WORST_CASE(Omega(n^3),O(n^3))
proof of input_bDblas6hKs.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(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), 2 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 187 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 14 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 244 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 116 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 171 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 4 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 102 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 42 ms]
    (38) CpxRNTS
    (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 373 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 107 ms]
    (44) CpxRNTS
    (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 1240 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 181 ms]
    (50) CpxRNTS
    (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (52) CpxRNTS
    (53) IntTrsBoundProof [UPPER BOUND(ID), 457 ms]
    (54) CpxRNTS
    (55) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
    (56) CpxRNTS
    (57) FinalProof [FINISHED, 0 ms]
    (58) BOUNDS(1, n^3)
(59) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(60) CdtProblem
    (61) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CpxRelTRS
    (63) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CpxRelTRS
    (65) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) typed CpxTrs
    (67) OrderProof [LOWER BOUND(ID), 11 ms]
    (68) typed CpxTrs
    (69) RewriteLemmaProof [LOWER BOUND(ID), 332 ms]
    (70) BEST
        (71) proven lower bound
            (72) LowerBoundPropagationProof [FINISHED, 0 ms]
            (73) BOUNDS(n^1, INF)
        (74) typed CpxTrs
            (75) RewriteLemmaProof [LOWER BOUND(ID), 99 ms]
            (76) typed CpxTrs
            (77) RewriteLemmaProof [LOWER BOUND(ID), 13 ms]
            (78) typed CpxTrs
            (79) RewriteLemmaProof [LOWER BOUND(ID), 790 ms]
            (80) BEST
                (81) proven lower bound
                    (82) LowerBoundPropagationProof [FINISHED, 0 ms]
                    (83) BOUNDS(n^2, INF)
                (84) typed CpxTrs
                    (85) RewriteLemmaProof [LOWER BOUND(ID), 665 ms]
                    (86) BEST
                        (87) proven lower bound
                            (88) LowerBoundPropagationProof [FINISHED, 0 ms]
                            (89) BOUNDS(n^3, INF)
                        (90) typed CpxTrs
                            (91) RewriteLemmaProof [LOWER BOUND(ID), 1396 ms]
                            (92) BOUNDS(1, INF)


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

(0)
Obligation:
The Runtime Complexity (parallel-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: PARALLEL_INNERMOST
----------------------------------------

(1) RelTrsToWeightedTrsProof (UPPER BOUND(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) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   null(nil) -> true
   null(add(z0, z1)) -> false
   tail(add(z0, z1)) -> z1
   tail(nil) -> nil
   head(add(z0, z1)) -> z0
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   reverse(nil) -> nil
   reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
   shuffle(z0) -> shuff(z0, nil)
   shuff(z0, z1) -> if(null(z0), z0, z1, app(z1, add(head(z0), nil)))
   if(true, z0, z1, z2) -> z1
   if(false, z0, z1, z2) -> shuff(reverse(tail(z0)), z2)
Tuples:
   NULL(nil) -> c
   NULL(add(z0, z1)) -> c1
   TAIL(add(z0, z1)) -> c2
   TAIL(nil) -> c3
   HEAD(add(z0, z1)) -> c4
   APP(nil, z0) -> c5
   APP(add(z0, z1), z2) -> c6(APP(z1, z2))
   REVERSE(nil) -> c7
   REVERSE(add(z0, z1)) -> c8(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
   SHUFFLE(z0) -> c9(SHUFF(z0, nil))
   SHUFF(z0, z1) -> c10(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), NULL(z0))
   SHUFF(z0, z1) -> c11(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
   IF(true, z0, z1, z2) -> c12
   IF(false, z0, z1, z2) -> c13(SHUFF(reverse(tail(z0)), z2), REVERSE(tail(z0)), TAIL(z0))
S tuples:
   NULL(nil) -> c
   NULL(add(z0, z1)) -> c1
   TAIL(add(z0, z1)) -> c2
   TAIL(nil) -> c3
   HEAD(add(z0, z1)) -> c4
   APP(nil, z0) -> c5
   APP(add(z0, z1), z2) -> c6(APP(z1, z2))
   REVERSE(nil) -> c7
   REVERSE(add(z0, z1)) -> c8(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
   SHUFFLE(z0) -> c9(SHUFF(z0, nil))
   SHUFF(z0, z1) -> c10(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), NULL(z0))
   SHUFF(z0, z1) -> c11(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
   IF(true, z0, z1, z2) -> c12
   IF(false, z0, z1, z2) -> c13(SHUFF(reverse(tail(z0)), z2), REVERSE(tail(z0)), TAIL(z0))
K tuples:none
Defined Rule Symbols:   null_1, tail_1, head_1, app_2, reverse_1, shuffle_1, shuff_2, if_4

Defined Pair Symbols:   NULL_1, TAIL_1, HEAD_1, APP_2, REVERSE_1, SHUFFLE_1, SHUFF_2, IF_4

Compound Symbols:   c, c1, c2, c3, c4, c5, c6_1, c7, c8_2, c9_1, c10_2, c11_3, c12, c13_3


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

(61) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(62)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   NULL(nil) -> c
   NULL(add(z0, z1)) -> c1
   TAIL(add(z0, z1)) -> c2
   TAIL(nil) -> c3
   HEAD(add(z0, z1)) -> c4
   APP(nil, z0) -> c5
   APP(add(z0, z1), z2) -> c6(APP(z1, z2))
   REVERSE(nil) -> c7
   REVERSE(add(z0, z1)) -> c8(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
   SHUFFLE(z0) -> c9(SHUFF(z0, nil))
   SHUFF(z0, z1) -> c10(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), NULL(z0))
   SHUFF(z0, z1) -> c11(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
   IF(true, z0, z1, z2) -> c12
   IF(false, z0, z1, z2) -> c13(SHUFF(reverse(tail(z0)), z2), REVERSE(tail(z0)), TAIL(z0))

The (relative) TRS S consists of the following rules:

   null(nil) -> true
   null(add(z0, z1)) -> false
   tail(add(z0, z1)) -> z1
   tail(nil) -> nil
   head(add(z0, z1)) -> z0
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   reverse(nil) -> nil
   reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
   shuffle(z0) -> shuff(z0, nil)
   shuff(z0, z1) -> if(null(z0), z0, z1, app(z1, add(head(z0), nil)))
   if(true, z0, z1, z2) -> z1
   if(false, z0, z1, z2) -> shuff(reverse(tail(z0)), z2)

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

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

(64)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   NULL(nil) -> c
   NULL(add(z0, z1)) -> c1
   TAIL(add(z0, z1)) -> c2
   TAIL(nil) -> c3
   HEAD(add(z0, z1)) -> c4
   APP(nil, z0) -> c5
   APP(add(z0, z1), z2) -> c6(APP(z1, z2))
   REVERSE(nil) -> c7
   REVERSE(add(z0, z1)) -> c8(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
   SHUFFLE(z0) -> c9(SHUFF(z0, nil))
   SHUFF(z0, z1) -> c10(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), NULL(z0))
   SHUFF(z0, z1) -> c11(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
   IF(true, z0, z1, z2) -> c12
   IF(false, z0, z1, z2) -> c13(SHUFF(reverse(tail(z0)), z2), REVERSE(tail(z0)), TAIL(z0))

The (relative) TRS S consists of the following rules:

   null(nil) -> true
   null(add(z0, z1)) -> false
   tail(add(z0, z1)) -> z1
   tail(nil) -> nil
   head(add(z0, z1)) -> z0
   app(nil, z0) -> z0
   app(add(z0, z1), z2) -> add(z0, app(z1, z2))
   reverse(nil) -> nil
   reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
   shuffle(z0) -> shuff(z0, nil)
   shuff(z0, z1) -> if(null(z0), z0, z1, app(z1, add(head(z0), nil)))
   if(true, z0, z1, z2) -> z1
   if(false, z0, z1, z2) -> shuff(reverse(tail(z0)), z2)

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

(65) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(66)
Obligation:
Innermost TRS:
Rules:
NULL(nil) -> c
NULL(add(z0, z1)) -> c1
TAIL(add(z0, z1)) -> c2
TAIL(nil) -> c3
HEAD(add(z0, z1)) -> c4
APP(nil, z0) -> c5
APP(add(z0, z1), z2) -> c6(APP(z1, z2))
REVERSE(nil) -> c7
REVERSE(add(z0, z1)) -> c8(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(z0) -> c9(SHUFF(z0, nil))
SHUFF(z0, z1) -> c10(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), NULL(z0))
SHUFF(z0, z1) -> c11(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
IF(true, z0, z1, z2) -> c12
IF(false, z0, z1, z2) -> c13(SHUFF(reverse(tail(z0)), z2), REVERSE(tail(z0)), TAIL(z0))
null(nil) -> true
null(add(z0, z1)) -> false
tail(add(z0, z1)) -> z1
tail(nil) -> nil
head(add(z0, z1)) -> z0
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(z0) -> shuff(z0, nil)
shuff(z0, z1) -> if(null(z0), z0, z1, app(z1, add(head(z0), nil)))
if(true, z0, z1, z2) -> z1
if(false, z0, z1, z2) -> shuff(reverse(tail(z0)), z2)

Types:
NULL :: nil:add -> c:c1
nil :: nil:add
c :: c:c1
add :: head -> nil:add -> nil:add
c1 :: c:c1
TAIL :: nil:add -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
HEAD :: nil:add -> c4
c4 :: c4
APP :: nil:add -> nil:add -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
REVERSE :: nil:add -> c7:c8
c7 :: c7:c8
c8 :: c5:c6 -> c7:c8 -> c7:c8
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c9
c9 :: c10:c11 -> c9
SHUFF :: nil:add -> nil:add -> c10:c11
c10 :: c12:c13 -> c:c1 -> c10:c11
IF :: true:false -> nil:add -> nil:add -> nil:add -> c12:c13
null :: nil:add -> true:false
app :: nil:add -> nil:add -> nil:add
head :: nil:add -> head
c11 :: c12:c13 -> c5:c6 -> c4 -> c10:c11
true :: true:false
c12 :: c12:c13
false :: true:false
c13 :: c10:c11 -> c7:c8 -> c2:c3 -> c12:c13
tail :: 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_c:c11_14 :: c:c1
hole_nil:add2_14 :: nil:add
hole_head3_14 :: head
hole_c2:c34_14 :: c2:c3
hole_c45_14 :: c4
hole_c5:c66_14 :: c5:c6
hole_c7:c87_14 :: c7:c8
hole_c98_14 :: c9
hole_c10:c119_14 :: c10:c11
hole_c12:c1310_14 :: c12:c13
hole_true:false11_14 :: true:false
gen_nil:add12_14 :: Nat -> nil:add
gen_c5:c613_14 :: Nat -> c5:c6
gen_c7:c814_14 :: Nat -> c7:c8

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

(67) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
APP, REVERSE, reverse, SHUFF, app, shuff

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

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

(68)
Obligation:
Innermost TRS:
Rules:
NULL(nil) -> c
NULL(add(z0, z1)) -> c1
TAIL(add(z0, z1)) -> c2
TAIL(nil) -> c3
HEAD(add(z0, z1)) -> c4
APP(nil, z0) -> c5
APP(add(z0, z1), z2) -> c6(APP(z1, z2))
REVERSE(nil) -> c7
REVERSE(add(z0, z1)) -> c8(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(z0) -> c9(SHUFF(z0, nil))
SHUFF(z0, z1) -> c10(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), NULL(z0))
SHUFF(z0, z1) -> c11(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
IF(true, z0, z1, z2) -> c12
IF(false, z0, z1, z2) -> c13(SHUFF(reverse(tail(z0)), z2), REVERSE(tail(z0)), TAIL(z0))
null(nil) -> true
null(add(z0, z1)) -> false
tail(add(z0, z1)) -> z1
tail(nil) -> nil
head(add(z0, z1)) -> z0
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(z0) -> shuff(z0, nil)
shuff(z0, z1) -> if(null(z0), z0, z1, app(z1, add(head(z0), nil)))
if(true, z0, z1, z2) -> z1
if(false, z0, z1, z2) -> shuff(reverse(tail(z0)), z2)

Types:
NULL :: nil:add -> c:c1
nil :: nil:add
c :: c:c1
add :: head -> nil:add -> nil:add
c1 :: c:c1
TAIL :: nil:add -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
HEAD :: nil:add -> c4
c4 :: c4
APP :: nil:add -> nil:add -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
REVERSE :: nil:add -> c7:c8
c7 :: c7:c8
c8 :: c5:c6 -> c7:c8 -> c7:c8
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c9
c9 :: c10:c11 -> c9
SHUFF :: nil:add -> nil:add -> c10:c11
c10 :: c12:c13 -> c:c1 -> c10:c11
IF :: true:false -> nil:add -> nil:add -> nil:add -> c12:c13
null :: nil:add -> true:false
app :: nil:add -> nil:add -> nil:add
head :: nil:add -> head
c11 :: c12:c13 -> c5:c6 -> c4 -> c10:c11
true :: true:false
c12 :: c12:c13
false :: true:false
c13 :: c10:c11 -> c7:c8 -> c2:c3 -> c12:c13
tail :: 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_c:c11_14 :: c:c1
hole_nil:add2_14 :: nil:add
hole_head3_14 :: head
hole_c2:c34_14 :: c2:c3
hole_c45_14 :: c4
hole_c5:c66_14 :: c5:c6
hole_c7:c87_14 :: c7:c8
hole_c98_14 :: c9
hole_c10:c119_14 :: c10:c11
hole_c12:c1310_14 :: c12:c13
hole_true:false11_14 :: true:false
gen_nil:add12_14 :: Nat -> nil:add
gen_c5:c613_14 :: Nat -> c5:c6
gen_c7:c814_14 :: Nat -> c7:c8


Generator Equations:
gen_nil:add12_14(0) <=> nil
gen_nil:add12_14(+(x, 1)) <=> add(hole_head3_14, gen_nil:add12_14(x))
gen_c5:c613_14(0) <=> c5
gen_c5:c613_14(+(x, 1)) <=> c6(gen_c5:c613_14(x))
gen_c7:c814_14(0) <=> c7
gen_c7:c814_14(+(x, 1)) <=> c8(c5, gen_c7:c814_14(x))


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

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

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

(69) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
APP(gen_nil:add12_14(n16_14), gen_nil:add12_14(b)) -> gen_c5:c613_14(n16_14), rt in Omega(1 + n16_14)

Induction Base:
APP(gen_nil:add12_14(0), gen_nil:add12_14(b)) ->_R^Omega(1)
c5

Induction Step:
APP(gen_nil:add12_14(+(n16_14, 1)), gen_nil:add12_14(b)) ->_R^Omega(1)
c6(APP(gen_nil:add12_14(n16_14), gen_nil:add12_14(b))) ->_IH
c6(gen_c5:c613_14(c17_14))

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

(70)
Complex Obligation (BEST)

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

(71)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
NULL(nil) -> c
NULL(add(z0, z1)) -> c1
TAIL(add(z0, z1)) -> c2
TAIL(nil) -> c3
HEAD(add(z0, z1)) -> c4
APP(nil, z0) -> c5
APP(add(z0, z1), z2) -> c6(APP(z1, z2))
REVERSE(nil) -> c7
REVERSE(add(z0, z1)) -> c8(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(z0) -> c9(SHUFF(z0, nil))
SHUFF(z0, z1) -> c10(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), NULL(z0))
SHUFF(z0, z1) -> c11(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
IF(true, z0, z1, z2) -> c12
IF(false, z0, z1, z2) -> c13(SHUFF(reverse(tail(z0)), z2), REVERSE(tail(z0)), TAIL(z0))
null(nil) -> true
null(add(z0, z1)) -> false
tail(add(z0, z1)) -> z1
tail(nil) -> nil
head(add(z0, z1)) -> z0
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(z0) -> shuff(z0, nil)
shuff(z0, z1) -> if(null(z0), z0, z1, app(z1, add(head(z0), nil)))
if(true, z0, z1, z2) -> z1
if(false, z0, z1, z2) -> shuff(reverse(tail(z0)), z2)

Types:
NULL :: nil:add -> c:c1
nil :: nil:add
c :: c:c1
add :: head -> nil:add -> nil:add
c1 :: c:c1
TAIL :: nil:add -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
HEAD :: nil:add -> c4
c4 :: c4
APP :: nil:add -> nil:add -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
REVERSE :: nil:add -> c7:c8
c7 :: c7:c8
c8 :: c5:c6 -> c7:c8 -> c7:c8
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c9
c9 :: c10:c11 -> c9
SHUFF :: nil:add -> nil:add -> c10:c11
c10 :: c12:c13 -> c:c1 -> c10:c11
IF :: true:false -> nil:add -> nil:add -> nil:add -> c12:c13
null :: nil:add -> true:false
app :: nil:add -> nil:add -> nil:add
head :: nil:add -> head
c11 :: c12:c13 -> c5:c6 -> c4 -> c10:c11
true :: true:false
c12 :: c12:c13
false :: true:false
c13 :: c10:c11 -> c7:c8 -> c2:c3 -> c12:c13
tail :: 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_c:c11_14 :: c:c1
hole_nil:add2_14 :: nil:add
hole_head3_14 :: head
hole_c2:c34_14 :: c2:c3
hole_c45_14 :: c4
hole_c5:c66_14 :: c5:c6
hole_c7:c87_14 :: c7:c8
hole_c98_14 :: c9
hole_c10:c119_14 :: c10:c11
hole_c12:c1310_14 :: c12:c13
hole_true:false11_14 :: true:false
gen_nil:add12_14 :: Nat -> nil:add
gen_c5:c613_14 :: Nat -> c5:c6
gen_c7:c814_14 :: Nat -> c7:c8


Generator Equations:
gen_nil:add12_14(0) <=> nil
gen_nil:add12_14(+(x, 1)) <=> add(hole_head3_14, gen_nil:add12_14(x))
gen_c5:c613_14(0) <=> c5
gen_c5:c613_14(+(x, 1)) <=> c6(gen_c5:c613_14(x))
gen_c7:c814_14(0) <=> c7
gen_c7:c814_14(+(x, 1)) <=> c8(c5, gen_c7:c814_14(x))


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

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

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

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

(73)
BOUNDS(n^1, INF)

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

(74)
Obligation:
Innermost TRS:
Rules:
NULL(nil) -> c
NULL(add(z0, z1)) -> c1
TAIL(add(z0, z1)) -> c2
TAIL(nil) -> c3
HEAD(add(z0, z1)) -> c4
APP(nil, z0) -> c5
APP(add(z0, z1), z2) -> c6(APP(z1, z2))
REVERSE(nil) -> c7
REVERSE(add(z0, z1)) -> c8(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(z0) -> c9(SHUFF(z0, nil))
SHUFF(z0, z1) -> c10(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), NULL(z0))
SHUFF(z0, z1) -> c11(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
IF(true, z0, z1, z2) -> c12
IF(false, z0, z1, z2) -> c13(SHUFF(reverse(tail(z0)), z2), REVERSE(tail(z0)), TAIL(z0))
null(nil) -> true
null(add(z0, z1)) -> false
tail(add(z0, z1)) -> z1
tail(nil) -> nil
head(add(z0, z1)) -> z0
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(z0) -> shuff(z0, nil)
shuff(z0, z1) -> if(null(z0), z0, z1, app(z1, add(head(z0), nil)))
if(true, z0, z1, z2) -> z1
if(false, z0, z1, z2) -> shuff(reverse(tail(z0)), z2)

Types:
NULL :: nil:add -> c:c1
nil :: nil:add
c :: c:c1
add :: head -> nil:add -> nil:add
c1 :: c:c1
TAIL :: nil:add -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
HEAD :: nil:add -> c4
c4 :: c4
APP :: nil:add -> nil:add -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
REVERSE :: nil:add -> c7:c8
c7 :: c7:c8
c8 :: c5:c6 -> c7:c8 -> c7:c8
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c9
c9 :: c10:c11 -> c9
SHUFF :: nil:add -> nil:add -> c10:c11
c10 :: c12:c13 -> c:c1 -> c10:c11
IF :: true:false -> nil:add -> nil:add -> nil:add -> c12:c13
null :: nil:add -> true:false
app :: nil:add -> nil:add -> nil:add
head :: nil:add -> head
c11 :: c12:c13 -> c5:c6 -> c4 -> c10:c11
true :: true:false
c12 :: c12:c13
false :: true:false
c13 :: c10:c11 -> c7:c8 -> c2:c3 -> c12:c13
tail :: 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_c:c11_14 :: c:c1
hole_nil:add2_14 :: nil:add
hole_head3_14 :: head
hole_c2:c34_14 :: c2:c3
hole_c45_14 :: c4
hole_c5:c66_14 :: c5:c6
hole_c7:c87_14 :: c7:c8
hole_c98_14 :: c9
hole_c10:c119_14 :: c10:c11
hole_c12:c1310_14 :: c12:c13
hole_true:false11_14 :: true:false
gen_nil:add12_14 :: Nat -> nil:add
gen_c5:c613_14 :: Nat -> c5:c6
gen_c7:c814_14 :: Nat -> c7:c8


Lemmas:
APP(gen_nil:add12_14(n16_14), gen_nil:add12_14(b)) -> gen_c5:c613_14(n16_14), rt in Omega(1 + n16_14)


Generator Equations:
gen_nil:add12_14(0) <=> nil
gen_nil:add12_14(+(x, 1)) <=> add(hole_head3_14, gen_nil:add12_14(x))
gen_c5:c613_14(0) <=> c5
gen_c5:c613_14(+(x, 1)) <=> c6(gen_c5:c613_14(x))
gen_c7:c814_14(0) <=> c7
gen_c7:c814_14(+(x, 1)) <=> c8(c5, gen_c7:c814_14(x))


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

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

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

(75) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
app(gen_nil:add12_14(n700_14), gen_nil:add12_14(b)) -> gen_nil:add12_14(+(n700_14, b)), rt in Omega(0)

Induction Base:
app(gen_nil:add12_14(0), gen_nil:add12_14(b)) ->_R^Omega(0)
gen_nil:add12_14(b)

Induction Step:
app(gen_nil:add12_14(+(n700_14, 1)), gen_nil:add12_14(b)) ->_R^Omega(0)
add(hole_head3_14, app(gen_nil:add12_14(n700_14), gen_nil:add12_14(b))) ->_IH
add(hole_head3_14, gen_nil:add12_14(+(b, c701_14)))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(76)
Obligation:
Innermost TRS:
Rules:
NULL(nil) -> c
NULL(add(z0, z1)) -> c1
TAIL(add(z0, z1)) -> c2
TAIL(nil) -> c3
HEAD(add(z0, z1)) -> c4
APP(nil, z0) -> c5
APP(add(z0, z1), z2) -> c6(APP(z1, z2))
REVERSE(nil) -> c7
REVERSE(add(z0, z1)) -> c8(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(z0) -> c9(SHUFF(z0, nil))
SHUFF(z0, z1) -> c10(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), NULL(z0))
SHUFF(z0, z1) -> c11(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
IF(true, z0, z1, z2) -> c12
IF(false, z0, z1, z2) -> c13(SHUFF(reverse(tail(z0)), z2), REVERSE(tail(z0)), TAIL(z0))
null(nil) -> true
null(add(z0, z1)) -> false
tail(add(z0, z1)) -> z1
tail(nil) -> nil
head(add(z0, z1)) -> z0
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(z0) -> shuff(z0, nil)
shuff(z0, z1) -> if(null(z0), z0, z1, app(z1, add(head(z0), nil)))
if(true, z0, z1, z2) -> z1
if(false, z0, z1, z2) -> shuff(reverse(tail(z0)), z2)

Types:
NULL :: nil:add -> c:c1
nil :: nil:add
c :: c:c1
add :: head -> nil:add -> nil:add
c1 :: c:c1
TAIL :: nil:add -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
HEAD :: nil:add -> c4
c4 :: c4
APP :: nil:add -> nil:add -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
REVERSE :: nil:add -> c7:c8
c7 :: c7:c8
c8 :: c5:c6 -> c7:c8 -> c7:c8
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c9
c9 :: c10:c11 -> c9
SHUFF :: nil:add -> nil:add -> c10:c11
c10 :: c12:c13 -> c:c1 -> c10:c11
IF :: true:false -> nil:add -> nil:add -> nil:add -> c12:c13
null :: nil:add -> true:false
app :: nil:add -> nil:add -> nil:add
head :: nil:add -> head
c11 :: c12:c13 -> c5:c6 -> c4 -> c10:c11
true :: true:false
c12 :: c12:c13
false :: true:false
c13 :: c10:c11 -> c7:c8 -> c2:c3 -> c12:c13
tail :: 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_c:c11_14 :: c:c1
hole_nil:add2_14 :: nil:add
hole_head3_14 :: head
hole_c2:c34_14 :: c2:c3
hole_c45_14 :: c4
hole_c5:c66_14 :: c5:c6
hole_c7:c87_14 :: c7:c8
hole_c98_14 :: c9
hole_c10:c119_14 :: c10:c11
hole_c12:c1310_14 :: c12:c13
hole_true:false11_14 :: true:false
gen_nil:add12_14 :: Nat -> nil:add
gen_c5:c613_14 :: Nat -> c5:c6
gen_c7:c814_14 :: Nat -> c7:c8


Lemmas:
APP(gen_nil:add12_14(n16_14), gen_nil:add12_14(b)) -> gen_c5:c613_14(n16_14), rt in Omega(1 + n16_14)
app(gen_nil:add12_14(n700_14), gen_nil:add12_14(b)) -> gen_nil:add12_14(+(n700_14, b)), rt in Omega(0)


Generator Equations:
gen_nil:add12_14(0) <=> nil
gen_nil:add12_14(+(x, 1)) <=> add(hole_head3_14, gen_nil:add12_14(x))
gen_c5:c613_14(0) <=> c5
gen_c5:c613_14(+(x, 1)) <=> c6(gen_c5:c613_14(x))
gen_c7:c814_14(0) <=> c7
gen_c7:c814_14(+(x, 1)) <=> c8(c5, gen_c7:c814_14(x))


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

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

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

(77) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
reverse(gen_nil:add12_14(n1833_14)) -> gen_nil:add12_14(n1833_14), rt in Omega(0)

Induction Base:
reverse(gen_nil:add12_14(0)) ->_R^Omega(0)
nil

Induction Step:
reverse(gen_nil:add12_14(+(n1833_14, 1))) ->_R^Omega(0)
app(reverse(gen_nil:add12_14(n1833_14)), add(hole_head3_14, nil)) ->_IH
app(gen_nil:add12_14(c1834_14), add(hole_head3_14, nil)) ->_L^Omega(0)
gen_nil:add12_14(+(n1833_14, +(0, 1)))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(78)
Obligation:
Innermost TRS:
Rules:
NULL(nil) -> c
NULL(add(z0, z1)) -> c1
TAIL(add(z0, z1)) -> c2
TAIL(nil) -> c3
HEAD(add(z0, z1)) -> c4
APP(nil, z0) -> c5
APP(add(z0, z1), z2) -> c6(APP(z1, z2))
REVERSE(nil) -> c7
REVERSE(add(z0, z1)) -> c8(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(z0) -> c9(SHUFF(z0, nil))
SHUFF(z0, z1) -> c10(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), NULL(z0))
SHUFF(z0, z1) -> c11(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
IF(true, z0, z1, z2) -> c12
IF(false, z0, z1, z2) -> c13(SHUFF(reverse(tail(z0)), z2), REVERSE(tail(z0)), TAIL(z0))
null(nil) -> true
null(add(z0, z1)) -> false
tail(add(z0, z1)) -> z1
tail(nil) -> nil
head(add(z0, z1)) -> z0
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(z0) -> shuff(z0, nil)
shuff(z0, z1) -> if(null(z0), z0, z1, app(z1, add(head(z0), nil)))
if(true, z0, z1, z2) -> z1
if(false, z0, z1, z2) -> shuff(reverse(tail(z0)), z2)

Types:
NULL :: nil:add -> c:c1
nil :: nil:add
c :: c:c1
add :: head -> nil:add -> nil:add
c1 :: c:c1
TAIL :: nil:add -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
HEAD :: nil:add -> c4
c4 :: c4
APP :: nil:add -> nil:add -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
REVERSE :: nil:add -> c7:c8
c7 :: c7:c8
c8 :: c5:c6 -> c7:c8 -> c7:c8
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c9
c9 :: c10:c11 -> c9
SHUFF :: nil:add -> nil:add -> c10:c11
c10 :: c12:c13 -> c:c1 -> c10:c11
IF :: true:false -> nil:add -> nil:add -> nil:add -> c12:c13
null :: nil:add -> true:false
app :: nil:add -> nil:add -> nil:add
head :: nil:add -> head
c11 :: c12:c13 -> c5:c6 -> c4 -> c10:c11
true :: true:false
c12 :: c12:c13
false :: true:false
c13 :: c10:c11 -> c7:c8 -> c2:c3 -> c12:c13
tail :: 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_c:c11_14 :: c:c1
hole_nil:add2_14 :: nil:add
hole_head3_14 :: head
hole_c2:c34_14 :: c2:c3
hole_c45_14 :: c4
hole_c5:c66_14 :: c5:c6
hole_c7:c87_14 :: c7:c8
hole_c98_14 :: c9
hole_c10:c119_14 :: c10:c11
hole_c12:c1310_14 :: c12:c13
hole_true:false11_14 :: true:false
gen_nil:add12_14 :: Nat -> nil:add
gen_c5:c613_14 :: Nat -> c5:c6
gen_c7:c814_14 :: Nat -> c7:c8


Lemmas:
APP(gen_nil:add12_14(n16_14), gen_nil:add12_14(b)) -> gen_c5:c613_14(n16_14), rt in Omega(1 + n16_14)
app(gen_nil:add12_14(n700_14), gen_nil:add12_14(b)) -> gen_nil:add12_14(+(n700_14, b)), rt in Omega(0)
reverse(gen_nil:add12_14(n1833_14)) -> gen_nil:add12_14(n1833_14), rt in Omega(0)


Generator Equations:
gen_nil:add12_14(0) <=> nil
gen_nil:add12_14(+(x, 1)) <=> add(hole_head3_14, gen_nil:add12_14(x))
gen_c5:c613_14(0) <=> c5
gen_c5:c613_14(+(x, 1)) <=> c6(gen_c5:c613_14(x))
gen_c7:c814_14(0) <=> c7
gen_c7:c814_14(+(x, 1)) <=> c8(c5, gen_c7:c814_14(x))


The following defined symbols remain to be analysed:
REVERSE, SHUFF, shuff

They will be analysed ascendingly in the following order:
REVERSE < SHUFF

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

(79) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
REVERSE(gen_nil:add12_14(n2211_14)) -> *15_14, rt in Omega(n2211_14 + n2211_14^2)

Induction Base:
REVERSE(gen_nil:add12_14(0))

Induction Step:
REVERSE(gen_nil:add12_14(+(n2211_14, 1))) ->_R^Omega(1)
c8(APP(reverse(gen_nil:add12_14(n2211_14)), add(hole_head3_14, nil)), REVERSE(gen_nil:add12_14(n2211_14))) ->_L^Omega(0)
c8(APP(gen_nil:add12_14(n2211_14), add(hole_head3_14, nil)), REVERSE(gen_nil:add12_14(n2211_14))) ->_L^Omega(1 + n2211_14)
c8(gen_c5:c613_14(n2211_14), REVERSE(gen_nil:add12_14(n2211_14))) ->_IH
c8(gen_c5:c613_14(n2211_14), *15_14)

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

(80)
Complex Obligation (BEST)

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

(81)
Obligation:
Proved the lower bound n^2 for the following obligation:

Innermost TRS:
Rules:
NULL(nil) -> c
NULL(add(z0, z1)) -> c1
TAIL(add(z0, z1)) -> c2
TAIL(nil) -> c3
HEAD(add(z0, z1)) -> c4
APP(nil, z0) -> c5
APP(add(z0, z1), z2) -> c6(APP(z1, z2))
REVERSE(nil) -> c7
REVERSE(add(z0, z1)) -> c8(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(z0) -> c9(SHUFF(z0, nil))
SHUFF(z0, z1) -> c10(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), NULL(z0))
SHUFF(z0, z1) -> c11(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
IF(true, z0, z1, z2) -> c12
IF(false, z0, z1, z2) -> c13(SHUFF(reverse(tail(z0)), z2), REVERSE(tail(z0)), TAIL(z0))
null(nil) -> true
null(add(z0, z1)) -> false
tail(add(z0, z1)) -> z1
tail(nil) -> nil
head(add(z0, z1)) -> z0
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(z0) -> shuff(z0, nil)
shuff(z0, z1) -> if(null(z0), z0, z1, app(z1, add(head(z0), nil)))
if(true, z0, z1, z2) -> z1
if(false, z0, z1, z2) -> shuff(reverse(tail(z0)), z2)

Types:
NULL :: nil:add -> c:c1
nil :: nil:add
c :: c:c1
add :: head -> nil:add -> nil:add
c1 :: c:c1
TAIL :: nil:add -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
HEAD :: nil:add -> c4
c4 :: c4
APP :: nil:add -> nil:add -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
REVERSE :: nil:add -> c7:c8
c7 :: c7:c8
c8 :: c5:c6 -> c7:c8 -> c7:c8
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c9
c9 :: c10:c11 -> c9
SHUFF :: nil:add -> nil:add -> c10:c11
c10 :: c12:c13 -> c:c1 -> c10:c11
IF :: true:false -> nil:add -> nil:add -> nil:add -> c12:c13
null :: nil:add -> true:false
app :: nil:add -> nil:add -> nil:add
head :: nil:add -> head
c11 :: c12:c13 -> c5:c6 -> c4 -> c10:c11
true :: true:false
c12 :: c12:c13
false :: true:false
c13 :: c10:c11 -> c7:c8 -> c2:c3 -> c12:c13
tail :: 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_c:c11_14 :: c:c1
hole_nil:add2_14 :: nil:add
hole_head3_14 :: head
hole_c2:c34_14 :: c2:c3
hole_c45_14 :: c4
hole_c5:c66_14 :: c5:c6
hole_c7:c87_14 :: c7:c8
hole_c98_14 :: c9
hole_c10:c119_14 :: c10:c11
hole_c12:c1310_14 :: c12:c13
hole_true:false11_14 :: true:false
gen_nil:add12_14 :: Nat -> nil:add
gen_c5:c613_14 :: Nat -> c5:c6
gen_c7:c814_14 :: Nat -> c7:c8


Lemmas:
APP(gen_nil:add12_14(n16_14), gen_nil:add12_14(b)) -> gen_c5:c613_14(n16_14), rt in Omega(1 + n16_14)
app(gen_nil:add12_14(n700_14), gen_nil:add12_14(b)) -> gen_nil:add12_14(+(n700_14, b)), rt in Omega(0)
reverse(gen_nil:add12_14(n1833_14)) -> gen_nil:add12_14(n1833_14), rt in Omega(0)


Generator Equations:
gen_nil:add12_14(0) <=> nil
gen_nil:add12_14(+(x, 1)) <=> add(hole_head3_14, gen_nil:add12_14(x))
gen_c5:c613_14(0) <=> c5
gen_c5:c613_14(+(x, 1)) <=> c6(gen_c5:c613_14(x))
gen_c7:c814_14(0) <=> c7
gen_c7:c814_14(+(x, 1)) <=> c8(c5, gen_c7:c814_14(x))


The following defined symbols remain to be analysed:
REVERSE, SHUFF, shuff

They will be analysed ascendingly in the following order:
REVERSE < SHUFF

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

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

(83)
BOUNDS(n^2, INF)

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

(84)
Obligation:
Innermost TRS:
Rules:
NULL(nil) -> c
NULL(add(z0, z1)) -> c1
TAIL(add(z0, z1)) -> c2
TAIL(nil) -> c3
HEAD(add(z0, z1)) -> c4
APP(nil, z0) -> c5
APP(add(z0, z1), z2) -> c6(APP(z1, z2))
REVERSE(nil) -> c7
REVERSE(add(z0, z1)) -> c8(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(z0) -> c9(SHUFF(z0, nil))
SHUFF(z0, z1) -> c10(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), NULL(z0))
SHUFF(z0, z1) -> c11(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
IF(true, z0, z1, z2) -> c12
IF(false, z0, z1, z2) -> c13(SHUFF(reverse(tail(z0)), z2), REVERSE(tail(z0)), TAIL(z0))
null(nil) -> true
null(add(z0, z1)) -> false
tail(add(z0, z1)) -> z1
tail(nil) -> nil
head(add(z0, z1)) -> z0
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(z0) -> shuff(z0, nil)
shuff(z0, z1) -> if(null(z0), z0, z1, app(z1, add(head(z0), nil)))
if(true, z0, z1, z2) -> z1
if(false, z0, z1, z2) -> shuff(reverse(tail(z0)), z2)

Types:
NULL :: nil:add -> c:c1
nil :: nil:add
c :: c:c1
add :: head -> nil:add -> nil:add
c1 :: c:c1
TAIL :: nil:add -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
HEAD :: nil:add -> c4
c4 :: c4
APP :: nil:add -> nil:add -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
REVERSE :: nil:add -> c7:c8
c7 :: c7:c8
c8 :: c5:c6 -> c7:c8 -> c7:c8
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c9
c9 :: c10:c11 -> c9
SHUFF :: nil:add -> nil:add -> c10:c11
c10 :: c12:c13 -> c:c1 -> c10:c11
IF :: true:false -> nil:add -> nil:add -> nil:add -> c12:c13
null :: nil:add -> true:false
app :: nil:add -> nil:add -> nil:add
head :: nil:add -> head
c11 :: c12:c13 -> c5:c6 -> c4 -> c10:c11
true :: true:false
c12 :: c12:c13
false :: true:false
c13 :: c10:c11 -> c7:c8 -> c2:c3 -> c12:c13
tail :: 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_c:c11_14 :: c:c1
hole_nil:add2_14 :: nil:add
hole_head3_14 :: head
hole_c2:c34_14 :: c2:c3
hole_c45_14 :: c4
hole_c5:c66_14 :: c5:c6
hole_c7:c87_14 :: c7:c8
hole_c98_14 :: c9
hole_c10:c119_14 :: c10:c11
hole_c12:c1310_14 :: c12:c13
hole_true:false11_14 :: true:false
gen_nil:add12_14 :: Nat -> nil:add
gen_c5:c613_14 :: Nat -> c5:c6
gen_c7:c814_14 :: Nat -> c7:c8


Lemmas:
APP(gen_nil:add12_14(n16_14), gen_nil:add12_14(b)) -> gen_c5:c613_14(n16_14), rt in Omega(1 + n16_14)
app(gen_nil:add12_14(n700_14), gen_nil:add12_14(b)) -> gen_nil:add12_14(+(n700_14, b)), rt in Omega(0)
reverse(gen_nil:add12_14(n1833_14)) -> gen_nil:add12_14(n1833_14), rt in Omega(0)
REVERSE(gen_nil:add12_14(n2211_14)) -> *15_14, rt in Omega(n2211_14 + n2211_14^2)


Generator Equations:
gen_nil:add12_14(0) <=> nil
gen_nil:add12_14(+(x, 1)) <=> add(hole_head3_14, gen_nil:add12_14(x))
gen_c5:c613_14(0) <=> c5
gen_c5:c613_14(+(x, 1)) <=> c6(gen_c5:c613_14(x))
gen_c7:c814_14(0) <=> c7
gen_c7:c814_14(+(x, 1)) <=> c8(c5, gen_c7:c814_14(x))


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

(85) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
SHUFF(gen_nil:add12_14(n6537_14), gen_nil:add12_14(b)) -> *15_14, rt in Omega(n6537_14 + n6537_14^2 + n6537_14^3)

Induction Base:
SHUFF(gen_nil:add12_14(0), gen_nil:add12_14(b))

Induction Step:
SHUFF(gen_nil:add12_14(+(n6537_14, 1)), gen_nil:add12_14(b)) ->_R^Omega(1)
c10(IF(null(gen_nil:add12_14(+(n6537_14, 1))), gen_nil:add12_14(+(n6537_14, 1)), gen_nil:add12_14(b), app(gen_nil:add12_14(b), add(head(gen_nil:add12_14(+(n6537_14, 1))), nil))), NULL(gen_nil:add12_14(+(n6537_14, 1)))) ->_R^Omega(0)
c10(IF(false, gen_nil:add12_14(+(1, n6537_14)), gen_nil:add12_14(b), app(gen_nil:add12_14(b), add(head(gen_nil:add12_14(+(1, n6537_14))), nil))), NULL(gen_nil:add12_14(+(1, n6537_14)))) ->_R^Omega(0)
c10(IF(false, gen_nil:add12_14(+(1, n6537_14)), gen_nil:add12_14(b), app(gen_nil:add12_14(b), add(hole_head3_14, nil))), NULL(gen_nil:add12_14(+(1, n6537_14)))) ->_L^Omega(0)
c10(IF(false, gen_nil:add12_14(+(1, n6537_14)), gen_nil:add12_14(+(0, 1)), gen_nil:add12_14(+(b, +(0, 1)))), NULL(gen_nil:add12_14(+(1, n6537_14)))) ->_R^Omega(1)
c10(c13(SHUFF(reverse(tail(gen_nil:add12_14(+(1, n6537_14)))), gen_nil:add12_14(+(1, b))), REVERSE(tail(gen_nil:add12_14(+(1, n6537_14)))), TAIL(gen_nil:add12_14(+(1, n6537_14)))), NULL(gen_nil:add12_14(+(1, n6537_14)))) ->_R^Omega(0)
c10(c13(SHUFF(reverse(gen_nil:add12_14(n6537_14)), gen_nil:add12_14(+(1, b))), REVERSE(tail(gen_nil:add12_14(+(1, n6537_14)))), TAIL(gen_nil:add12_14(+(1, n6537_14)))), NULL(gen_nil:add12_14(+(1, n6537_14)))) ->_L^Omega(0)
c10(c13(SHUFF(gen_nil:add12_14(n6537_14), gen_nil:add12_14(+(1, b))), REVERSE(tail(gen_nil:add12_14(+(1, n6537_14)))), TAIL(gen_nil:add12_14(+(1, n6537_14)))), NULL(gen_nil:add12_14(+(1, n6537_14)))) ->_IH
c10(c13(*15_14, REVERSE(tail(gen_nil:add12_14(+(1, n6537_14)))), TAIL(gen_nil:add12_14(+(1, n6537_14)))), NULL(gen_nil:add12_14(+(1, n6537_14)))) ->_R^Omega(0)
c10(c13(*15_14, REVERSE(gen_nil:add12_14(n6537_14)), TAIL(gen_nil:add12_14(+(1, n6537_14)))), NULL(gen_nil:add12_14(+(1, n6537_14)))) ->_L^Omega(n6537_14 + n6537_14^2)
c10(c13(*15_14, *15_14, TAIL(gen_nil:add12_14(+(1, n6537_14)))), NULL(gen_nil:add12_14(+(1, n6537_14)))) ->_R^Omega(1)
c10(c13(*15_14, *15_14, c2), NULL(gen_nil:add12_14(+(1, n6537_14)))) ->_R^Omega(1)
c10(c13(*15_14, *15_14, c2), c1)

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

(86)
Complex Obligation (BEST)

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

(87)
Obligation:
Proved the lower bound n^3 for the following obligation:

Innermost TRS:
Rules:
NULL(nil) -> c
NULL(add(z0, z1)) -> c1
TAIL(add(z0, z1)) -> c2
TAIL(nil) -> c3
HEAD(add(z0, z1)) -> c4
APP(nil, z0) -> c5
APP(add(z0, z1), z2) -> c6(APP(z1, z2))
REVERSE(nil) -> c7
REVERSE(add(z0, z1)) -> c8(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(z0) -> c9(SHUFF(z0, nil))
SHUFF(z0, z1) -> c10(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), NULL(z0))
SHUFF(z0, z1) -> c11(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
IF(true, z0, z1, z2) -> c12
IF(false, z0, z1, z2) -> c13(SHUFF(reverse(tail(z0)), z2), REVERSE(tail(z0)), TAIL(z0))
null(nil) -> true
null(add(z0, z1)) -> false
tail(add(z0, z1)) -> z1
tail(nil) -> nil
head(add(z0, z1)) -> z0
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(z0) -> shuff(z0, nil)
shuff(z0, z1) -> if(null(z0), z0, z1, app(z1, add(head(z0), nil)))
if(true, z0, z1, z2) -> z1
if(false, z0, z1, z2) -> shuff(reverse(tail(z0)), z2)

Types:
NULL :: nil:add -> c:c1
nil :: nil:add
c :: c:c1
add :: head -> nil:add -> nil:add
c1 :: c:c1
TAIL :: nil:add -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
HEAD :: nil:add -> c4
c4 :: c4
APP :: nil:add -> nil:add -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
REVERSE :: nil:add -> c7:c8
c7 :: c7:c8
c8 :: c5:c6 -> c7:c8 -> c7:c8
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c9
c9 :: c10:c11 -> c9
SHUFF :: nil:add -> nil:add -> c10:c11
c10 :: c12:c13 -> c:c1 -> c10:c11
IF :: true:false -> nil:add -> nil:add -> nil:add -> c12:c13
null :: nil:add -> true:false
app :: nil:add -> nil:add -> nil:add
head :: nil:add -> head
c11 :: c12:c13 -> c5:c6 -> c4 -> c10:c11
true :: true:false
c12 :: c12:c13
false :: true:false
c13 :: c10:c11 -> c7:c8 -> c2:c3 -> c12:c13
tail :: 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_c:c11_14 :: c:c1
hole_nil:add2_14 :: nil:add
hole_head3_14 :: head
hole_c2:c34_14 :: c2:c3
hole_c45_14 :: c4
hole_c5:c66_14 :: c5:c6
hole_c7:c87_14 :: c7:c8
hole_c98_14 :: c9
hole_c10:c119_14 :: c10:c11
hole_c12:c1310_14 :: c12:c13
hole_true:false11_14 :: true:false
gen_nil:add12_14 :: Nat -> nil:add
gen_c5:c613_14 :: Nat -> c5:c6
gen_c7:c814_14 :: Nat -> c7:c8


Lemmas:
APP(gen_nil:add12_14(n16_14), gen_nil:add12_14(b)) -> gen_c5:c613_14(n16_14), rt in Omega(1 + n16_14)
app(gen_nil:add12_14(n700_14), gen_nil:add12_14(b)) -> gen_nil:add12_14(+(n700_14, b)), rt in Omega(0)
reverse(gen_nil:add12_14(n1833_14)) -> gen_nil:add12_14(n1833_14), rt in Omega(0)
REVERSE(gen_nil:add12_14(n2211_14)) -> *15_14, rt in Omega(n2211_14 + n2211_14^2)


Generator Equations:
gen_nil:add12_14(0) <=> nil
gen_nil:add12_14(+(x, 1)) <=> add(hole_head3_14, gen_nil:add12_14(x))
gen_c5:c613_14(0) <=> c5
gen_c5:c613_14(+(x, 1)) <=> c6(gen_c5:c613_14(x))
gen_c7:c814_14(0) <=> c7
gen_c7:c814_14(+(x, 1)) <=> c8(c5, gen_c7:c814_14(x))


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

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

(89)
BOUNDS(n^3, INF)

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

(90)
Obligation:
Innermost TRS:
Rules:
NULL(nil) -> c
NULL(add(z0, z1)) -> c1
TAIL(add(z0, z1)) -> c2
TAIL(nil) -> c3
HEAD(add(z0, z1)) -> c4
APP(nil, z0) -> c5
APP(add(z0, z1), z2) -> c6(APP(z1, z2))
REVERSE(nil) -> c7
REVERSE(add(z0, z1)) -> c8(APP(reverse(z1), add(z0, nil)), REVERSE(z1))
SHUFFLE(z0) -> c9(SHUFF(z0, nil))
SHUFF(z0, z1) -> c10(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), NULL(z0))
SHUFF(z0, z1) -> c11(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
IF(true, z0, z1, z2) -> c12
IF(false, z0, z1, z2) -> c13(SHUFF(reverse(tail(z0)), z2), REVERSE(tail(z0)), TAIL(z0))
null(nil) -> true
null(add(z0, z1)) -> false
tail(add(z0, z1)) -> z1
tail(nil) -> nil
head(add(z0, z1)) -> z0
app(nil, z0) -> z0
app(add(z0, z1), z2) -> add(z0, app(z1, z2))
reverse(nil) -> nil
reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil))
shuffle(z0) -> shuff(z0, nil)
shuff(z0, z1) -> if(null(z0), z0, z1, app(z1, add(head(z0), nil)))
if(true, z0, z1, z2) -> z1
if(false, z0, z1, z2) -> shuff(reverse(tail(z0)), z2)

Types:
NULL :: nil:add -> c:c1
nil :: nil:add
c :: c:c1
add :: head -> nil:add -> nil:add
c1 :: c:c1
TAIL :: nil:add -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
HEAD :: nil:add -> c4
c4 :: c4
APP :: nil:add -> nil:add -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
REVERSE :: nil:add -> c7:c8
c7 :: c7:c8
c8 :: c5:c6 -> c7:c8 -> c7:c8
reverse :: nil:add -> nil:add
SHUFFLE :: nil:add -> c9
c9 :: c10:c11 -> c9
SHUFF :: nil:add -> nil:add -> c10:c11
c10 :: c12:c13 -> c:c1 -> c10:c11
IF :: true:false -> nil:add -> nil:add -> nil:add -> c12:c13
null :: nil:add -> true:false
app :: nil:add -> nil:add -> nil:add
head :: nil:add -> head
c11 :: c12:c13 -> c5:c6 -> c4 -> c10:c11
true :: true:false
c12 :: c12:c13
false :: true:false
c13 :: c10:c11 -> c7:c8 -> c2:c3 -> c12:c13
tail :: 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_c:c11_14 :: c:c1
hole_nil:add2_14 :: nil:add
hole_head3_14 :: head
hole_c2:c34_14 :: c2:c3
hole_c45_14 :: c4
hole_c5:c66_14 :: c5:c6
hole_c7:c87_14 :: c7:c8
hole_c98_14 :: c9
hole_c10:c119_14 :: c10:c11
hole_c12:c1310_14 :: c12:c13
hole_true:false11_14 :: true:false
gen_nil:add12_14 :: Nat -> nil:add
gen_c5:c613_14 :: Nat -> c5:c6
gen_c7:c814_14 :: Nat -> c7:c8


Lemmas:
APP(gen_nil:add12_14(n16_14), gen_nil:add12_14(b)) -> gen_c5:c613_14(n16_14), rt in Omega(1 + n16_14)
app(gen_nil:add12_14(n700_14), gen_nil:add12_14(b)) -> gen_nil:add12_14(+(n700_14, b)), rt in Omega(0)
reverse(gen_nil:add12_14(n1833_14)) -> gen_nil:add12_14(n1833_14), rt in Omega(0)
REVERSE(gen_nil:add12_14(n2211_14)) -> *15_14, rt in Omega(n2211_14 + n2211_14^2)
SHUFF(gen_nil:add12_14(n6537_14), gen_nil:add12_14(b)) -> *15_14, rt in Omega(n6537_14 + n6537_14^2 + n6537_14^3)


Generator Equations:
gen_nil:add12_14(0) <=> nil
gen_nil:add12_14(+(x, 1)) <=> add(hole_head3_14, gen_nil:add12_14(x))
gen_c5:c613_14(0) <=> c5
gen_c5:c613_14(+(x, 1)) <=> c6(gen_c5:c613_14(x))
gen_c7:c814_14(0) <=> c7
gen_c7:c814_14(+(x, 1)) <=> c8(c5, gen_c7:c814_14(x))


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

(91) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
shuff(gen_nil:add12_14(n18157_14), gen_nil:add12_14(b)) -> *15_14, rt in Omega(0)

Induction Base:
shuff(gen_nil:add12_14(0), gen_nil:add12_14(b))

Induction Step:
shuff(gen_nil:add12_14(+(n18157_14, 1)), gen_nil:add12_14(b)) ->_R^Omega(0)
if(null(gen_nil:add12_14(+(n18157_14, 1))), gen_nil:add12_14(+(n18157_14, 1)), gen_nil:add12_14(b), app(gen_nil:add12_14(b), add(head(gen_nil:add12_14(+(n18157_14, 1))), nil))) ->_R^Omega(0)
if(false, gen_nil:add12_14(+(1, n18157_14)), gen_nil:add12_14(b), app(gen_nil:add12_14(b), add(head(gen_nil:add12_14(+(1, n18157_14))), nil))) ->_R^Omega(0)
if(false, gen_nil:add12_14(+(1, n18157_14)), gen_nil:add12_14(b), app(gen_nil:add12_14(b), add(hole_head3_14, nil))) ->_L^Omega(0)
if(false, gen_nil:add12_14(+(1, n18157_14)), gen_nil:add12_14(+(0, 1)), gen_nil:add12_14(+(b, +(0, 1)))) ->_R^Omega(0)
shuff(reverse(tail(gen_nil:add12_14(+(1, n18157_14)))), gen_nil:add12_14(+(1, b))) ->_R^Omega(0)
shuff(reverse(gen_nil:add12_14(n18157_14)), gen_nil:add12_14(+(1, b))) ->_L^Omega(0)
shuff(gen_nil:add12_14(n18157_14), gen_nil:add12_14(+(1, b))) ->_IH
*15_14

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(92)
BOUNDS(1, INF)