), .(hole_car2_0, nil)) ->_L^Omega(1 + n495_0)
gen_nil:.4_0(+(n495_0, +(0, 1)))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(62)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
rev(nil) -> nil
rev(.(x, y)) -> ++(rev(y), .(x, nil))
car(.(x, y)) -> x
cdr(.(x, y)) -> y
null(nil) -> true
null(.(x, y)) -> false
++(nil, y) -> y
++(.(x, y), z) -> .(x, ++(y, z))

Types:
rev :: nil:. -> nil:.
nil :: nil:.
. :: car -> nil:. -> nil:.
++ :: nil:. -> nil:. -> nil:.
car :: nil:. -> car
cdr :: nil:. -> nil:.
null :: nil:. -> true:false
true :: true:false
false :: true:false
hole_nil:.1_0 :: nil:.
hole_car2_0 :: car
hole_true:false3_0 :: true:false
gen_nil:.4_0 :: Nat -> nil:.


Lemmas:
++(gen_nil:.4_0(n6_0), gen_nil:.4_0(b)) -> gen_nil:.4_0(+(n6_0, b)), rt in Omega(1 + n6_0)


Generator Equations:
gen_nil:.4_0(0) <=> nil
gen_nil:.4_0(+(x, 1)) <=> .(hole_car2_0, gen_nil:.4_0(x))


The following defined symbols remain to be analysed:
rev
----------------------------------------

(63) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(64)
BOUNDS(n^2, INF)