MAYBE
proof of input_M45Ufeizzo.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) InliningProof [UPPER BOUND(ID), 100 ms]
        (16) CpxRNTS
        (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
        (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 77 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 59 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 2731 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 490 ms]
        (32) CpxRNTS
    (33) CompletionProof [UPPER BOUND(ID), 0 ms]
    (34) CpxTypedWeightedCompleteTrs
        (35) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (36) CpxRNTS
(37) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(38) CdtProblem
    (39) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CdtProblem
    (43) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms]
    (52) CdtProblem
    (53) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

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

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:

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

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

(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(4)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

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

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

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

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

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

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

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

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

(9) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   hd_1
   append_2
   ifappend_3

(c) The following functions are completely defined:

   is_empty_1
   tl_1

Due to the following rules being added:

   tl(v0) -> nil [0]

And the following fresh constants:    const

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

(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   is_empty(nil) -> true [1]
   is_empty(cons(x, l)) -> false [1]
   hd(cons(x, l)) -> x [1]
   tl(cons(x, l)) -> cons(x, l) [1]
   append(l1, l2) -> ifappend(l1, l2, is_empty(l1)) [1]
   ifappend(l1, l2, true) -> l2 [1]
   ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2)) [1]
   tl(v0) -> nil [0]

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 -> true:false -> nil:cons
const :: hd

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

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

(12)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   is_empty(nil) -> true [1]
   is_empty(cons(x, l)) -> false [1]
   hd(cons(x, l)) -> x [1]
   tl(cons(x, l)) -> cons(x, l) [1]
   append(nil, l2) -> ifappend(nil, l2, true) [2]
   append(cons(x', l'), l2) -> ifappend(cons(x', l'), l2, false) [2]
   ifappend(l1, l2, true) -> l2 [1]
   ifappend(cons(x'', l''), l2, false) -> cons(hd(cons(x'', l'')), append(cons(x'', l''), l2)) [2]
   ifappend(l1, l2, false) -> cons(hd(l1), append(nil, l2)) [1]
   tl(v0) -> nil [0]

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 -> true:false -> nil:cons
const :: hd

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

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

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

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

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

   append(z, z') -{ 2 }-> ifappend(0, l2, 1) :|: z' = l2, z = 0, l2 >= 0
   append(z, z') -{ 2 }-> ifappend(1 + x' + l', l2, 0) :|: x' >= 0, l' >= 0, z' = l2, l2 >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> l2 :|: z = l1, z' = l2, l1 >= 0, l2 >= 0, z'' = 1
   ifappend(z, z', z'') -{ 1 }-> 1 + hd(l1) + append(0, l2) :|: z'' = 0, z = l1, z' = l2, l1 >= 0, l2 >= 0
   ifappend(z, z', z'') -{ 2 }-> 1 + hd(1 + x'' + l'') + append(1 + x'' + l'', l2) :|: z'' = 0, l'' >= 0, z' = l2, x'' >= 0, l2 >= 0, z = 1 + x'' + l''
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   tl(z) -{ 1 }-> 1 + x + l :|: x >= 0, l >= 0, z = 1 + x + l


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

(15) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l

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

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

   append(z, z') -{ 2 }-> ifappend(0, l2, 1) :|: z' = l2, z = 0, l2 >= 0
   append(z, z') -{ 2 }-> ifappend(1 + x' + l', l2, 0) :|: x' >= 0, l' >= 0, z' = l2, l2 >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> l2 :|: z = l1, z' = l2, l1 >= 0, l2 >= 0, z'' = 1
   ifappend(z, z', z'') -{ 2 }-> 1 + x + append(0, l2) :|: z'' = 0, z = l1, z' = l2, l1 >= 0, l2 >= 0, x >= 0, l >= 0, l1 = 1 + x + l
   ifappend(z, z', z'') -{ 3 }-> 1 + x + append(1 + x'' + l'', l2) :|: z'' = 0, l'' >= 0, z' = l2, x'' >= 0, l2 >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   tl(z) -{ 1 }-> 1 + x + l :|: x >= 0, l >= 0, z = 1 + x + l


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

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

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

   append(z, z') -{ 2 }-> ifappend(0, z', 1) :|: z = 0, z' >= 0
   append(z, z') -{ 2 }-> ifappend(1 + x' + l', z', 0) :|: x' >= 0, l' >= 0, z' >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z >= 0, z' >= 0, z'' = 1
   ifappend(z, z', z'') -{ 2 }-> 1 + x + append(0, z') :|: z'' = 0, z >= 0, z' >= 0, x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 3 }-> 1 + x + append(1 + x'' + l'', z') :|: z'' = 0, l'' >= 0, x'' >= 0, z' >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: z >= 0
   tl(z) -{ 1 }-> 1 + x + l :|: x >= 0, l >= 0, z = 1 + x + l


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

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

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

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

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

   append(z, z') -{ 2 }-> ifappend(0, z', 1) :|: z = 0, z' >= 0
   append(z, z') -{ 2 }-> ifappend(1 + x' + l', z', 0) :|: x' >= 0, l' >= 0, z' >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z >= 0, z' >= 0, z'' = 1
   ifappend(z, z', z'') -{ 2 }-> 1 + x + append(0, z') :|: z'' = 0, z >= 0, z' >= 0, x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 3 }-> 1 + x + append(1 + x'' + l'', z') :|: z'' = 0, l'' >= 0, x'' >= 0, z' >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: z >= 0
   tl(z) -{ 1 }-> 1 + x + l :|: x >= 0, l >= 0, z = 1 + x + l

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

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

(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') -{ 2 }-> ifappend(0, z', 1) :|: z = 0, z' >= 0
   append(z, z') -{ 2 }-> ifappend(1 + x' + l', z', 0) :|: x' >= 0, l' >= 0, z' >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z >= 0, z' >= 0, z'' = 1
   ifappend(z, z', z'') -{ 2 }-> 1 + x + append(0, z') :|: z'' = 0, z >= 0, z' >= 0, x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 3 }-> 1 + x + append(1 + x'' + l'', z') :|: z'' = 0, l'' >= 0, x'' >= 0, z' >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: z >= 0
   tl(z) -{ 1 }-> 1 + x + l :|: x >= 0, l >= 0, z = 1 + x + l

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

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

(23) 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

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

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

   append(z, z') -{ 2 }-> ifappend(0, z', 1) :|: z = 0, z' >= 0
   append(z, z') -{ 2 }-> ifappend(1 + x' + l', z', 0) :|: x' >= 0, l' >= 0, z' >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z >= 0, z' >= 0, z'' = 1
   ifappend(z, z', z'') -{ 2 }-> 1 + x + append(0, z') :|: z'' = 0, z >= 0, z' >= 0, x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 3 }-> 1 + x + append(1 + x'' + l'', z') :|: z'' = 0, l'' >= 0, x'' >= 0, z' >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: z >= 0
   tl(z) -{ 1 }-> 1 + x + 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]

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

(25) 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

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

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

   append(z, z') -{ 2 }-> ifappend(0, z', 1) :|: z = 0, z' >= 0
   append(z, z') -{ 2 }-> ifappend(1 + x' + l', z', 0) :|: x' >= 0, l' >= 0, z' >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z >= 0, z' >= 0, z'' = 1
   ifappend(z, z', z'') -{ 2 }-> 1 + x + append(0, z') :|: z'' = 0, z >= 0, z' >= 0, x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 3 }-> 1 + x + append(1 + x'' + l'', z') :|: z'' = 0, l'' >= 0, x'' >= 0, z' >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: z >= 0
   tl(z) -{ 1 }-> 1 + x + 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]

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

(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') -{ 2 }-> ifappend(0, z', 1) :|: z = 0, z' >= 0
   append(z, z') -{ 2 }-> ifappend(1 + x' + l', z', 0) :|: x' >= 0, l' >= 0, z' >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z >= 0, z' >= 0, z'' = 1
   ifappend(z, z', z'') -{ 2 }-> 1 + x + append(0, z') :|: z'' = 0, z >= 0, z' >= 0, x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 3 }-> 1 + x + append(1 + x'' + l'', z') :|: z'' = 0, l'' >= 0, x'' >= 0, z' >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: z >= 0
   tl(z) -{ 1 }-> 1 + x + 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]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: ifappend
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

Computed SIZE bound using CoFloCo for: append
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

   append(z, z') -{ 2 }-> ifappend(0, z', 1) :|: z = 0, z' >= 0
   append(z, z') -{ 2 }-> ifappend(1 + x' + l', z', 0) :|: x' >= 0, l' >= 0, z' >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z >= 0, z' >= 0, z'' = 1
   ifappend(z, z', z'') -{ 2 }-> 1 + x + append(0, z') :|: z'' = 0, z >= 0, z' >= 0, x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 3 }-> 1 + x + append(1 + x'' + l'', z') :|: z'' = 0, l'' >= 0, x'' >= 0, z' >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: z >= 0
   tl(z) -{ 1 }-> 1 + x + 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: INF
  append: runtime: ?, size: INF

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: ifappend
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

   append(z, z') -{ 2 }-> ifappend(0, z', 1) :|: z = 0, z' >= 0
   append(z, z') -{ 2 }-> ifappend(1 + x' + l', z', 0) :|: x' >= 0, l' >= 0, z' >= 0, z = 1 + x' + l'
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 1 }-> z' :|: z >= 0, z' >= 0, z'' = 1
   ifappend(z, z', z'') -{ 2 }-> 1 + x + append(0, z') :|: z'' = 0, z >= 0, z' >= 0, x >= 0, l >= 0, z = 1 + x + l
   ifappend(z, z', z'') -{ 3 }-> 1 + x + append(1 + x'' + l'', z') :|: z'' = 0, l'' >= 0, x'' >= 0, z' >= 0, z = 1 + x'' + l'', x >= 0, l >= 0, 1 + x'' + l'' = 1 + x + l
   is_empty(z) -{ 1 }-> 1 :|: z = 0
   is_empty(z) -{ 1 }-> 0 :|: x >= 0, l >= 0, z = 1 + x + l
   tl(z) -{ 0 }-> 0 :|: z >= 0
   tl(z) -{ 1 }-> 1 + x + 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: INF, size: INF
  append: runtime: ?, size: INF

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

(33) 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:

   hd(v0) -> null_hd [0]
   tl(v0) -> null_tl [0]
   is_empty(v0) -> null_is_empty [0]
   ifappend(v0, v1, v2) -> null_ifappend [0]

And the following fresh constants:    null_hd, null_tl, null_is_empty, null_ifappend

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

(34)
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:

   is_empty(nil) -> true [1]
   is_empty(cons(x, l)) -> false [1]
   hd(cons(x, l)) -> x [1]
   tl(cons(x, l)) -> cons(x, l) [1]
   append(l1, l2) -> ifappend(l1, l2, is_empty(l1)) [1]
   ifappend(l1, l2, true) -> l2 [1]
   ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2)) [1]
   hd(v0) -> null_hd [0]
   tl(v0) -> null_tl [0]
   is_empty(v0) -> null_is_empty [0]
   ifappend(v0, v1, v2) -> null_ifappend [0]

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

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:

   nil => 0
   true => 2
   false => 1
   null_hd => 0
   null_tl => 0
   null_is_empty => 0
   null_ifappend => 0

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

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

   append(z, z') -{ 1 }-> ifappend(l1, l2, is_empty(l1)) :|: z = l1, z' = l2, l1 >= 0, l2 >= 0
   hd(z) -{ 1 }-> x :|: x >= 0, l >= 0, z = 1 + x + l
   hd(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   ifappend(z, z', z'') -{ 1 }-> l2 :|: z = l1, z' = l2, z'' = 2, l1 >= 0, l2 >= 0
   ifappend(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   ifappend(z, z', z'') -{ 1 }-> 1 + hd(l1) + append(tl(l1), l2) :|: z = l1, z' = l2, l1 >= 0, l2 >= 0, z'' = 1
   is_empty(z) -{ 1 }-> 2 :|: z = 0
   is_empty(z) -{ 1 }-> 1 :|: x >= 0, l >= 0, z = 1 + x + l
   is_empty(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   tl(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   tl(z) -{ 1 }-> 1 + x + l :|: x >= 0, l >= 0, z = 1 + x + l

Only complete derivations are relevant for the runtime complexity.

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

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> cons(z0, z1)
   append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
   ifappend(z0, z1, true) -> z1
   ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))
Tuples:
   IS_EMPTY(nil) -> c
   IS_EMPTY(cons(z0, z1)) -> c1
   HD(cons(z0, z1)) -> c2
   TL(cons(z0, z1)) -> c3
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
   IFAPPEND(z0, z1, true) -> c5
   IFAPPEND(z0, z1, false) -> c6(HD(z0))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
S tuples:
   IS_EMPTY(nil) -> c
   IS_EMPTY(cons(z0, z1)) -> c1
   HD(cons(z0, z1)) -> c2
   TL(cons(z0, z1)) -> c3
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
   IFAPPEND(z0, z1, true) -> c5
   IFAPPEND(z0, z1, false) -> c6(HD(z0))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
K tuples:none
Defined Rule Symbols:   is_empty_1, hd_1, tl_1, append_2, ifappend_3

Defined Pair Symbols:   IS_EMPTY_1, HD_1, TL_1, APPEND_2, IFAPPEND_3

Compound Symbols:   c, c1, c2, c3, c4_2, c5, c6_1, c7_2


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

(39) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   IS_EMPTY(cons(z0, z1)) -> c1
   HD(cons(z0, z1)) -> c2
   IFAPPEND(z0, z1, false) -> c6(HD(z0))
   IFAPPEND(z0, z1, true) -> c5
   IS_EMPTY(nil) -> c
   TL(cons(z0, z1)) -> c3

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> cons(z0, z1)
   append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
   ifappend(z0, z1, true) -> z1
   ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))
Tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
S tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
K tuples:none
Defined Rule Symbols:   is_empty_1, hd_1, tl_1, append_2, ifappend_3

Defined Pair Symbols:   APPEND_2, IFAPPEND_3

Compound Symbols:   c4_2, c7_2


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

(41) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> cons(z0, z1)
   append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
   ifappend(z0, z1, true) -> z1
   ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))
Tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1))
S tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1))
K tuples:none
Defined Rule Symbols:   is_empty_1, hd_1, tl_1, append_2, ifappend_3

Defined Pair Symbols:   APPEND_2, IFAPPEND_3

Compound Symbols:   c4_1, c7_1


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

(43) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   hd(cons(z0, z1)) -> z0
   append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
   ifappend(z0, z1, true) -> z1
   ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false
   tl(cons(z0, z1)) -> cons(z0, z1)
Tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1))
S tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1))
K tuples:none
Defined Rule Symbols:   is_empty_1, tl_1

Defined Pair Symbols:   APPEND_2, IFAPPEND_3

Compound Symbols:   c4_1, c7_1


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

(45) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0))) by 
   APPEND(nil, x1) -> c4(IFAPPEND(nil, x1, true))
   APPEND(cons(z0, z1), x1) -> c4(IFAPPEND(cons(z0, z1), x1, false))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false
   tl(cons(z0, z1)) -> cons(z0, z1)
Tuples:
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1))
   APPEND(nil, x1) -> c4(IFAPPEND(nil, x1, true))
   APPEND(cons(z0, z1), x1) -> c4(IFAPPEND(cons(z0, z1), x1, false))
S tuples:
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1))
   APPEND(nil, x1) -> c4(IFAPPEND(nil, x1, true))
   APPEND(cons(z0, z1), x1) -> c4(IFAPPEND(cons(z0, z1), x1, false))
K tuples:none
Defined Rule Symbols:   is_empty_1, tl_1

Defined Pair Symbols:   IFAPPEND_3, APPEND_2

Compound Symbols:   c7_1, c4_1


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

(47) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   APPEND(nil, x1) -> c4(IFAPPEND(nil, x1, true))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false
   tl(cons(z0, z1)) -> cons(z0, z1)
Tuples:
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1))
   APPEND(cons(z0, z1), x1) -> c4(IFAPPEND(cons(z0, z1), x1, false))
S tuples:
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1))
   APPEND(cons(z0, z1), x1) -> c4(IFAPPEND(cons(z0, z1), x1, false))
K tuples:none
Defined Rule Symbols:   is_empty_1, tl_1

Defined Pair Symbols:   IFAPPEND_3, APPEND_2

Compound Symbols:   c7_1, c4_1


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

(49) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   tl(cons(z0, z1)) -> cons(z0, z1)
Tuples:
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1))
   APPEND(cons(z0, z1), x1) -> c4(IFAPPEND(cons(z0, z1), x1, false))
S tuples:
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1))
   APPEND(cons(z0, z1), x1) -> c4(IFAPPEND(cons(z0, z1), x1, false))
K tuples:none
Defined Rule Symbols:   tl_1

Defined Pair Symbols:   IFAPPEND_3, APPEND_2

Compound Symbols:   c7_1, c4_1


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

(51) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1)) by 
   IFAPPEND(cons(z0, z1), x1, false) -> c7(APPEND(cons(z0, z1), x1))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   tl(cons(z0, z1)) -> cons(z0, z1)
Tuples:
   APPEND(cons(z0, z1), x1) -> c4(IFAPPEND(cons(z0, z1), x1, false))
   IFAPPEND(cons(z0, z1), x1, false) -> c7(APPEND(cons(z0, z1), x1))
S tuples:
   APPEND(cons(z0, z1), x1) -> c4(IFAPPEND(cons(z0, z1), x1, false))
   IFAPPEND(cons(z0, z1), x1, false) -> c7(APPEND(cons(z0, z1), x1))
K tuples:none
Defined Rule Symbols:   tl_1

Defined Pair Symbols:   APPEND_2, IFAPPEND_3

Compound Symbols:   c4_1, c7_1


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

(53) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   tl(cons(z0, z1)) -> cons(z0, z1)

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   APPEND(cons(z0, z1), x1) -> c4(IFAPPEND(cons(z0, z1), x1, false))
   IFAPPEND(cons(z0, z1), x1, false) -> c7(APPEND(cons(z0, z1), x1))
S tuples:
   APPEND(cons(z0, z1), x1) -> c4(IFAPPEND(cons(z0, z1), x1, false))
   IFAPPEND(cons(z0, z1), x1, false) -> c7(APPEND(cons(z0, z1), x1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   APPEND_2, IFAPPEND_3

Compound Symbols:   c4_1, c7_1