/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^3), O(n^3))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^3, n^3).

(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), 201 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), 414 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 86 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 87 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 97 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 13 ms]
    (38) CpxRNTS
    (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 272 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 103 ms]
    (44) CpxRNTS
    (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 1424 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 131 ms]
    (50) CpxRNTS
    (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (52) CpxRNTS
    (53) IntTrsBoundProof [UPPER BOUND(ID), 477 ms]
    (54) CpxRNTS
    (55) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (56) CpxRNTS
    (57) FinalProof [FINISHED, 0 ms]
    (58) BOUNDS(1, n^3)
(59) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(60) CpxTRS
    (61) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) typed CpxTrs
    (63) OrderProof [LOWER BOUND(ID), 0 ms]
    (64) typed CpxTrs
    (65) RewriteLemmaProof [LOWER BOUND(ID), 271 ms]
    (66) BEST
        (67) proven lower bound
            (68) LowerBoundPropagationProof [FINISHED, 0 ms]
            (69) BOUNDS(n^1, INF)
        (70) typed CpxTrs
            (71) RewriteLemmaProof [LOWER BOUND(ID), 7 ms]
            (72) BEST
                (73) proven lower bound
                    (74) LowerBoundPropagationProof [FINISHED, 0 ms]
                    (75) BOUNDS(n^2, INF)
                (76) typed CpxTrs
                    (77) RewriteLemmaProof [LOWER BOUND(ID), 723 ms]
                    (78) proven lower bound
                    (79) LowerBoundPropagationProof [FINISHED, 0 ms]
                    (80) BOUNDS(n^3, INF)


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, n^3).


The TRS R consists of the following rules:

   null(nil) -> true
   null(add(n, x)) -> false
   tail(add(n, x)) -> x
   tail(nil) -> nil
   head(add(n, x)) -> n
   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   reverse(nil) -> nil
   reverse(add(n, x)) -> app(reverse(x), add(n, nil))
   shuffle(x) -> shuff(x, nil)
   shuff(x, y) -> if(null(x), x, y, app(y, add(head(x), nil)))
   if(true, x, y, z) -> y
   if(false, x, y, z) -> shuff(reverse(tail(x)), z)

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^3).


The TRS R consists of the following rules:

   null(nil) -> true [1]
   null(add(n, x)) -> false [1]
   tail(add(n, x)) -> x [1]
   tail(nil) -> nil [1]
   head(add(n, x)) -> n [1]
   app(nil, y) -> y [1]
   app(add(n, x), y) -> add(n, app(x, y)) [1]
   reverse(nil) -> nil [1]
   reverse(add(n, x)) -> app(reverse(x), add(n, nil)) [1]
   shuffle(x) -> shuff(x, nil) [1]
   shuff(x, y) -> if(null(x), x, y, app(y, add(head(x), nil))) [1]
   if(true, x, y, z) -> y [1]
   if(false, x, y, z) -> shuff(reverse(tail(x)), z) [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:

   null(nil) -> true [1]
   null(add(n, x)) -> false [1]
   tail(add(n, x)) -> x [1]
   tail(nil) -> nil [1]
   head(add(n, x)) -> n [1]
   app(nil, y) -> y [1]
   app(add(n, x), y) -> add(n, app(x, y)) [1]
   reverse(nil) -> nil [1]
   reverse(add(n, x)) -> app(reverse(x), add(n, nil)) [1]
   shuffle(x) -> shuff(x, nil) [1]
   shuff(x, y) -> if(null(x), x, y, app(y, add(head(x), nil))) [1]
   if(true, x, y, z) -> y [1]
   if(false, x, y, z) -> shuff(reverse(tail(x)), z) [1]

The TRS has the following type information:
null :: nil:add -> true:false
nil :: nil:add
true :: true:false
add :: head -> nil:add -> nil:add
false :: true:false
tail :: nil:add -> nil:add
head :: nil:add -> head
app :: nil:add -> nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
shuff :: nil:add -> nil:add -> nil:add
if :: true:false -> nil:add -> nil:add -> nil:add -> nil:add

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:

   shuffle_1
   shuff_2
   if_4

(c) The following functions are completely defined:

   null_1
   app_2
   head_1
   reverse_1
   tail_1

Due to the following rules being added:

   head(v0) -> const [0]

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:

   null(nil) -> true [1]
   null(add(n, x)) -> false [1]
   tail(add(n, x)) -> x [1]
   tail(nil) -> nil [1]
   head(add(n, x)) -> n [1]
   app(nil, y) -> y [1]
   app(add(n, x), y) -> add(n, app(x, y)) [1]
   reverse(nil) -> nil [1]
   reverse(add(n, x)) -> app(reverse(x), add(n, nil)) [1]
   shuffle(x) -> shuff(x, nil) [1]
   shuff(x, y) -> if(null(x), x, y, app(y, add(head(x), nil))) [1]
   if(true, x, y, z) -> y [1]
   if(false, x, y, z) -> shuff(reverse(tail(x)), z) [1]
   head(v0) -> const [0]

The TRS has the following type information:
null :: nil:add -> true:false
nil :: nil:add
true :: true:false
add :: const -> nil:add -> nil:add
false :: true:false
tail :: nil:add -> nil:add
head :: nil:add -> const
app :: nil:add -> nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
shuff :: nil:add -> nil:add -> nil:add
if :: true:false -> nil:add -> nil:add -> nil:add -> nil:add
const :: const

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:

   null(nil) -> true [1]
   null(add(n, x)) -> false [1]
   tail(add(n, x)) -> x [1]
   tail(nil) -> nil [1]
   head(add(n, x)) -> n [1]
   app(nil, y) -> y [1]
   app(add(n, x), y) -> add(n, app(x, y)) [1]
   reverse(nil) -> nil [1]
   reverse(add(n, nil)) -> app(nil, add(n, nil)) [2]
   reverse(add(n, add(n', x'))) -> app(app(reverse(x'), add(n', nil)), add(n, nil)) [2]
   shuffle(x) -> shuff(x, nil) [1]
   shuff(nil, y) -> if(true, nil, y, app(y, add(const, nil))) [2]
   shuff(add(n'', x''), y) -> if(false, add(n'', x''), y, app(y, add(n'', nil))) [3]
   shuff(add(n'', x''), y) -> if(false, add(n'', x''), y, app(y, add(const, nil))) [2]
   if(true, x, y, z) -> y [1]
   if(false, add(n1, x1), y, z) -> shuff(reverse(x1), z) [2]
   if(false, nil, y, z) -> shuff(reverse(nil), z) [2]
   head(v0) -> const [0]

The TRS has the following type information:
null :: nil:add -> true:false
nil :: nil:add
true :: true:false
add :: const -> nil:add -> nil:add
false :: true:false
tail :: nil:add -> nil:add
head :: nil:add -> const
app :: nil:add -> nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
shuff :: nil:add -> nil:add -> nil:add
if :: true:false -> nil:add -> nil:add -> nil:add -> nil:add
const :: const

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:

   app(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, y) :|: n >= 0, z'' = y, z' = 1 + n + x, x >= 0, y >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   if(z', z'', z1, z2) -{ 1 }-> y :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(x1), z) :|: z1 = y, x1 >= 0, z >= 0, z2 = z, y >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(0), z) :|: z'' = 0, z1 = y, z >= 0, z2 = z, y >= 0, z' = 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 2 }-> app(0, 1 + n + 0) :|: n >= 0, z' = 1 + n + 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 2 }-> if(1, 0, y, app(y, 1 + 0 + 0)) :|: z'' = y, y >= 0, z' = 0
   shuff(z', z'') -{ 3 }-> if(0, 1 + n'' + x'', y, app(y, 1 + n'' + 0)) :|: z' = 1 + n'' + x'', z'' = y, y >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 2 }-> if(0, 1 + n'' + x'', y, app(y, 1 + 0 + 0)) :|: z' = 1 + n'' + x'', z'' = y, y >= 0, n'' >= 0, x'' >= 0
   shuffle(z') -{ 1 }-> shuff(x, 0) :|: z' = x, x >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0


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

(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:

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(x1), z2) :|: x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 2 }-> app(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 2 }-> if(1, 0, z'', app(z'', 1 + 0 + 0)) :|: z'' >= 0, z' = 0
   shuff(z', z'') -{ 3 }-> if(0, 1 + n'' + x'', z'', app(z'', 1 + n'' + 0)) :|: z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 2 }-> if(0, 1 + n'' + x'', z'', app(z'', 1 + 0 + 0)) :|: z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuffle(z') -{ 1 }-> shuff(z', 0) :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0


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

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

   { head }
   { app }
   { null }
   { tail }
   { reverse }
   { shuff, if }
   { shuffle }

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(x1), z2) :|: x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 2 }-> app(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 2 }-> if(1, 0, z'', app(z'', 1 + 0 + 0)) :|: z'' >= 0, z' = 0
   shuff(z', z'') -{ 3 }-> if(0, 1 + n'' + x'', z'', app(z'', 1 + n'' + 0)) :|: z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 2 }-> if(0, 1 + n'' + x'', z'', app(z'', 1 + 0 + 0)) :|: z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuffle(z') -{ 1 }-> shuff(z', 0) :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {head}, {app}, {null}, {tail}, {reverse}, {shuff,if}, {shuffle}

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

(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:

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(x1), z2) :|: x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 2 }-> app(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 2 }-> if(1, 0, z'', app(z'', 1 + 0 + 0)) :|: z'' >= 0, z' = 0
   shuff(z', z'') -{ 3 }-> if(0, 1 + n'' + x'', z'', app(z'', 1 + n'' + 0)) :|: z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 2 }-> if(0, 1 + n'' + x'', z'', app(z'', 1 + 0 + 0)) :|: z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuffle(z') -{ 1 }-> shuff(z', 0) :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {head}, {app}, {null}, {tail}, {reverse}, {shuff,if}, {shuffle}

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

(17) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(x1), z2) :|: x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 2 }-> app(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 2 }-> if(1, 0, z'', app(z'', 1 + 0 + 0)) :|: z'' >= 0, z' = 0
   shuff(z', z'') -{ 3 }-> if(0, 1 + n'' + x'', z'', app(z'', 1 + n'' + 0)) :|: z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 2 }-> if(0, 1 + n'' + x'', z'', app(z'', 1 + 0 + 0)) :|: z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuffle(z') -{ 1 }-> shuff(z', 0) :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {head}, {app}, {null}, {tail}, {reverse}, {shuff,if}, {shuffle}
Previous analysis results are:
  head: runtime: ?, size: O(n^1) [z']

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: head
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:

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(x1), z2) :|: x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 2 }-> app(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 2 }-> if(1, 0, z'', app(z'', 1 + 0 + 0)) :|: z'' >= 0, z' = 0
   shuff(z', z'') -{ 3 }-> if(0, 1 + n'' + x'', z'', app(z'', 1 + n'' + 0)) :|: z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 2 }-> if(0, 1 + n'' + x'', z'', app(z'', 1 + 0 + 0)) :|: z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuffle(z') -{ 1 }-> shuff(z', 0) :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {app}, {null}, {tail}, {reverse}, {shuff,if}, {shuffle}
Previous analysis results are:
  head: runtime: O(1) [1], size: O(n^1) [z']

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

(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:

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(x1), z2) :|: x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 2 }-> app(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 2 }-> if(1, 0, z'', app(z'', 1 + 0 + 0)) :|: z'' >= 0, z' = 0
   shuff(z', z'') -{ 3 }-> if(0, 1 + n'' + x'', z'', app(z'', 1 + n'' + 0)) :|: z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 2 }-> if(0, 1 + n'' + x'', z'', app(z'', 1 + 0 + 0)) :|: z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuffle(z') -{ 1 }-> shuff(z', 0) :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {app}, {null}, {tail}, {reverse}, {shuff,if}, {shuffle}
Previous analysis results are:
  head: runtime: O(1) [1], size: O(n^1) [z']

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: app
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:

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(x1), z2) :|: x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 2 }-> app(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 2 }-> if(1, 0, z'', app(z'', 1 + 0 + 0)) :|: z'' >= 0, z' = 0
   shuff(z', z'') -{ 3 }-> if(0, 1 + n'' + x'', z'', app(z'', 1 + n'' + 0)) :|: z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 2 }-> if(0, 1 + n'' + x'', z'', app(z'', 1 + 0 + 0)) :|: z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuffle(z') -{ 1 }-> shuff(z', 0) :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {app}, {null}, {tail}, {reverse}, {shuff,if}, {shuffle}
Previous analysis results are:
  head: runtime: O(1) [1], size: O(n^1) [z']
  app: runtime: ?, size: O(n^1) [z' + z'']

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(x1), z2) :|: x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 2 }-> app(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 2 }-> if(1, 0, z'', app(z'', 1 + 0 + 0)) :|: z'' >= 0, z' = 0
   shuff(z', z'') -{ 3 }-> if(0, 1 + n'' + x'', z'', app(z'', 1 + n'' + 0)) :|: z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 2 }-> if(0, 1 + n'' + x'', z'', app(z'', 1 + 0 + 0)) :|: z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuffle(z') -{ 1 }-> shuff(z', 0) :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {null}, {tail}, {reverse}, {shuff,if}, {shuffle}
Previous analysis results are:
  head: runtime: O(1) [1], size: O(n^1) [z']
  app: runtime: O(n^1) [1 + 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:

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(x1), z2) :|: x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 3 + z'' }-> if(1, 0, z'', s'') :|: s'' >= 0, s'' <= z'' + (1 + 0 + 0), z'' >= 0, z' = 0
   shuff(z', z'') -{ 4 + z'' }-> if(0, 1 + n'' + x'', z'', s1) :|: s1 >= 0, s1 <= z'' + (1 + n'' + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 3 + z'' }-> if(0, 1 + n'' + x'', z'', s2) :|: s2 >= 0, s2 <= z'' + (1 + 0 + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuffle(z') -{ 1 }-> shuff(z', 0) :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {null}, {tail}, {reverse}, {shuff,if}, {shuffle}
Previous analysis results are:
  head: runtime: O(1) [1], size: O(n^1) [z']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']

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

(29) 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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(x1), z2) :|: x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 3 + z'' }-> if(1, 0, z'', s'') :|: s'' >= 0, s'' <= z'' + (1 + 0 + 0), z'' >= 0, z' = 0
   shuff(z', z'') -{ 4 + z'' }-> if(0, 1 + n'' + x'', z'', s1) :|: s1 >= 0, s1 <= z'' + (1 + n'' + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 3 + z'' }-> if(0, 1 + n'' + x'', z'', s2) :|: s2 >= 0, s2 <= z'' + (1 + 0 + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuffle(z') -{ 1 }-> shuff(z', 0) :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {null}, {tail}, {reverse}, {shuff,if}, {shuffle}
Previous analysis results are:
  head: runtime: O(1) [1], size: O(n^1) [z']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  null: runtime: ?, size: O(1) [1]

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

(31) 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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(x1), z2) :|: x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 3 + z'' }-> if(1, 0, z'', s'') :|: s'' >= 0, s'' <= z'' + (1 + 0 + 0), z'' >= 0, z' = 0
   shuff(z', z'') -{ 4 + z'' }-> if(0, 1 + n'' + x'', z'', s1) :|: s1 >= 0, s1 <= z'' + (1 + n'' + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 3 + z'' }-> if(0, 1 + n'' + x'', z'', s2) :|: s2 >= 0, s2 <= z'' + (1 + 0 + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuffle(z') -{ 1 }-> shuff(z', 0) :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {tail}, {reverse}, {shuff,if}, {shuffle}
Previous analysis results are:
  head: runtime: O(1) [1], size: O(n^1) [z']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  null: runtime: O(1) [1], size: O(1) [1]

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

(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:

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(x1), z2) :|: x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 3 + z'' }-> if(1, 0, z'', s'') :|: s'' >= 0, s'' <= z'' + (1 + 0 + 0), z'' >= 0, z' = 0
   shuff(z', z'') -{ 4 + z'' }-> if(0, 1 + n'' + x'', z'', s1) :|: s1 >= 0, s1 <= z'' + (1 + n'' + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 3 + z'' }-> if(0, 1 + n'' + x'', z'', s2) :|: s2 >= 0, s2 <= z'' + (1 + 0 + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuffle(z') -{ 1 }-> shuff(z', 0) :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {tail}, {reverse}, {shuff,if}, {shuffle}
Previous analysis results are:
  head: runtime: O(1) [1], size: O(n^1) [z']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  null: runtime: O(1) [1], size: O(1) [1]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: tail
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:

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(x1), z2) :|: x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 3 + z'' }-> if(1, 0, z'', s'') :|: s'' >= 0, s'' <= z'' + (1 + 0 + 0), z'' >= 0, z' = 0
   shuff(z', z'') -{ 4 + z'' }-> if(0, 1 + n'' + x'', z'', s1) :|: s1 >= 0, s1 <= z'' + (1 + n'' + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 3 + z'' }-> if(0, 1 + n'' + x'', z'', s2) :|: s2 >= 0, s2 <= z'' + (1 + 0 + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuffle(z') -{ 1 }-> shuff(z', 0) :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {tail}, {reverse}, {shuff,if}, {shuffle}
Previous analysis results are:
  head: runtime: O(1) [1], size: O(n^1) [z']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  null: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: ?, size: O(n^1) [z']

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: tail
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:

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(x1), z2) :|: x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 3 + z'' }-> if(1, 0, z'', s'') :|: s'' >= 0, s'' <= z'' + (1 + 0 + 0), z'' >= 0, z' = 0
   shuff(z', z'') -{ 4 + z'' }-> if(0, 1 + n'' + x'', z'', s1) :|: s1 >= 0, s1 <= z'' + (1 + n'' + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 3 + z'' }-> if(0, 1 + n'' + x'', z'', s2) :|: s2 >= 0, s2 <= z'' + (1 + 0 + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuffle(z') -{ 1 }-> shuff(z', 0) :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {reverse}, {shuff,if}, {shuffle}
Previous analysis results are:
  head: runtime: O(1) [1], size: O(n^1) [z']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  null: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z']

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(x1), z2) :|: x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 3 + z'' }-> if(1, 0, z'', s'') :|: s'' >= 0, s'' <= z'' + (1 + 0 + 0), z'' >= 0, z' = 0
   shuff(z', z'') -{ 4 + z'' }-> if(0, 1 + n'' + x'', z'', s1) :|: s1 >= 0, s1 <= z'' + (1 + n'' + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 3 + z'' }-> if(0, 1 + n'' + x'', z'', s2) :|: s2 >= 0, s2 <= z'' + (1 + 0 + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuffle(z') -{ 1 }-> shuff(z', 0) :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {reverse}, {shuff,if}, {shuffle}
Previous analysis results are:
  head: runtime: O(1) [1], size: O(n^1) [z']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  null: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z']

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(x1), z2) :|: x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 3 + z'' }-> if(1, 0, z'', s'') :|: s'' >= 0, s'' <= z'' + (1 + 0 + 0), z'' >= 0, z' = 0
   shuff(z', z'') -{ 4 + z'' }-> if(0, 1 + n'' + x'', z'', s1) :|: s1 >= 0, s1 <= z'' + (1 + n'' + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 3 + z'' }-> if(0, 1 + n'' + x'', z'', s2) :|: s2 >= 0, s2 <= z'' + (1 + 0 + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuffle(z') -{ 1 }-> shuff(z', 0) :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {reverse}, {shuff,if}, {shuffle}
Previous analysis results are:
  head: runtime: O(1) [1], size: O(n^1) [z']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  null: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  reverse: runtime: ?, size: O(n^1) [z']

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(x1), z2) :|: x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> shuff(reverse(0), z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x'), 1 + n' + 0), 1 + n + 0) :|: n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 3 + z'' }-> if(1, 0, z'', s'') :|: s'' >= 0, s'' <= z'' + (1 + 0 + 0), z'' >= 0, z' = 0
   shuff(z', z'') -{ 4 + z'' }-> if(0, 1 + n'' + x'', z'', s1) :|: s1 >= 0, s1 <= z'' + (1 + n'' + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 3 + z'' }-> if(0, 1 + n'' + x'', z'', s2) :|: s2 >= 0, s2 <= z'' + (1 + 0 + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuffle(z') -{ 1 }-> shuff(z', 0) :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {shuff,if}, {shuffle}
Previous analysis results are:
  head: runtime: O(1) [1], size: O(n^1) [z']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  null: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  reverse: runtime: O(n^2) [4 + 3*z' + 2*z'^2], size: O(n^1) [z']

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 6 + 3*x1 + 2*x1^2 }-> shuff(s6, z2) :|: s6 >= 0, s6 <= x1, x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 6 }-> shuff(s7, z2) :|: s7 >= 0, s7 <= 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 8 + s3 + s4 + 3*x' + 2*x'^2 }-> s5 :|: s3 >= 0, s3 <= x', s4 >= 0, s4 <= s3 + (1 + n' + 0), s5 >= 0, s5 <= s4 + (1 + n + 0), n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 3 + z'' }-> if(1, 0, z'', s'') :|: s'' >= 0, s'' <= z'' + (1 + 0 + 0), z'' >= 0, z' = 0
   shuff(z', z'') -{ 4 + z'' }-> if(0, 1 + n'' + x'', z'', s1) :|: s1 >= 0, s1 <= z'' + (1 + n'' + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 3 + z'' }-> if(0, 1 + n'' + x'', z'', s2) :|: s2 >= 0, s2 <= z'' + (1 + 0 + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuffle(z') -{ 1 }-> shuff(z', 0) :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {shuff,if}, {shuffle}
Previous analysis results are:
  head: runtime: O(1) [1], size: O(n^1) [z']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  null: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  reverse: runtime: O(n^2) [4 + 3*z' + 2*z'^2], size: O(n^1) [z']

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed SIZE bound using KoAT for: if
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: z''^2 + z1 + 2*z2

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

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 6 + 3*x1 + 2*x1^2 }-> shuff(s6, z2) :|: s6 >= 0, s6 <= x1, x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 6 }-> shuff(s7, z2) :|: s7 >= 0, s7 <= 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 8 + s3 + s4 + 3*x' + 2*x'^2 }-> s5 :|: s3 >= 0, s3 <= x', s4 >= 0, s4 <= s3 + (1 + n' + 0), s5 >= 0, s5 <= s4 + (1 + n + 0), n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 3 + z'' }-> if(1, 0, z'', s'') :|: s'' >= 0, s'' <= z'' + (1 + 0 + 0), z'' >= 0, z' = 0
   shuff(z', z'') -{ 4 + z'' }-> if(0, 1 + n'' + x'', z'', s1) :|: s1 >= 0, s1 <= z'' + (1 + n'' + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 3 + z'' }-> if(0, 1 + n'' + x'', z'', s2) :|: s2 >= 0, s2 <= z'' + (1 + 0 + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuffle(z') -{ 1 }-> shuff(z', 0) :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {shuff,if}, {shuffle}
Previous analysis results are:
  head: runtime: O(1) [1], size: O(n^1) [z']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  null: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  reverse: runtime: O(n^2) [4 + 3*z' + 2*z'^2], size: O(n^1) [z']
  shuff: runtime: ?, size: O(n^2) [z'^2 + z'']
  if: runtime: ?, size: O(n^2) [z''^2 + z1 + 2*z2]

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: shuff
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 20 + 47*z' + 6*z'*z'' + 26*z'^2 + 16*z'^3 + 3*z''

Computed RUNTIME bound using KoAT for: if
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 53 + 50*z'' + 6*z''*z2 + 28*z''^2 + 16*z''^3 + 6*z2

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 6 + 3*x1 + 2*x1^2 }-> shuff(s6, z2) :|: s6 >= 0, s6 <= x1, x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 6 }-> shuff(s7, z2) :|: s7 >= 0, s7 <= 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 8 + s3 + s4 + 3*x' + 2*x'^2 }-> s5 :|: s3 >= 0, s3 <= x', s4 >= 0, s4 <= s3 + (1 + n' + 0), s5 >= 0, s5 <= s4 + (1 + n + 0), n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 3 + z'' }-> if(1, 0, z'', s'') :|: s'' >= 0, s'' <= z'' + (1 + 0 + 0), z'' >= 0, z' = 0
   shuff(z', z'') -{ 4 + z'' }-> if(0, 1 + n'' + x'', z'', s1) :|: s1 >= 0, s1 <= z'' + (1 + n'' + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 3 + z'' }-> if(0, 1 + n'' + x'', z'', s2) :|: s2 >= 0, s2 <= z'' + (1 + 0 + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuffle(z') -{ 1 }-> shuff(z', 0) :|: z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {shuffle}
Previous analysis results are:
  head: runtime: O(1) [1], size: O(n^1) [z']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  null: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  reverse: runtime: O(n^2) [4 + 3*z' + 2*z'^2], size: O(n^1) [z']
  shuff: runtime: O(n^3) [20 + 47*z' + 6*z'*z'' + 26*z'^2 + 16*z'^3 + 3*z''], size: O(n^2) [z'^2 + z'']
  if: runtime: O(n^3) [53 + 50*z'' + 6*z''*z2 + 28*z''^2 + 16*z''^3 + 6*z2], size: O(n^2) [z''^2 + z1 + 2*z2]

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

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

(52)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 26 + 47*s6 + 6*s6*z2 + 26*s6^2 + 16*s6^3 + 3*x1 + 2*x1^2 + 3*z2 }-> s12 :|: s12 >= 0, s12 <= s6 * s6 + z2, s6 >= 0, s6 <= x1, x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 26 + 47*s7 + 6*s7*z2 + 26*s7^2 + 16*s7^3 + 3*z2 }-> s13 :|: s13 >= 0, s13 <= s7 * s7 + z2, s7 >= 0, s7 <= 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 8 + s3 + s4 + 3*x' + 2*x'^2 }-> s5 :|: s3 >= 0, s3 <= x', s4 >= 0, s4 <= s3 + (1 + n' + 0), s5 >= 0, s5 <= s4 + (1 + n + 0), n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 151 + 154*n'' + 6*n''*s1 + 152*n''*x'' + 48*n''*x''^2 + 76*n''^2 + 48*n''^2*x'' + 16*n''^3 + 12*s1 + 6*s1*x'' + 154*x'' + 76*x''^2 + 16*x''^3 + z'' }-> s10 :|: s10 >= 0, s10 <= 2 * s1 + (1 + n'' + x'') * (1 + n'' + x'') + z'', s1 >= 0, s1 <= z'' + (1 + n'' + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 150 + 154*n'' + 6*n''*s2 + 152*n''*x'' + 48*n''*x''^2 + 76*n''^2 + 48*n''^2*x'' + 16*n''^3 + 12*s2 + 6*s2*x'' + 154*x'' + 76*x''^2 + 16*x''^3 + z'' }-> s11 :|: s11 >= 0, s11 <= 2 * s2 + (1 + n'' + x'') * (1 + n'' + x'') + z'', s2 >= 0, s2 <= z'' + (1 + 0 + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 56 + 6*s'' + z'' }-> s9 :|: s9 >= 0, s9 <= 2 * s'' + 0 * 0 + z'', s'' >= 0, s'' <= z'' + (1 + 0 + 0), z'' >= 0, z' = 0
   shuffle(z') -{ 21 + 47*z' + 26*z'^2 + 16*z'^3 }-> s8 :|: s8 >= 0, s8 <= z' * z' + 0, z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {shuffle}
Previous analysis results are:
  head: runtime: O(1) [1], size: O(n^1) [z']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  null: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  reverse: runtime: O(n^2) [4 + 3*z' + 2*z'^2], size: O(n^1) [z']
  shuff: runtime: O(n^3) [20 + 47*z' + 6*z'*z'' + 26*z'^2 + 16*z'^3 + 3*z''], size: O(n^2) [z'^2 + z'']
  if: runtime: O(n^3) [53 + 50*z'' + 6*z''*z2 + 28*z''^2 + 16*z''^3 + 6*z2], size: O(n^2) [z''^2 + z1 + 2*z2]

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: shuffle
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: z'^2

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

(54)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 26 + 47*s6 + 6*s6*z2 + 26*s6^2 + 16*s6^3 + 3*x1 + 2*x1^2 + 3*z2 }-> s12 :|: s12 >= 0, s12 <= s6 * s6 + z2, s6 >= 0, s6 <= x1, x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 26 + 47*s7 + 6*s7*z2 + 26*s7^2 + 16*s7^3 + 3*z2 }-> s13 :|: s13 >= 0, s13 <= s7 * s7 + z2, s7 >= 0, s7 <= 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 8 + s3 + s4 + 3*x' + 2*x'^2 }-> s5 :|: s3 >= 0, s3 <= x', s4 >= 0, s4 <= s3 + (1 + n' + 0), s5 >= 0, s5 <= s4 + (1 + n + 0), n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 151 + 154*n'' + 6*n''*s1 + 152*n''*x'' + 48*n''*x''^2 + 76*n''^2 + 48*n''^2*x'' + 16*n''^3 + 12*s1 + 6*s1*x'' + 154*x'' + 76*x''^2 + 16*x''^3 + z'' }-> s10 :|: s10 >= 0, s10 <= 2 * s1 + (1 + n'' + x'') * (1 + n'' + x'') + z'', s1 >= 0, s1 <= z'' + (1 + n'' + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 150 + 154*n'' + 6*n''*s2 + 152*n''*x'' + 48*n''*x''^2 + 76*n''^2 + 48*n''^2*x'' + 16*n''^3 + 12*s2 + 6*s2*x'' + 154*x'' + 76*x''^2 + 16*x''^3 + z'' }-> s11 :|: s11 >= 0, s11 <= 2 * s2 + (1 + n'' + x'') * (1 + n'' + x'') + z'', s2 >= 0, s2 <= z'' + (1 + 0 + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 56 + 6*s'' + z'' }-> s9 :|: s9 >= 0, s9 <= 2 * s'' + 0 * 0 + z'', s'' >= 0, s'' <= z'' + (1 + 0 + 0), z'' >= 0, z' = 0
   shuffle(z') -{ 21 + 47*z' + 26*z'^2 + 16*z'^3 }-> s8 :|: s8 >= 0, s8 <= z' * z' + 0, z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {shuffle}
Previous analysis results are:
  head: runtime: O(1) [1], size: O(n^1) [z']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  null: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  reverse: runtime: O(n^2) [4 + 3*z' + 2*z'^2], size: O(n^1) [z']
  shuff: runtime: O(n^3) [20 + 47*z' + 6*z'*z'' + 26*z'^2 + 16*z'^3 + 3*z''], size: O(n^2) [z'^2 + z'']
  if: runtime: O(n^3) [53 + 50*z'' + 6*z''*z2 + 28*z''^2 + 16*z''^3 + 6*z2], size: O(n^2) [z''^2 + z1 + 2*z2]
  shuffle: runtime: ?, size: O(n^2) [z'^2]

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: shuffle
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 21 + 47*z' + 26*z'^2 + 16*z'^3

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

(56)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s :|: s >= 0, s <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   head(z') -{ 1 }-> n :|: n >= 0, z' = 1 + n + x, x >= 0
   head(z') -{ 0 }-> 0 :|: z' >= 0
   if(z', z'', z1, z2) -{ 26 + 47*s6 + 6*s6*z2 + 26*s6^2 + 16*s6^3 + 3*x1 + 2*x1^2 + 3*z2 }-> s12 :|: s12 >= 0, s12 <= s6 * s6 + z2, s6 >= 0, s6 <= x1, x1 >= 0, z2 >= 0, z1 >= 0, z'' = 1 + n1 + x1, n1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 26 + 47*s7 + 6*s7*z2 + 26*s7^2 + 16*s7^3 + 3*z2 }-> s13 :|: s13 >= 0, s13 <= s7 * s7 + z2, s7 >= 0, s7 <= 0, z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   null(z') -{ 1 }-> 1 :|: z' = 0
   null(z') -{ 1 }-> 0 :|: n >= 0, z' = 1 + n + x, x >= 0
   reverse(z') -{ 3 }-> s' :|: s' >= 0, s' <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 8 + s3 + s4 + 3*x' + 2*x'^2 }-> s5 :|: s3 >= 0, s3 <= x', s4 >= 0, s4 <= s3 + (1 + n' + 0), s5 >= 0, s5 <= s4 + (1 + n + 0), n >= 0, x' >= 0, n' >= 0, z' = 1 + n + (1 + n' + x')
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuff(z', z'') -{ 151 + 154*n'' + 6*n''*s1 + 152*n''*x'' + 48*n''*x''^2 + 76*n''^2 + 48*n''^2*x'' + 16*n''^3 + 12*s1 + 6*s1*x'' + 154*x'' + 76*x''^2 + 16*x''^3 + z'' }-> s10 :|: s10 >= 0, s10 <= 2 * s1 + (1 + n'' + x'') * (1 + n'' + x'') + z'', s1 >= 0, s1 <= z'' + (1 + n'' + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 150 + 154*n'' + 6*n''*s2 + 152*n''*x'' + 48*n''*x''^2 + 76*n''^2 + 48*n''^2*x'' + 16*n''^3 + 12*s2 + 6*s2*x'' + 154*x'' + 76*x''^2 + 16*x''^3 + z'' }-> s11 :|: s11 >= 0, s11 <= 2 * s2 + (1 + n'' + x'') * (1 + n'' + x'') + z'', s2 >= 0, s2 <= z'' + (1 + 0 + 0), z' = 1 + n'' + x'', z'' >= 0, n'' >= 0, x'' >= 0
   shuff(z', z'') -{ 56 + 6*s'' + z'' }-> s9 :|: s9 >= 0, s9 <= 2 * s'' + 0 * 0 + z'', s'' >= 0, s'' <= z'' + (1 + 0 + 0), z'' >= 0, z' = 0
   shuffle(z') -{ 21 + 47*z' + 26*z'^2 + 16*z'^3 }-> s8 :|: s8 >= 0, s8 <= z' * z' + 0, z' >= 0
   tail(z') -{ 1 }-> x :|: n >= 0, z' = 1 + n + x, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: 
Previous analysis results are:
  head: runtime: O(1) [1], size: O(n^1) [z']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  null: runtime: O(1) [1], size: O(1) [1]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  reverse: runtime: O(n^2) [4 + 3*z' + 2*z'^2], size: O(n^1) [z']
  shuff: runtime: O(n^3) [20 + 47*z' + 6*z'*z'' + 26*z'^2 + 16*z'^3 + 3*z''], size: O(n^2) [z'^2 + z'']
  if: runtime: O(n^3) [53 + 50*z'' + 6*z''*z2 + 28*z''^2 + 16*z''^3 + 6*z2], size: O(n^2) [z''^2 + z1 + 2*z2]
  shuffle: runtime: O(n^3) [21 + 47*z' + 26*z'^2 + 16*z'^3], size: O(n^2) [z'^2]

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

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

(58)
BOUNDS(1, n^3)

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

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

(60)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   null(nil) -> true
   null(add(n, x)) -> false
   tail(add(n, x)) -> x
   tail(nil) -> nil
   head(add(n, x)) -> n
   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   reverse(nil) -> nil
   reverse(add(n, x)) -> app(reverse(x), add(n, nil))
   shuffle(x) -> shuff(x, nil)
   shuff(x, y) -> if(null(x), x, y, app(y, add(head(x), nil)))
   if(true, x, y, z) -> y
   if(false, x, y, z) -> shuff(reverse(tail(x)), z)

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

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

(62)
Obligation:
Innermost TRS:
Rules:
null(nil) -> true
null(add(n, x)) -> false
tail(add(n, x)) -> x
tail(nil) -> nil
head(add(n, x)) -> n
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(x) -> shuff(x, nil)
shuff(x, y) -> if(null(x), x, y, app(y, add(head(x), nil)))
if(true, x, y, z) -> y
if(false, x, y, z) -> shuff(reverse(tail(x)), z)

Types:
null :: nil:add -> true:false
nil :: nil:add
true :: true:false
add :: head -> nil:add -> nil:add
false :: true:false
tail :: nil:add -> nil:add
head :: nil:add -> head
app :: nil:add -> nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
shuff :: nil:add -> nil:add -> nil:add
if :: true:false -> nil:add -> nil:add -> nil:add -> nil:add
hole_true:false1_0 :: true:false
hole_nil:add2_0 :: nil:add
hole_head3_0 :: head
gen_nil:add4_0 :: Nat -> nil:add

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

(63) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
app, reverse, shuff

They will be analysed ascendingly in the following order:
app < reverse
app < shuff
reverse < shuff

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

(64)
Obligation:
Innermost TRS:
Rules:
null(nil) -> true
null(add(n, x)) -> false
tail(add(n, x)) -> x
tail(nil) -> nil
head(add(n, x)) -> n
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(x) -> shuff(x, nil)
shuff(x, y) -> if(null(x), x, y, app(y, add(head(x), nil)))
if(true, x, y, z) -> y
if(false, x, y, z) -> shuff(reverse(tail(x)), z)

Types:
null :: nil:add -> true:false
nil :: nil:add
true :: true:false
add :: head -> nil:add -> nil:add
false :: true:false
tail :: nil:add -> nil:add
head :: nil:add -> head
app :: nil:add -> nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
shuff :: nil:add -> nil:add -> nil:add
if :: true:false -> nil:add -> nil:add -> nil:add -> nil:add
hole_true:false1_0 :: true:false
hole_nil:add2_0 :: nil:add
hole_head3_0 :: head
gen_nil:add4_0 :: Nat -> nil:add


Generator Equations:
gen_nil:add4_0(0) <=> nil
gen_nil:add4_0(+(x, 1)) <=> add(hole_head3_0, gen_nil:add4_0(x))


The following defined symbols remain to be analysed:
app, reverse, shuff

They will be analysed ascendingly in the following order:
app < reverse
app < shuff
reverse < shuff

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

(65) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
app(gen_nil:add4_0(n6_0), gen_nil:add4_0(b)) -> gen_nil:add4_0(+(n6_0, b)), rt in Omega(1 + n6_0)

Induction Base:
app(gen_nil:add4_0(0), gen_nil:add4_0(b)) ->_R^Omega(1)
gen_nil:add4_0(b)

Induction Step:
app(gen_nil:add4_0(+(n6_0, 1)), gen_nil:add4_0(b)) ->_R^Omega(1)
add(hole_head3_0, app(gen_nil:add4_0(n6_0), gen_nil:add4_0(b))) ->_IH
add(hole_head3_0, gen_nil:add4_0(+(b, c7_0)))

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

(66)
Complex Obligation (BEST)

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

(67)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
null(nil) -> true
null(add(n, x)) -> false
tail(add(n, x)) -> x
tail(nil) -> nil
head(add(n, x)) -> n
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(x) -> shuff(x, nil)
shuff(x, y) -> if(null(x), x, y, app(y, add(head(x), nil)))
if(true, x, y, z) -> y
if(false, x, y, z) -> shuff(reverse(tail(x)), z)

Types:
null :: nil:add -> true:false
nil :: nil:add
true :: true:false
add :: head -> nil:add -> nil:add
false :: true:false
tail :: nil:add -> nil:add
head :: nil:add -> head
app :: nil:add -> nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
shuff :: nil:add -> nil:add -> nil:add
if :: true:false -> nil:add -> nil:add -> nil:add -> nil:add
hole_true:false1_0 :: true:false
hole_nil:add2_0 :: nil:add
hole_head3_0 :: head
gen_nil:add4_0 :: Nat -> nil:add


Generator Equations:
gen_nil:add4_0(0) <=> nil
gen_nil:add4_0(+(x, 1)) <=> add(hole_head3_0, gen_nil:add4_0(x))


The following defined symbols remain to be analysed:
app, reverse, shuff

They will be analysed ascendingly in the following order:
app < reverse
app < shuff
reverse < shuff

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

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

(69)
BOUNDS(n^1, INF)

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

(70)
Obligation:
Innermost TRS:
Rules:
null(nil) -> true
null(add(n, x)) -> false
tail(add(n, x)) -> x
tail(nil) -> nil
head(add(n, x)) -> n
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(x) -> shuff(x, nil)
shuff(x, y) -> if(null(x), x, y, app(y, add(head(x), nil)))
if(true, x, y, z) -> y
if(false, x, y, z) -> shuff(reverse(tail(x)), z)

Types:
null :: nil:add -> true:false
nil :: nil:add
true :: true:false
add :: head -> nil:add -> nil:add
false :: true:false
tail :: nil:add -> nil:add
head :: nil:add -> head
app :: nil:add -> nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
shuff :: nil:add -> nil:add -> nil:add
if :: true:false -> nil:add -> nil:add -> nil:add -> nil:add
hole_true:false1_0 :: true:false
hole_nil:add2_0 :: nil:add
hole_head3_0 :: head
gen_nil:add4_0 :: Nat -> nil:add


Lemmas:
app(gen_nil:add4_0(n6_0), gen_nil:add4_0(b)) -> gen_nil:add4_0(+(n6_0, b)), rt in Omega(1 + n6_0)


Generator Equations:
gen_nil:add4_0(0) <=> nil
gen_nil:add4_0(+(x, 1)) <=> add(hole_head3_0, gen_nil:add4_0(x))


The following defined symbols remain to be analysed:
reverse, shuff

They will be analysed ascendingly in the following order:
reverse < shuff

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

(71) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
reverse(gen_nil:add4_0(n665_0)) -> gen_nil:add4_0(n665_0), rt in Omega(1 + n665_0 + n665_0^2)

Induction Base:
reverse(gen_nil:add4_0(0)) ->_R^Omega(1)
nil

Induction Step:
reverse(gen_nil:add4_0(+(n665_0, 1))) ->_R^Omega(1)
app(reverse(gen_nil:add4_0(n665_0)), add(hole_head3_0, nil)) ->_IH
app(gen_nil:add4_0(c666_0), add(hole_head3_0, nil)) ->_L^Omega(1 + n665_0)
gen_nil:add4_0(+(n665_0, +(0, 1)))

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

(72)
Complex Obligation (BEST)

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

(73)
Obligation:
Proved the lower bound n^2 for the following obligation:

Innermost TRS:
Rules:
null(nil) -> true
null(add(n, x)) -> false
tail(add(n, x)) -> x
tail(nil) -> nil
head(add(n, x)) -> n
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(x) -> shuff(x, nil)
shuff(x, y) -> if(null(x), x, y, app(y, add(head(x), nil)))
if(true, x, y, z) -> y
if(false, x, y, z) -> shuff(reverse(tail(x)), z)

Types:
null :: nil:add -> true:false
nil :: nil:add
true :: true:false
add :: head -> nil:add -> nil:add
false :: true:false
tail :: nil:add -> nil:add
head :: nil:add -> head
app :: nil:add -> nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
shuff :: nil:add -> nil:add -> nil:add
if :: true:false -> nil:add -> nil:add -> nil:add -> nil:add
hole_true:false1_0 :: true:false
hole_nil:add2_0 :: nil:add
hole_head3_0 :: head
gen_nil:add4_0 :: Nat -> nil:add


Lemmas:
app(gen_nil:add4_0(n6_0), gen_nil:add4_0(b)) -> gen_nil:add4_0(+(n6_0, b)), rt in Omega(1 + n6_0)


Generator Equations:
gen_nil:add4_0(0) <=> nil
gen_nil:add4_0(+(x, 1)) <=> add(hole_head3_0, gen_nil:add4_0(x))


The following defined symbols remain to be analysed:
reverse, shuff

They will be analysed ascendingly in the following order:
reverse < shuff

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

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

(75)
BOUNDS(n^2, INF)

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

(76)
Obligation:
Innermost TRS:
Rules:
null(nil) -> true
null(add(n, x)) -> false
tail(add(n, x)) -> x
tail(nil) -> nil
head(add(n, x)) -> n
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(x) -> shuff(x, nil)
shuff(x, y) -> if(null(x), x, y, app(y, add(head(x), nil)))
if(true, x, y, z) -> y
if(false, x, y, z) -> shuff(reverse(tail(x)), z)

Types:
null :: nil:add -> true:false
nil :: nil:add
true :: true:false
add :: head -> nil:add -> nil:add
false :: true:false
tail :: nil:add -> nil:add
head :: nil:add -> head
app :: nil:add -> nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
shuff :: nil:add -> nil:add -> nil:add
if :: true:false -> nil:add -> nil:add -> nil:add -> nil:add
hole_true:false1_0 :: true:false
hole_nil:add2_0 :: nil:add
hole_head3_0 :: head
gen_nil:add4_0 :: Nat -> nil:add


Lemmas:
app(gen_nil:add4_0(n6_0), gen_nil:add4_0(b)) -> gen_nil:add4_0(+(n6_0, b)), rt in Omega(1 + n6_0)
reverse(gen_nil:add4_0(n665_0)) -> gen_nil:add4_0(n665_0), rt in Omega(1 + n665_0 + n665_0^2)


Generator Equations:
gen_nil:add4_0(0) <=> nil
gen_nil:add4_0(+(x, 1)) <=> add(hole_head3_0, gen_nil:add4_0(x))


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

(77) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
shuff(gen_nil:add4_0(n952_0), gen_nil:add4_0(b)) -> *5_0, rt in Omega(b*n952_0 + n952_0 + n952_0^2 + n952_0^3)

Induction Base:
shuff(gen_nil:add4_0(0), gen_nil:add4_0(b))

Induction Step:
shuff(gen_nil:add4_0(+(n952_0, 1)), gen_nil:add4_0(b)) ->_R^Omega(1)
if(null(gen_nil:add4_0(+(n952_0, 1))), gen_nil:add4_0(+(n952_0, 1)), gen_nil:add4_0(b), app(gen_nil:add4_0(b), add(head(gen_nil:add4_0(+(n952_0, 1))), nil))) ->_R^Omega(1)
if(false, gen_nil:add4_0(+(1, n952_0)), gen_nil:add4_0(b), app(gen_nil:add4_0(b), add(head(gen_nil:add4_0(+(1, n952_0))), nil))) ->_R^Omega(1)
if(false, gen_nil:add4_0(+(1, n952_0)), gen_nil:add4_0(b), app(gen_nil:add4_0(b), add(hole_head3_0, nil))) ->_L^Omega(1 + b)
if(false, gen_nil:add4_0(+(1, n952_0)), gen_nil:add4_0(+(0, 1)), gen_nil:add4_0(+(b, +(0, 1)))) ->_R^Omega(1)
shuff(reverse(tail(gen_nil:add4_0(+(1, n952_0)))), gen_nil:add4_0(+(1, b))) ->_R^Omega(1)
shuff(reverse(gen_nil:add4_0(n952_0)), gen_nil:add4_0(+(1, b))) ->_L^Omega(1 + n952_0 + n952_0^2)
shuff(gen_nil:add4_0(n952_0), gen_nil:add4_0(+(1, b))) ->_IH
*5_0

We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3).
----------------------------------------

(78)
Obligation:
Proved the lower bound n^3 for the following obligation:

Innermost TRS:
Rules:
null(nil) -> true
null(add(n, x)) -> false
tail(add(n, x)) -> x
tail(nil) -> nil
head(add(n, x)) -> n
app(nil, y) -> y
app(add(n, x), y) -> add(n, app(x, y))
reverse(nil) -> nil
reverse(add(n, x)) -> app(reverse(x), add(n, nil))
shuffle(x) -> shuff(x, nil)
shuff(x, y) -> if(null(x), x, y, app(y, add(head(x), nil)))
if(true, x, y, z) -> y
if(false, x, y, z) -> shuff(reverse(tail(x)), z)

Types:
null :: nil:add -> true:false
nil :: nil:add
true :: true:false
add :: head -> nil:add -> nil:add
false :: true:false
tail :: nil:add -> nil:add
head :: nil:add -> head
app :: nil:add -> nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
shuff :: nil:add -> nil:add -> nil:add
if :: true:false -> nil:add -> nil:add -> nil:add -> nil:add
hole_true:false1_0 :: true:false
hole_nil:add2_0 :: nil:add
hole_head3_0 :: head
gen_nil:add4_0 :: Nat -> nil:add


Lemmas:
app(gen_nil:add4_0(n6_0), gen_nil:add4_0(b)) -> gen_nil:add4_0(+(n6_0, b)), rt in Omega(1 + n6_0)
reverse(gen_nil:add4_0(n665_0)) -> gen_nil:add4_0(n665_0), rt in Omega(1 + n665_0 + n665_0^2)


Generator Equations:
gen_nil:add4_0(0) <=> nil
gen_nil:add4_0(+(x, 1)) <=> add(hole_head3_0, gen_nil:add4_0(x))


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

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

(80)
BOUNDS(n^3, INF)