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


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1, n^1).

(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), 0 ms]
    (16) CpxRNTS
    (17) ResultPropagationProof [UPPER BOUND(ID), 2 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 151 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 57 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 668 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 240 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 139 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 42 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 128 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 44 ms]
    (40) CpxRNTS
    (41) FinalProof [FINISHED, 0 ms]
    (42) BOUNDS(1, n^1)
(43) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(44) TRS for Loop Detection
    (45) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (46) BEST
        (47) proven lower bound
            (48) LowerBoundPropagationProof [FINISHED, 0 ms]
            (49) BOUNDS(n^1, INF)
        (50) TRS for Loop Detection


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   is_empty(nil) -> true
   is_empty(cons(x, l)) -> false
   hd(cons(x, l)) -> x
   tl(cons(x, l)) -> l
   append(l1, l2) -> ifappend(l1, l2, l1)
   ifappend(l1, l2, nil) -> l2
   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2))

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

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

As the TRS does not nest defined symbols, we have rc = irc.
----------------------------------------

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


The TRS R consists of the following rules:

   is_empty(nil) -> true
   is_empty(cons(x, l)) -> false
   hd(cons(x, l)) -> x
   tl(cons(x, l)) -> l
   append(l1, l2) -> ifappend(l1, l2, l1)
   ifappend(l1, l2, nil) -> l2
   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2))

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^1).


The TRS R consists of the following rules:

   is_empty(nil) -> true [1]
   is_empty(cons(x, l)) -> false [1]
   hd(cons(x, l)) -> x [1]
   tl(cons(x, l)) -> l [1]
   append(l1, l2) -> ifappend(l1, l2, l1) [1]
   ifappend(l1, l2, nil) -> l2 [1]
   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2)) [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:

   is_empty(nil) -> true [1]
   is_empty(cons(x, l)) -> false [1]
   hd(cons(x, l)) -> x [1]
   tl(cons(x, l)) -> l [1]
   append(l1, l2) -> ifappend(l1, l2, l1) [1]
   ifappend(l1, l2, nil) -> l2 [1]
   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2)) [1]

The TRS has the following type information:
is_empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: hd -> nil:cons -> nil:cons
false :: true:false
hd :: nil:cons -> hd
tl :: nil:cons -> nil:cons
append :: nil:cons -> nil:cons -> nil:cons
ifappend :: nil:cons -> nil:cons -> nil:cons -> nil:cons

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:

   is_empty_1
   hd_1
   tl_1
   append_2
   ifappend_3

(c) The following functions are completely defined:
none

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:

   is_empty(nil) -> true [1]
   is_empty(cons(x, l)) -> false [1]
   hd(cons(x, l)) -> x [1]
   tl(cons(x, l)) -> l [1]
   append(l1, l2) -> ifappend(l1, l2, l1) [1]
   ifappend(l1, l2, nil) -> l2 [1]
   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2)) [1]

The TRS has the following type information:
is_empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: hd -> nil:cons -> nil:cons
false :: true:false
hd :: nil:cons -> hd
tl :: nil:cons -> nil:cons
append :: nil:cons -> nil:cons -> nil:cons
ifappend :: nil:cons -> nil:cons -> nil:cons -> nil:cons
const :: hd

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:

   is_empty(nil) -> true [1]
   is_empty(cons(x, l)) -> false [1]
   hd(cons(x, l)) -> x [1]
   tl(cons(x, l)) -> l [1]
   append(l1, l2) -> ifappend(l1, l2, l1) [1]
   ifappend(l1, l2, nil) -> l2 [1]
   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2)) [1]

The TRS has the following type information:
is_empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: hd -> nil:cons -> nil:cons
false :: true:false
hd :: nil:cons -> hd
tl :: nil:cons -> nil:cons
append :: nil:cons -> nil:cons -> nil:cons
ifappend :: nil:cons -> nil:cons -> nil:cons -> nil:cons
const :: hd

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:

   append(z, z') -{ 1 }-> ifappend(l1, l2, l1) :|: z = l1, z' = l2, l1 >= 0, l2 >= 0
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> l2 :|: z'' = 0, z = l1, z' = l2, l1 >= 0, l2 >= 0
   ifappend(z, z', z'') -{ 1 }-> 1 + x + append(l, l2) :|: z'' = 1 + x + l, z = l1, x >= 0, l >= 0, z' = l2, l1 >= 0, l2 >= 0
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l


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

(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:

   append(z, z') -{ 1 }-> ifappend(z, z', z) :|: z >= 0, z' >= 0
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
   ifappend(z, z', z'') -{ 1 }-> 1 + x + append(l, z') :|: z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l


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

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

   { is_empty }
   { ifappend, append }
   { tl }
   { hd }

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

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

   append(z, z') -{ 1 }-> ifappend(z, z', z) :|: z >= 0, z' >= 0
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
   ifappend(z, z', z'') -{ 1 }-> 1 + x + append(l, z') :|: z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l

Function symbols to be analyzed: {is_empty}, {ifappend,append}, {tl}, {hd}

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

(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:

   append(z, z') -{ 1 }-> ifappend(z, z', z) :|: z >= 0, z' >= 0
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
   ifappend(z, z', z'') -{ 1 }-> 1 + x + append(l, z') :|: z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l

Function symbols to be analyzed: {is_empty}, {ifappend,append}, {tl}, {hd}

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   append(z, z') -{ 1 }-> ifappend(z, z', z) :|: z >= 0, z' >= 0
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
   ifappend(z, z', z'') -{ 1 }-> 1 + x + append(l, z') :|: z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l

Function symbols to be analyzed: {is_empty}, {ifappend,append}, {tl}, {hd}
Previous analysis results are:
  is_empty: runtime: ?, size: O(1) [1]

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   append(z, z') -{ 1 }-> ifappend(z, z', z) :|: z >= 0, z' >= 0
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
   ifappend(z, z', z'') -{ 1 }-> 1 + x + append(l, z') :|: z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l

Function symbols to be analyzed: {ifappend,append}, {tl}, {hd}
Previous analysis results are:
  is_empty: runtime: O(1) [1], size: O(1) [1]

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

(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:

   append(z, z') -{ 1 }-> ifappend(z, z', z) :|: z >= 0, z' >= 0
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
   ifappend(z, z', z'') -{ 1 }-> 1 + x + append(l, z') :|: z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l

Function symbols to be analyzed: {ifappend,append}, {tl}, {hd}
Previous analysis results are:
  is_empty: runtime: O(1) [1], size: O(1) [1]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   append(z, z') -{ 1 }-> ifappend(z, z', z) :|: z >= 0, z' >= 0
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
   ifappend(z, z', z'') -{ 1 }-> 1 + x + append(l, z') :|: z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: ifappend
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + 2*z''

Computed RUNTIME bound using CoFloCo for: append
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 4 + 2*z

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

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

   append(z, z') -{ 1 }-> ifappend(z, z', z) :|: z >= 0, z' >= 0
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
   ifappend(z, z', z'') -{ 1 }-> 1 + x + append(l, z') :|: z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l

Function symbols to be analyzed: {tl}, {hd}
Previous analysis results are:
  is_empty: runtime: O(1) [1], size: O(1) [1]
  ifappend: runtime: O(n^1) [3 + 2*z''], size: O(n^1) [z' + z'']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']

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

(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:

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z' + z, z >= 0, z' >= 0
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
   ifappend(z, z', z'') -{ 5 + 2*l }-> 1 + x + s' :|: s' >= 0, s' <= l + z', z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l

Function symbols to be analyzed: {tl}, {hd}
Previous analysis results are:
  is_empty: runtime: O(1) [1], size: O(1) [1]
  ifappend: runtime: O(n^1) [3 + 2*z''], size: O(n^1) [z' + z'']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: tl
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:

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z' + z, z >= 0, z' >= 0
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
   ifappend(z, z', z'') -{ 5 + 2*l }-> 1 + x + s' :|: s' >= 0, s' <= l + z', z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l

Function symbols to be analyzed: {tl}, {hd}
Previous analysis results are:
  is_empty: runtime: O(1) [1], size: O(1) [1]
  ifappend: runtime: O(n^1) [3 + 2*z''], size: O(n^1) [z' + z'']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
  tl: runtime: ?, size: O(n^1) [z]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: tl
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:

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z' + z, z >= 0, z' >= 0
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
   ifappend(z, z', z'') -{ 5 + 2*l }-> 1 + x + s' :|: s' >= 0, s' <= l + z', z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l

Function symbols to be analyzed: {hd}
Previous analysis results are:
  is_empty: runtime: O(1) [1], size: O(1) [1]
  ifappend: runtime: O(n^1) [3 + 2*z''], size: O(n^1) [z' + z'']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
  tl: 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:

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z' + z, z >= 0, z' >= 0
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
   ifappend(z, z', z'') -{ 5 + 2*l }-> 1 + x + s' :|: s' >= 0, s' <= l + z', z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l

Function symbols to be analyzed: {hd}
Previous analysis results are:
  is_empty: runtime: O(1) [1], size: O(1) [1]
  ifappend: runtime: O(n^1) [3 + 2*z''], size: O(n^1) [z' + z'']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
  tl: runtime: O(1) [1], size: O(n^1) [z]

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(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z' + z, z >= 0, z' >= 0
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
   ifappend(z, z', z'') -{ 5 + 2*l }-> 1 + x + s' :|: s' >= 0, s' <= l + z', z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l

Function symbols to be analyzed: {hd}
Previous analysis results are:
  is_empty: runtime: O(1) [1], size: O(1) [1]
  ifappend: runtime: O(n^1) [3 + 2*z''], size: O(n^1) [z' + z'']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
  tl: runtime: O(1) [1], size: O(n^1) [z]
  hd: runtime: ?, size: O(n^1) [z]

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(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   append(z, z') -{ 4 + 2*z }-> s :|: s >= 0, s <= z' + z, z >= 0, z' >= 0
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z'' = 0, z >= 0, z' >= 0
   ifappend(z, z', z'') -{ 5 + 2*l }-> 1 + x + s' :|: s' >= 0, s' <= l + z', z'' = 1 + x + l, x >= 0, l >= 0, z >= 0, z' >= 0
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 1 }-> l :|: x >= 0, l >= 0, z = 1 + x + l

Function symbols to be analyzed: 
Previous analysis results are:
  is_empty: runtime: O(1) [1], size: O(1) [1]
  ifappend: runtime: O(n^1) [3 + 2*z''], size: O(n^1) [z' + z'']
  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
  tl: runtime: O(1) [1], size: O(n^1) [z]
  hd: runtime: O(1) [1], size: O(n^1) [z]

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(41) FinalProof (FINISHED)
Computed overall runtime complexity
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(42)
BOUNDS(1, n^1)

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(43) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(44)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   is_empty(nil) -> true
   is_empty(cons(x, l)) -> false
   hd(cons(x, l)) -> x
   tl(cons(x, l)) -> l
   append(l1, l2) -> ifappend(l1, l2, l1)
   ifappend(l1, l2, nil) -> l2
   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2))

S is empty.
Rewrite Strategy: FULL
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(45) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

append(cons(x3_0, l4_0), l2) ->^+ cons(x3_0, append(l4_0, l2))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1].

The pumping substitution is [l4_0 / cons(x3_0, l4_0)].

The result substitution is [ ].




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(46)
Complex Obligation (BEST)

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(47)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   is_empty(nil) -> true
   is_empty(cons(x, l)) -> false
   hd(cons(x, l)) -> x
   tl(cons(x, l)) -> l
   append(l1, l2) -> ifappend(l1, l2, l1)
   ifappend(l1, l2, nil) -> l2
   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2))

S is empty.
Rewrite Strategy: FULL
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(48) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(49)
BOUNDS(n^1, INF)

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(50)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   is_empty(nil) -> true
   is_empty(cons(x, l)) -> false
   hd(cons(x, l)) -> x
   tl(cons(x, l)) -> l
   append(l1, l2) -> ifappend(l1, l2, l1)
   ifappend(l1, l2, nil) -> l2
   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2))

S is empty.
Rewrite Strategy: FULL