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


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


WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxRNTS
    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 152 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 581 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 216 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 127 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 42 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 127 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
    (38) CpxRNTS
    (39) FinalProof [FINISHED, 0 ms]
    (40) BOUNDS(1, n^1)
(41) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(42) CpxTRS
    (43) SlicingProof [LOWER BOUND(ID), 0 ms]
    (44) CpxTRS
    (45) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) typed CpxTrs
    (47) OrderProof [LOWER BOUND(ID), 0 ms]
    (48) typed CpxTrs
    (49) RewriteLemmaProof [LOWER BOUND(ID), 195 ms]
    (50) BEST
        (51) proven lower bound
            (52) LowerBoundPropagationProof [FINISHED, 0 ms]
            (53) BOUNDS(n^1, INF)
        (54) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (innermost) 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: INNERMOST
----------------------------------------

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

(2)
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
----------------------------------------

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

(4)
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
----------------------------------------

(5) 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

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

(6)
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
----------------------------------------

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

(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) 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

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

(10)
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


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

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

(12)
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


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

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

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

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

(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

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

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

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

(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) 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

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

(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}
Previous analysis results are:
  is_empty: runtime: ?, size: O(1) [1]

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

(19) 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

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

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

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

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

(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) 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'

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

(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]
  ifappend: runtime: ?, size: O(n^1) [z' + z'']
  append: runtime: ?, size: O(n^1) [z + z']

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

(25) 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

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

(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: {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']

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

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

(28)
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']

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

(29) 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

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

(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']
  tl: runtime: ?, size: O(n^1) [z]

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

(31) 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

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

(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: {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]

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

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

(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) 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

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

(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]
  hd: runtime: ?, size: O(n^1) [z]

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

(37) 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

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

(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: 
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]

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

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

(40)
BOUNDS(1, n^1)

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

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

(42)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


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

(43) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
ifappend/0

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

(44)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


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(l2, l1)
   ifappend(l2, nil) -> l2
   ifappend(l2, cons(x, l)) -> cons(x, append(l, l2))

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

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

(46)
Obligation:
Innermost TRS:
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(l2, l1)
ifappend(l2, nil) -> l2
ifappend(l2, cons(x, l)) -> cons(x, append(l, l2))

Types:
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
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_hd3_0 :: hd
gen_nil:cons4_0 :: Nat -> nil:cons

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

(47) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
append, ifappend

They will be analysed ascendingly in the following order:
append = ifappend

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

(48)
Obligation:
Innermost TRS:
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(l2, l1)
ifappend(l2, nil) -> l2
ifappend(l2, cons(x, l)) -> cons(x, append(l, l2))

Types:
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
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_hd3_0 :: hd
gen_nil:cons4_0 :: Nat -> nil:cons


Generator Equations:
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(hole_hd3_0, gen_nil:cons4_0(x))


The following defined symbols remain to be analysed:
ifappend, append

They will be analysed ascendingly in the following order:
append = ifappend

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

(49) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ifappend(gen_nil:cons4_0(a), gen_nil:cons4_0(n6_0)) -> gen_nil:cons4_0(+(n6_0, a)), rt in Omega(1 + n6_0)

Induction Base:
ifappend(gen_nil:cons4_0(a), gen_nil:cons4_0(0)) ->_R^Omega(1)
gen_nil:cons4_0(a)

Induction Step:
ifappend(gen_nil:cons4_0(a), gen_nil:cons4_0(+(n6_0, 1))) ->_R^Omega(1)
cons(hole_hd3_0, append(gen_nil:cons4_0(n6_0), gen_nil:cons4_0(a))) ->_R^Omega(1)
cons(hole_hd3_0, ifappend(gen_nil:cons4_0(a), gen_nil:cons4_0(n6_0))) ->_IH
cons(hole_hd3_0, gen_nil:cons4_0(+(a, c7_0)))

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

(50)
Complex Obligation (BEST)

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

(51)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
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(l2, l1)
ifappend(l2, nil) -> l2
ifappend(l2, cons(x, l)) -> cons(x, append(l, l2))

Types:
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
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_hd3_0 :: hd
gen_nil:cons4_0 :: Nat -> nil:cons


Generator Equations:
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(hole_hd3_0, gen_nil:cons4_0(x))


The following defined symbols remain to be analysed:
ifappend, append

They will be analysed ascendingly in the following order:
append = ifappend

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

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

(53)
BOUNDS(n^1, INF)

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

(54)
Obligation:
Innermost TRS:
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(l2, l1)
ifappend(l2, nil) -> l2
ifappend(l2, cons(x, l)) -> cons(x, append(l, l2))

Types:
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
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_hd3_0 :: hd
gen_nil:cons4_0 :: Nat -> nil:cons


Lemmas:
ifappend(gen_nil:cons4_0(a), gen_nil:cons4_0(n6_0)) -> gen_nil:cons4_0(+(n6_0, a)), rt in Omega(1 + n6_0)


Generator Equations:
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(hole_hd3_0, gen_nil:cons4_0(x))


The following defined symbols remain to be analysed:
append

They will be analysed ascendingly in the following order:
append = ifappend