WORST_CASE(Omega(n^3),O(n^3))
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 1742 ms]
(2) CpxRelTRS
(3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 13 ms]
    (12) CpxRNTS
    (13) InliningProof [UPPER BOUND(ID), 773 ms]
    (14) CpxRNTS
    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 89 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 47 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 170 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 312 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 115 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 178 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 32 ms]
    (42) CpxRNTS
    (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 148 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
    (48) CpxRNTS
    (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 352 ms]
    (52) CpxRNTS
    (53) IntTrsBoundProof [UPPER BOUND(ID), 54 ms]
    (54) CpxRNTS
    (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 128 ms]
    (58) CpxRNTS
    (59) IntTrsBoundProof [UPPER BOUND(ID), 23 ms]
    (60) CpxRNTS
    (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (62) CpxRNTS
    (63) IntTrsBoundProof [UPPER BOUND(ID), 51 ms]
    (64) CpxRNTS
    (65) IntTrsBoundProof [UPPER BOUND(ID), 25 ms]
    (66) CpxRNTS
    (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (68) CpxRNTS
    (69) IntTrsBoundProof [UPPER BOUND(ID), 375 ms]
    (70) CpxRNTS
    (71) IntTrsBoundProof [UPPER BOUND(ID), 36 ms]
    (72) CpxRNTS
    (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (74) CpxRNTS
    (75) IntTrsBoundProof [UPPER BOUND(ID), 1758 ms]
    (76) CpxRNTS
    (77) IntTrsBoundProof [UPPER BOUND(ID), 468 ms]
    (78) CpxRNTS
    (79) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (80) CpxRNTS
    (81) IntTrsBoundProof [UPPER BOUND(ID), 373 ms]
    (82) CpxRNTS
    (83) IntTrsBoundProof [UPPER BOUND(ID), 87 ms]
    (84) CpxRNTS
    (85) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (86) CpxRNTS
    (87) IntTrsBoundProof [UPPER BOUND(ID), 298 ms]
    (88) CpxRNTS
    (89) IntTrsBoundProof [UPPER BOUND(ID), 22 ms]
    (90) CpxRNTS
    (91) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (92) CpxRNTS
    (93) IntTrsBoundProof [UPPER BOUND(ID), 12.8 s]
    (94) CpxRNTS
    (95) IntTrsBoundProof [UPPER BOUND(ID), 1398 ms]
    (96) CpxRNTS
    (97) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (98) CpxRNTS
    (99) IntTrsBoundProof [UPPER BOUND(ID), 398 ms]
    (100) CpxRNTS
    (101) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (102) CpxRNTS
    (103) FinalProof [FINISHED, 0 ms]
    (104) BOUNDS(1, n^3)
(105) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(106) CpxRelTRS
    (107) SlicingProof [LOWER BOUND(ID), 0 ms]
    (108) CpxRelTRS
    (109) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) typed CpxTrs
    (111) OrderProof [LOWER BOUND(ID), 0 ms]
    (112) typed CpxTrs
    (113) RewriteLemmaProof [LOWER BOUND(ID), 273 ms]
    (114) BEST
        (115) proven lower bound
            (116) LowerBoundPropagationProof [FINISHED, 0 ms]
            (117) BOUNDS(n^1, INF)
        (118) typed CpxTrs
            (119) RewriteLemmaProof [LOWER BOUND(ID), 76 ms]
            (120) typed CpxTrs
            (121) RewriteLemmaProof [LOWER BOUND(ID), 43 ms]
            (122) typed CpxTrs
            (123) RewriteLemmaProof [LOWER BOUND(ID), 662 ms]
            (124) BEST
                (125) proven lower bound
                    (126) LowerBoundPropagationProof [FINISHED, 0 ms]
                    (127) BOUNDS(n^2, INF)
                (128) typed CpxTrs
                    (129) RewriteLemmaProof [LOWER BOUND(ID), 550 ms]
                    (130) BEST
                        (131) proven lower bound
                            (132) LowerBoundPropagationProof [FINISHED, 0 ms]
                            (133) BOUNDS(n^3, INF)
                        (134) typed CpxTrs
                            (135) RewriteLemmaProof [LOWER BOUND(ID), 1232 ms]
                            (136) BOUNDS(1, INF)


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

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


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

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^3, n^3).


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

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   NULL(nil) -> c [1]
   NULL(add(z0, z1)) -> c1 [1]
   TAIL(add(z0, z1)) -> c2 [1]
   TAIL(nil) -> c3 [1]
   HEAD(add(z0, z1)) -> c4 [1]
   APP(nil, z0) -> c5 [1]
   APP(add(z0, z1), z2) -> c6(APP(z1, z2)) [1]
   REVERSE(nil) -> c7 [1]
   REVERSE(add(z0, z1)) -> c8(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) [1]
   SHUFFLE(z0) -> c9(SHUFF(z0, nil)) [1]
   SHUFF(z0, z1) -> c10(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), NULL(z0)) [1]
   SHUFF(z0, z1) -> c11(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0)) [1]
   IF(true, z0, z1, z2) -> c12 [1]
   IF(false, z0, z1, z2) -> c13(SHUFF(reverse(tail(z0)), z2), REVERSE(tail(z0)), TAIL(z0)) [1]
   null(nil) -> true [0]
   null(add(z0, z1)) -> false [0]
   tail(add(z0, z1)) -> z1 [0]
   tail(nil) -> nil [0]
   head(add(z0, z1)) -> z0 [0]
   app(nil, z0) -> z0 [0]
   app(add(z0, z1), z2) -> add(z0, app(z1, z2)) [0]
   reverse(nil) -> nil [0]
   reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil)) [0]
   shuffle(z0) -> shuff(z0, nil) [0]
   shuff(z0, z1) -> if(null(z0), z0, z1, app(z1, add(head(z0), nil))) [0]
   if(true, z0, z1, z2) -> z1 [0]
   if(false, z0, z1, z2) -> shuff(reverse(tail(z0)), z2) [0]

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

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

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   NULL(nil) -> c [1]
   NULL(add(z0, z1)) -> c1 [1]
   TAIL(add(z0, z1)) -> c2 [1]
   TAIL(nil) -> c3 [1]
   HEAD(add(z0, z1)) -> c4 [1]
   APP(nil, z0) -> c5 [1]
   APP(add(z0, z1), z2) -> c6(APP(z1, z2)) [1]
   REVERSE(nil) -> c7 [1]
   REVERSE(add(z0, z1)) -> c8(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) [1]
   SHUFFLE(z0) -> c9(SHUFF(z0, nil)) [1]
   SHUFF(z0, z1) -> c10(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), NULL(z0)) [1]
   SHUFF(z0, z1) -> c11(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0)) [1]
   IF(true, z0, z1, z2) -> c12 [1]
   IF(false, z0, z1, z2) -> c13(SHUFF(reverse(tail(z0)), z2), REVERSE(tail(z0)), TAIL(z0)) [1]
   null(nil) -> true [0]
   null(add(z0, z1)) -> false [0]
   tail(add(z0, z1)) -> z1 [0]
   tail(nil) -> nil [0]
   head(add(z0, z1)) -> z0 [0]
   app(nil, z0) -> z0 [0]
   app(add(z0, z1), z2) -> add(z0, app(z1, z2)) [0]
   reverse(nil) -> nil [0]
   reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil)) [0]
   shuffle(z0) -> shuff(z0, nil) [0]
   shuff(z0, z1) -> if(null(z0), z0, z1, app(z1, add(head(z0), nil))) [0]
   if(true, z0, z1, z2) -> z1 [0]
   if(false, z0, z1, z2) -> shuff(reverse(tail(z0)), z2) [0]

The TRS has the following type information:
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

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

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

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

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

   NULL_1
   TAIL_1
   HEAD_1
   APP_2
   REVERSE_1
   SHUFFLE_1
   SHUFF_2
   IF_4

(c) The following functions are completely defined:

   null_1
   tail_1
   head_1
   app_2
   reverse_1
   shuffle_1
   shuff_2
   if_4

Due to the following rules being added:

   null(v0) -> null_null [0]
   tail(v0) -> nil [0]
   head(v0) -> const [0]
   app(v0, v1) -> nil [0]
   reverse(v0) -> nil [0]
   shuffle(v0) -> nil [0]
   shuff(v0, v1) -> nil [0]
   if(v0, v1, v2, v3) -> nil [0]

And the following fresh constants:    null_null, const, const1, const2

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

(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) -> c [1]
   NULL(add(z0, z1)) -> c1 [1]
   TAIL(add(z0, z1)) -> c2 [1]
   TAIL(nil) -> c3 [1]
   HEAD(add(z0, z1)) -> c4 [1]
   APP(nil, z0) -> c5 [1]
   APP(add(z0, z1), z2) -> c6(APP(z1, z2)) [1]
   REVERSE(nil) -> c7 [1]
   REVERSE(add(z0, z1)) -> c8(APP(reverse(z1), add(z0, nil)), REVERSE(z1)) [1]
   SHUFFLE(z0) -> c9(SHUFF(z0, nil)) [1]
   SHUFF(z0, z1) -> c10(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), NULL(z0)) [1]
   SHUFF(z0, z1) -> c11(IF(null(z0), z0, z1, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0)) [1]
   IF(true, z0, z1, z2) -> c12 [1]
   IF(false, z0, z1, z2) -> c13(SHUFF(reverse(tail(z0)), z2), REVERSE(tail(z0)), TAIL(z0)) [1]
   null(nil) -> true [0]
   null(add(z0, z1)) -> false [0]
   tail(add(z0, z1)) -> z1 [0]
   tail(nil) -> nil [0]
   head(add(z0, z1)) -> z0 [0]
   app(nil, z0) -> z0 [0]
   app(add(z0, z1), z2) -> add(z0, app(z1, z2)) [0]
   reverse(nil) -> nil [0]
   reverse(add(z0, z1)) -> app(reverse(z1), add(z0, nil)) [0]
   shuffle(z0) -> shuff(z0, nil) [0]
   shuff(z0, z1) -> if(null(z0), z0, z1, app(z1, add(head(z0), nil))) [0]
   if(true, z0, z1, z2) -> z1 [0]
   if(false, z0, z1, z2) -> shuff(reverse(tail(z0)), z2) [0]
   null(v0) -> null_null [0]
   tail(v0) -> nil [0]
   head(v0) -> const [0]
   app(v0, v1) -> nil [0]
   reverse(v0) -> nil [0]
   shuffle(v0) -> nil [0]
   shuff(v0, v1) -> nil [0]
   if(v0, v1, v2, v3) -> nil [0]

The TRS has the following type information:
NULL :: nil:add -> c:c1
nil :: nil:add
c :: c:c1
add :: const -> 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:null_null -> nil:add -> nil:add -> nil:add -> c12:c13
null :: nil:add -> true:false:null_null
app :: nil:add -> nil:add -> nil:add
head :: nil:add -> const
c11 :: c12:c13 -> c5:c6 -> c4 -> c10:c11
true :: true:false:null_null
c12 :: c12:c13
false :: true:false:null_null
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:null_null -> nil:add -> nil:add -> nil:add -> nil:add
null_null :: true:false:null_null
const :: const
const1 :: c9
const2 :: c10:c11

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

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

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

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

   NULL(nil) -> c [1]
   NULL(add(z0, z1)) -> c1 [1]
   TAIL(add(z0, z1)) -> c2 [1]
   TAIL(nil) -> c3 [1]
   HEAD(add(z0, z1)) -> c4 [1]
   APP(nil, z0) -> c5 [1]
   APP(add(z0, z1), z2) -> c6(APP(z1, z2)) [1]
   REVERSE(nil) -> c7 [1]
   REVERSE(add(z0, nil)) -> c8(APP(nil, add(z0, nil)), REVERSE(nil)) [1]
   REVERSE(add(z0, add(z0', z1'))) -> c8(APP(app(reverse(z1'), add(z0', nil)), add(z0, nil)), REVERSE(add(z0', z1'))) [1]
   REVERSE(add(z0, z1)) -> c8(APP(nil, add(z0, nil)), REVERSE(z1)) [1]
   SHUFFLE(z0) -> c9(SHUFF(z0, nil)) [1]
   SHUFF(nil, z1) -> c10(IF(true, nil, z1, app(z1, add(const, nil))), NULL(nil)) [1]
   SHUFF(add(z0'', z1''), z1) -> c10(IF(false, add(z0'', z1''), z1, app(z1, add(z0'', nil))), NULL(add(z0'', z1''))) [1]
   SHUFF(add(z0'', z1''), z1) -> c10(IF(false, add(z0'', z1''), z1, app(z1, add(const, nil))), NULL(add(z0'', z1''))) [1]
   SHUFF(add(z01, z11), z1) -> c10(IF(null_null, add(z01, z11), z1, app(z1, add(z01, nil))), NULL(add(z01, z11))) [1]
   SHUFF(z0, z1) -> c10(IF(null_null, z0, z1, app(z1, add(const, nil))), NULL(z0)) [1]
   SHUFF(nil, z1) -> c11(IF(true, nil, z1, app(z1, add(const, nil))), APP(z1, add(const, nil)), HEAD(nil)) [1]
   SHUFF(add(z02, z12), z1) -> c11(IF(false, add(z02, z12), z1, app(z1, add(z02, nil))), APP(z1, add(z02, nil)), HEAD(add(z02, z12))) [1]
   SHUFF(add(z02, z12), z1) -> c11(IF(false, add(z02, z12), z1, app(z1, add(z02, nil))), APP(z1, add(const, nil)), HEAD(add(z02, z12))) [1]
   SHUFF(add(z02, z12), z1) -> c11(IF(false, add(z02, z12), z1, app(z1, add(const, nil))), APP(z1, add(z02, nil)), HEAD(add(z02, z12))) [1]
   SHUFF(add(z02, z12), z1) -> c11(IF(false, add(z02, z12), z1, app(z1, add(const, nil))), APP(z1, add(const, nil)), HEAD(add(z02, z12))) [1]
   SHUFF(add(z03, z13), z1) -> c11(IF(null_null, add(z03, z13), z1, app(z1, add(z03, nil))), APP(z1, add(z03, nil)), HEAD(add(z03, z13))) [1]
   SHUFF(add(z03, z13), z1) -> c11(IF(null_null, add(z03, z13), z1, app(z1, add(z03, nil))), APP(z1, add(const, nil)), HEAD(add(z03, z13))) [1]
   SHUFF(add(z04, z14), z1) -> c11(IF(null_null, add(z04, z14), z1, app(z1, add(const, nil))), APP(z1, add(z04, nil)), HEAD(add(z04, z14))) [1]
   SHUFF(z0, z1) -> c11(IF(null_null, z0, z1, app(z1, add(const, nil))), APP(z1, add(const, nil)), HEAD(z0)) [1]
   IF(true, z0, z1, z2) -> c12 [1]
   IF(false, add(z05, z15), z1, z2) -> c13(SHUFF(reverse(z15), z2), REVERSE(z15), TAIL(add(z05, z15))) [1]
   IF(false, add(z05, z15), z1, z2) -> c13(SHUFF(reverse(z15), z2), REVERSE(nil), TAIL(add(z05, z15))) [1]
   IF(false, nil, z1, z2) -> c13(SHUFF(reverse(nil), z2), REVERSE(nil), TAIL(nil)) [1]
   IF(false, nil, z1, z2) -> c13(SHUFF(reverse(nil), z2), REVERSE(nil), TAIL(nil)) [1]
   IF(false, add(z06, z16), z1, z2) -> c13(SHUFF(reverse(nil), z2), REVERSE(z16), TAIL(add(z06, z16))) [1]
   IF(false, nil, z1, z2) -> c13(SHUFF(reverse(nil), z2), REVERSE(nil), TAIL(nil)) [1]
   IF(false, z0, z1, z2) -> c13(SHUFF(reverse(nil), z2), REVERSE(nil), TAIL(z0)) [1]
   null(nil) -> true [0]
   null(add(z0, z1)) -> false [0]
   tail(add(z0, z1)) -> z1 [0]
   tail(nil) -> nil [0]
   head(add(z0, z1)) -> z0 [0]
   app(nil, z0) -> z0 [0]
   app(add(z0, z1), z2) -> add(z0, app(z1, z2)) [0]
   reverse(nil) -> nil [0]
   reverse(add(z0, nil)) -> app(nil, add(z0, nil)) [0]
   reverse(add(z0, add(z07, z17))) -> app(app(reverse(z17), add(z07, nil)), add(z0, nil)) [0]
   reverse(add(z0, z1)) -> app(nil, add(z0, nil)) [0]
   shuffle(z0) -> shuff(z0, nil) [0]
   shuff(nil, z1) -> if(true, nil, z1, app(z1, add(const, nil))) [0]
   shuff(add(z08, z18), z1) -> if(false, add(z08, z18), z1, app(z1, add(z08, nil))) [0]
   shuff(add(z08, z18), z1) -> if(false, add(z08, z18), z1, app(z1, add(const, nil))) [0]
   shuff(add(z09, z19), z1) -> if(null_null, add(z09, z19), z1, app(z1, add(z09, nil))) [0]
   shuff(z0, z1) -> if(null_null, z0, z1, app(z1, add(const, nil))) [0]
   if(true, z0, z1, z2) -> z1 [0]
   if(false, add(z010, z110), z1, z2) -> shuff(reverse(z110), z2) [0]
   if(false, nil, z1, z2) -> shuff(reverse(nil), z2) [0]
   if(false, z0, z1, z2) -> shuff(reverse(nil), z2) [0]
   null(v0) -> null_null [0]
   tail(v0) -> nil [0]
   head(v0) -> const [0]
   app(v0, v1) -> nil [0]
   reverse(v0) -> nil [0]
   shuffle(v0) -> nil [0]
   shuff(v0, v1) -> nil [0]
   if(v0, v1, v2, v3) -> nil [0]

The TRS has the following type information:
NULL :: nil:add -> c:c1
nil :: nil:add
c :: c:c1
add :: const -> 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:null_null -> nil:add -> nil:add -> nil:add -> c12:c13
null :: nil:add -> true:false:null_null
app :: nil:add -> nil:add -> nil:add
head :: nil:add -> const
c11 :: c12:c13 -> c5:c6 -> c4 -> c10:c11
true :: true:false:null_null
c12 :: c12:c13
false :: true:false:null_null
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:null_null -> nil:add -> nil:add -> nil:add -> nil:add
null_null :: true:false:null_null
const :: const
const1 :: c9
const2 :: c10:c11

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

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

   nil => 0
   c => 0
   c1 => 1
   c2 => 0
   c3 => 1
   c4 => 0
   c5 => 0
   c7 => 0
   true => 2
   c12 => 0
   false => 1
   null_null => 0
   const => 0
   const1 => 0
   const2 => 0

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   APP(z, z') -{ 1 }-> 1 + APP(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z1 >= 0, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   IF(z, z', z'', z3) -{ 1 }-> 1 + SHUFF(reverse(z15), z2) + REVERSE(z15) + TAIL(1 + z05 + z15) :|: z15 >= 0, z1 >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 = z2, z2 >= 0, z'' = z1
   IF(z, z', z'', z3) -{ 1 }-> 1 + SHUFF(reverse(z15), z2) + REVERSE(0) + TAIL(1 + z05 + z15) :|: z15 >= 0, z1 >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 = z2, z2 >= 0, z'' = z1
   IF(z, z', z'', z3) -{ 1 }-> 1 + SHUFF(reverse(0), z2) + REVERSE(z16) + TAIL(1 + z06 + z16) :|: z1 >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 = z2, z2 >= 0, z'' = z1
   IF(z, z', z'', z3) -{ 1 }-> 1 + SHUFF(reverse(0), z2) + REVERSE(0) + TAIL(z0) :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   IF(z, z', z'', z3) -{ 1 }-> 1 + SHUFF(reverse(0), z2) + REVERSE(0) + TAIL(0) :|: z1 >= 0, z = 1, z3 = z2, z2 >= 0, z' = 0, z'' = z1
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + z0 + 0) + REVERSE(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + z0 + 0) + REVERSE(0) :|: z0 >= 0, z = 1 + z0 + 0
   SHUFF(z, z') -{ 1 }-> 1 + IF(2, 0, z1, app(z1, 1 + 0 + 0)) + NULL(0) :|: z1 >= 0, z' = z1, z = 0
   SHUFF(z, z') -{ 1 }-> 1 + IF(1, 1 + z0'' + z1'', z1, app(z1, 1 + z0'' + 0)) + NULL(1 + z0'' + z1'') :|: z = 1 + z0'' + z1'', z1 >= 0, z' = z1, z0'' >= 0, z1'' >= 0
   SHUFF(z, z') -{ 1 }-> 1 + IF(1, 1 + z0'' + z1'', z1, app(z1, 1 + 0 + 0)) + NULL(1 + z0'' + z1'') :|: z = 1 + z0'' + z1'', z1 >= 0, z' = z1, z0'' >= 0, z1'' >= 0
   SHUFF(z, z') -{ 1 }-> 1 + IF(0, z0, z1, app(z1, 1 + 0 + 0)) + NULL(z0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   SHUFF(z, z') -{ 1 }-> 1 + IF(0, 1 + z01 + z11, z1, app(z1, 1 + z01 + 0)) + NULL(1 + z01 + z11) :|: z11 >= 0, z1 >= 0, z01 >= 0, z = 1 + z01 + z11, z' = z1
   SHUFF(z, z') -{ 1 }-> 1 + IF(2, 0, z1, app(z1, 1 + 0 + 0)) + APP(z1, 1 + 0 + 0) + HEAD(0) :|: z1 >= 0, z' = z1, z = 0
   SHUFF(z, z') -{ 1 }-> 1 + IF(1, 1 + z02 + z12, z1, app(z1, 1 + z02 + 0)) + APP(z1, 1 + z02 + 0) + HEAD(1 + z02 + z12) :|: z = 1 + z02 + z12, z1 >= 0, z02 >= 0, z' = z1, z12 >= 0
   SHUFF(z, z') -{ 1 }-> 1 + IF(1, 1 + z02 + z12, z1, app(z1, 1 + z02 + 0)) + APP(z1, 1 + 0 + 0) + HEAD(1 + z02 + z12) :|: z = 1 + z02 + z12, z1 >= 0, z02 >= 0, z' = z1, z12 >= 0
   SHUFF(z, z') -{ 1 }-> 1 + IF(1, 1 + z02 + z12, z1, app(z1, 1 + 0 + 0)) + APP(z1, 1 + z02 + 0) + HEAD(1 + z02 + z12) :|: z = 1 + z02 + z12, z1 >= 0, z02 >= 0, z' = z1, z12 >= 0
   SHUFF(z, z') -{ 1 }-> 1 + IF(1, 1 + z02 + z12, z1, app(z1, 1 + 0 + 0)) + APP(z1, 1 + 0 + 0) + HEAD(1 + z02 + z12) :|: z = 1 + z02 + z12, z1 >= 0, z02 >= 0, z' = z1, z12 >= 0
   SHUFF(z, z') -{ 1 }-> 1 + IF(0, z0, z1, app(z1, 1 + 0 + 0)) + APP(z1, 1 + 0 + 0) + HEAD(z0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   SHUFF(z, z') -{ 1 }-> 1 + IF(0, 1 + z03 + z13, z1, app(z1, 1 + z03 + 0)) + APP(z1, 1 + z03 + 0) + HEAD(1 + z03 + z13) :|: z = 1 + z03 + z13, z1 >= 0, z' = z1, z03 >= 0, z13 >= 0
   SHUFF(z, z') -{ 1 }-> 1 + IF(0, 1 + z03 + z13, z1, app(z1, 1 + z03 + 0)) + APP(z1, 1 + 0 + 0) + HEAD(1 + z03 + z13) :|: z = 1 + z03 + z13, z1 >= 0, z' = z1, z03 >= 0, z13 >= 0
   SHUFF(z, z') -{ 1 }-> 1 + IF(0, 1 + z04 + z14, z1, app(z1, 1 + 0 + 0)) + APP(z1, 1 + z04 + 0) + HEAD(1 + z04 + z14) :|: z04 >= 0, z = 1 + z04 + z14, z1 >= 0, z' = z1, z14 >= 0
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z0, 0) :|: z = z0, z0 >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0
   app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   if(z, z', z'', z3) -{ 0 }-> z1 :|: z = 2, z1 >= 0, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z2) :|: z110 >= 0, z1 >= 0, z = 1, z' = 1 + z010 + z110, z3 = z2, z010 >= 0, z2 >= 0, z'' = z1
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z2) :|: z1 >= 0, z = 1, z3 = z2, z2 >= 0, z' = 0, z'' = z1
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z2) :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   if(z, z', z'', z3) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z3 = v3, v2 >= 0, v3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z0 >= 0, z = 1 + z0 + 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   shuff(z, z') -{ 0 }-> if(2, 0, z1, app(z1, 1 + 0 + 0)) :|: z1 >= 0, z' = z1, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z1, app(z1, 1 + z08 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z1 >= 0, z' = z1
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z1, app(z1, 1 + 0 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z1 >= 0, z' = z1
   shuff(z, z') -{ 0 }-> if(0, z0, z1, app(z1, 1 + 0 + 0)) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z1, app(z1, 1 + z09 + 0)) :|: z = 1 + z09 + z19, z1 >= 0, z19 >= 0, z' = z1, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   shuffle(z) -{ 0 }-> shuff(z0, 0) :|: z = z0, z0 >= 0
   shuffle(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

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

   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1

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

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

   APP(z, z') -{ 1 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   APP(z, z') -{ 1 }-> 1 + APP(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z1 >= 0, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z2) + REVERSE(z15) + 0 :|: z15 >= 0, z1 >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 = z2, z2 >= 0, z'' = z1, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z2) + REVERSE(0) + 0 :|: z15 >= 0, z1 >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 = z2, z2 >= 0, z'' = z1, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z2) + REVERSE(z16) + 0 :|: z1 >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 = z2, z2 >= 0, z'' = z1, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z2) + REVERSE(0) + 1 :|: z1 >= 0, z = 1, z3 = z2, z2 >= 0, z' = 0, z'' = z1, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z2) + REVERSE(0) + 1 :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1, z0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z2) + REVERSE(0) + 0 :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1, z1' >= 0, z0' >= 0, z0 = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + z0 + 0) + REVERSE(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + z0 + 0) + REVERSE(0) :|: z0 >= 0, z = 1 + z0 + 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z1, app(z1, 1 + 0 + 0)) + 0 :|: z1 >= 0, z' = z1, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z1, app(z1, 1 + z0'' + 0)) + 1 :|: z = 1 + z0'' + z1'', z1 >= 0, z' = z1, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z1, app(z1, 1 + 0 + 0)) + 1 :|: z = 1 + z0'' + z1'', z1 >= 0, z' = z1, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z0, z1, app(z1, 1 + 0 + 0)) + 1 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0, z1' >= 0, z0' >= 0, z0 = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z0, z1, app(z1, 1 + 0 + 0)) + 0 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0, z0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z1, app(z1, 1 + z01 + 0)) + 1 :|: z11 >= 0, z1 >= 0, z01 >= 0, z = 1 + z01 + z11, z' = z1, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1 }-> 1 + IF(2, 0, z1, app(z1, 1 + 0 + 0)) + APP(z1, 1 + 0 + 0) + HEAD(0) :|: z1 >= 0, z' = z1, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z1, app(z1, 1 + z02 + 0)) + APP(z1, 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z1 >= 0, z02 >= 0, z' = z1, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z1, app(z1, 1 + z02 + 0)) + APP(z1, 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z1 >= 0, z02 >= 0, z' = z1, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z1, app(z1, 1 + 0 + 0)) + APP(z1, 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z1 >= 0, z02 >= 0, z' = z1, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z1, app(z1, 1 + 0 + 0)) + APP(z1, 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z1 >= 0, z02 >= 0, z' = z1, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z0, z1, app(z1, 1 + 0 + 0)) + APP(z1, 1 + 0 + 0) + 0 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0, z1' >= 0, z0' >= 0, z0 = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z1, app(z1, 1 + z03 + 0)) + APP(z1, 1 + z03 + 0) + 0 :|: z = 1 + z03 + z13, z1 >= 0, z' = z1, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z1, app(z1, 1 + z03 + 0)) + APP(z1, 1 + 0 + 0) + 0 :|: z = 1 + z03 + z13, z1 >= 0, z' = z1, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z04 + z14, z1, app(z1, 1 + 0 + 0)) + APP(z1, 1 + z04 + 0) + 0 :|: z04 >= 0, z = 1 + z04 + z14, z1 >= 0, z' = z1, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z0, 0) :|: z = z0, z0 >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0
   app(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   if(z, z', z'', z3) -{ 0 }-> z1 :|: z = 2, z1 >= 0, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z2) :|: z110 >= 0, z1 >= 0, z = 1, z' = 1 + z010 + z110, z3 = z2, z010 >= 0, z2 >= 0, z'' = z1
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z2) :|: z1 >= 0, z = 1, z3 = z2, z2 >= 0, z' = 0, z'' = z1
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z2) :|: z1 >= 0, z = 1, z0 >= 0, z3 = z2, z' = z0, z2 >= 0, z'' = z1
   if(z, z', z'', z3) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z3 = v3, v2 >= 0, v3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z0 >= 0, z = 1 + z0 + 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   shuff(z, z') -{ 0 }-> if(2, 0, z1, app(z1, 1 + 0 + 0)) :|: z1 >= 0, z' = z1, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z1, app(z1, 1 + z08 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z1 >= 0, z' = z1
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z1, app(z1, 1 + 0 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z1 >= 0, z' = z1
   shuff(z, z') -{ 0 }-> if(0, z0, z1, app(z1, 1 + 0 + 0)) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z1, app(z1, 1 + z09 + 0)) :|: z = 1 + z09 + z19, z1 >= 0, z19 >= 0, z' = z1, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   shuffle(z) -{ 0 }-> shuff(z0, 0) :|: z = z0, z0 >= 0
   shuffle(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

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

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

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 1 }-> 1 + APP(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + z0 + 0) + REVERSE(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + (z - 1) + 0) + REVERSE(0) :|: z - 1 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + z0'' + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + 0 + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 1 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', app(z', 1 + z01 + 0)) + 1 :|: z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + HEAD(0) :|: z' >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + APP(z', 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + APP(z', 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + APP(z', 1 + z03 + 0) + 0 :|: z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z04 + z14, z', app(z', 1 + 0 + 0)) + APP(z', 1 + z04 + 0) + 0 :|: z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + z08 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + 0 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', app(z', 1 + z09 + 0)) :|: z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0


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

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

   { HEAD }
   { tail }
   { APP }
   { head }
   { NULL }
   { app }
   { null }
   { TAIL }
   { reverse }
   { shuff, if }
   { REVERSE }
   { shuffle }
   { IF, SHUFF }
   { SHUFFLE }

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

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

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 1 }-> 1 + APP(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + z0 + 0) + REVERSE(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + (z - 1) + 0) + REVERSE(0) :|: z - 1 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + z0'' + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + 0 + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 1 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', app(z', 1 + z01 + 0)) + 1 :|: z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + HEAD(0) :|: z' >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + APP(z', 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + APP(z', 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + APP(z', 1 + z03 + 0) + 0 :|: z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z04 + z14, z', app(z', 1 + 0 + 0)) + APP(z', 1 + z04 + 0) + 0 :|: z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + z08 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + 0 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', app(z', 1 + z09 + 0)) :|: z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {HEAD}, {tail}, {APP}, {head}, {NULL}, {app}, {null}, {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}

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

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

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

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 1 }-> 1 + APP(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + z0 + 0) + REVERSE(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + (z - 1) + 0) + REVERSE(0) :|: z - 1 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + z0'' + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + 0 + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 1 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', app(z', 1 + z01 + 0)) + 1 :|: z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + HEAD(0) :|: z' >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + APP(z', 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + APP(z', 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + APP(z', 1 + z03 + 0) + 0 :|: z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z04 + z14, z', app(z', 1 + 0 + 0)) + APP(z', 1 + z04 + 0) + 0 :|: z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + z08 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + 0 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', app(z', 1 + z09 + 0)) :|: z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {HEAD}, {tail}, {APP}, {head}, {NULL}, {app}, {null}, {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: HEAD
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 1 }-> 1 + APP(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + z0 + 0) + REVERSE(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + (z - 1) + 0) + REVERSE(0) :|: z - 1 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + z0'' + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + 0 + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 1 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', app(z', 1 + z01 + 0)) + 1 :|: z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + HEAD(0) :|: z' >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + APP(z', 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + APP(z', 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + APP(z', 1 + z03 + 0) + 0 :|: z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z04 + z14, z', app(z', 1 + 0 + 0)) + APP(z', 1 + z04 + 0) + 0 :|: z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + z08 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + 0 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', app(z', 1 + z09 + 0)) :|: z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {HEAD}, {tail}, {APP}, {head}, {NULL}, {app}, {null}, {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: ?, size: O(1) [0]

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

(23) 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

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

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

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 1 }-> 1 + APP(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + z0 + 0) + REVERSE(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + (z - 1) + 0) + REVERSE(0) :|: z - 1 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + z0'' + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + 0 + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 1 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', app(z', 1 + z01 + 0)) + 1 :|: z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + HEAD(0) :|: z' >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + APP(z', 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + APP(z', 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + APP(z', 1 + z03 + 0) + 0 :|: z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z04 + z14, z', app(z', 1 + 0 + 0)) + APP(z', 1 + z04 + 0) + 0 :|: z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + z08 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + 0 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', app(z', 1 + z09 + 0)) :|: z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {tail}, {APP}, {head}, {NULL}, {app}, {null}, {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]

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

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

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

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 1 }-> 1 + APP(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + z0 + 0) + REVERSE(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + (z - 1) + 0) + REVERSE(0) :|: z - 1 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + z0'' + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + 0 + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 1 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', app(z', 1 + z01 + 0)) + 1 :|: z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + s :|: s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + APP(z', 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + APP(z', 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + APP(z', 1 + z03 + 0) + 0 :|: z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z04 + z14, z', app(z', 1 + 0 + 0)) + APP(z', 1 + z04 + 0) + 0 :|: z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + z08 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + 0 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', app(z', 1 + z09 + 0)) :|: z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {tail}, {APP}, {head}, {NULL}, {app}, {null}, {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]

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

(27) 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

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

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

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 1 }-> 1 + APP(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + z0 + 0) + REVERSE(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + (z - 1) + 0) + REVERSE(0) :|: z - 1 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + z0'' + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + 0 + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 1 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', app(z', 1 + z01 + 0)) + 1 :|: z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + s :|: s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + APP(z', 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + APP(z', 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + APP(z', 1 + z03 + 0) + 0 :|: z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z04 + z14, z', app(z', 1 + 0 + 0)) + APP(z', 1 + z04 + 0) + 0 :|: z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + z08 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + 0 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', app(z', 1 + z09 + 0)) :|: z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {tail}, {APP}, {head}, {NULL}, {app}, {null}, {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: ?, size: O(n^1) [z]

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

(29) 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: 0

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

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

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 1 }-> 1 + APP(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + z0 + 0) + REVERSE(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + (z - 1) + 0) + REVERSE(0) :|: z - 1 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + z0'' + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + 0 + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 1 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', app(z', 1 + z01 + 0)) + 1 :|: z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + s :|: s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + APP(z', 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + APP(z', 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + APP(z', 1 + z03 + 0) + 0 :|: z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z04 + z14, z', app(z', 1 + 0 + 0)) + APP(z', 1 + z04 + 0) + 0 :|: z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + z08 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + 0 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', app(z', 1 + z09 + 0)) :|: z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {APP}, {head}, {NULL}, {app}, {null}, {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]

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

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

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

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 1 }-> 1 + APP(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + z0 + 0) + REVERSE(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + (z - 1) + 0) + REVERSE(0) :|: z - 1 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + z0'' + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + 0 + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 1 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', app(z', 1 + z01 + 0)) + 1 :|: z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + s :|: s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + APP(z', 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + APP(z', 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + APP(z', 1 + z03 + 0) + 0 :|: z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z04 + z14, z', app(z', 1 + 0 + 0)) + APP(z', 1 + z04 + 0) + 0 :|: z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + z08 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + 0 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', app(z', 1 + z09 + 0)) :|: z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {APP}, {head}, {NULL}, {app}, {null}, {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]

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

(33) 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

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

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 1 }-> 1 + APP(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + z0 + 0) + REVERSE(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + (z - 1) + 0) + REVERSE(0) :|: z - 1 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + z0'' + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + 0 + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 1 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', app(z', 1 + z01 + 0)) + 1 :|: z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + s :|: s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + APP(z', 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + APP(z', 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + APP(z', 1 + z03 + 0) + 0 :|: z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z04 + z14, z', app(z', 1 + 0 + 0)) + APP(z', 1 + z04 + 0) + 0 :|: z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + z08 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + 0 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', app(z', 1 + z09 + 0)) :|: z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {APP}, {head}, {NULL}, {app}, {null}, {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: ?, size: O(n^1) [z]

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

(35) 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

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

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

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 1 }-> 1 + APP(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + z0 + 0) + REVERSE(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(0, 1 + (z - 1) + 0) + REVERSE(0) :|: z - 1 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + z0'' + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + 0 + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 1 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', app(z', 1 + z01 + 0)) + 1 :|: z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + s :|: s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + APP(z', 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + APP(z', 1 + z02 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + APP(z', 1 + z03 + 0) + 0 :|: z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + APP(z', 1 + 0 + 0) + 0 :|: z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z04 + z14, z', app(z', 1 + 0 + 0)) + APP(z', 1 + z04 + 0) + 0 :|: z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + z08 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + 0 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', app(z', 1 + z09 + 0)) :|: z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {head}, {NULL}, {app}, {null}, {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]

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

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

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

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + z0'' + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + 0 + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 1 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', app(z', 1 + z01 + 0)) + 1 :|: z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + s2 + s :|: s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + s3 + 0 :|: s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + s4 + 0 :|: s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + s5 + 0 :|: s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + s6 + 0 :|: s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + s10 + 0 :|: s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + s7 + 0 :|: s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + s8 + 0 :|: s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', app(z', 1 + 0 + 0)) + s9 + 0 :|: s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + z08 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + 0 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', app(z', 1 + z09 + 0)) :|: z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {head}, {NULL}, {app}, {null}, {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]

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

(39) 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

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

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

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + z0'' + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + 0 + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 1 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', app(z', 1 + z01 + 0)) + 1 :|: z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + s2 + s :|: s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + s3 + 0 :|: s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + s4 + 0 :|: s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + s5 + 0 :|: s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + s6 + 0 :|: s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + s10 + 0 :|: s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + s7 + 0 :|: s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + s8 + 0 :|: s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', app(z', 1 + 0 + 0)) + s9 + 0 :|: s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + z08 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + 0 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', app(z', 1 + z09 + 0)) :|: z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {head}, {NULL}, {app}, {null}, {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: ?, size: O(n^1) [z]

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

(41) 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: 0

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

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

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + z0'' + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + 0 + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 1 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', app(z', 1 + z01 + 0)) + 1 :|: z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + s2 + s :|: s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + s3 + 0 :|: s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + s4 + 0 :|: s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + s5 + 0 :|: s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + s6 + 0 :|: s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + s10 + 0 :|: s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + s7 + 0 :|: s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + s8 + 0 :|: s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', app(z', 1 + 0 + 0)) + s9 + 0 :|: s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + z08 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + 0 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', app(z', 1 + z09 + 0)) :|: z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {NULL}, {app}, {null}, {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]

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

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

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

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + z0'' + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + 0 + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 1 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', app(z', 1 + z01 + 0)) + 1 :|: z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + s2 + s :|: s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + s3 + 0 :|: s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + s4 + 0 :|: s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + s5 + 0 :|: s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + s6 + 0 :|: s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + s10 + 0 :|: s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + s7 + 0 :|: s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + s8 + 0 :|: s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', app(z', 1 + 0 + 0)) + s9 + 0 :|: s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + z08 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + 0 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', app(z', 1 + z09 + 0)) :|: z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {NULL}, {app}, {null}, {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]

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

(45) 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

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

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

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + z0'' + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + 0 + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 1 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', app(z', 1 + z01 + 0)) + 1 :|: z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + s2 + s :|: s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + s3 + 0 :|: s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + s4 + 0 :|: s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + s5 + 0 :|: s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + s6 + 0 :|: s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + s10 + 0 :|: s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + s7 + 0 :|: s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + s8 + 0 :|: s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', app(z', 1 + 0 + 0)) + s9 + 0 :|: s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + z08 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + 0 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', app(z', 1 + z09 + 0)) :|: z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {NULL}, {app}, {null}, {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: ?, size: O(1) [1]

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

(47) 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

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

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

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + z0'' + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + 0 + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 1 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', app(z', 1 + z01 + 0)) + 1 :|: z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + s2 + s :|: s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + s3 + 0 :|: s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + s4 + 0 :|: s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + s5 + 0 :|: s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + s6 + 0 :|: s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + s10 + 0 :|: s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + s7 + 0 :|: s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + s8 + 0 :|: s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', app(z', 1 + 0 + 0)) + s9 + 0 :|: s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + z08 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + 0 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', app(z', 1 + z09 + 0)) :|: z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {app}, {null}, {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]

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

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

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

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + z0'' + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + 0 + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 1 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', app(z', 1 + z01 + 0)) + 1 :|: z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + s2 + s :|: s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + s3 + 0 :|: s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + s4 + 0 :|: s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + s5 + 0 :|: s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + s6 + 0 :|: s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + s10 + 0 :|: s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + s7 + 0 :|: s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + s8 + 0 :|: s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', app(z', 1 + 0 + 0)) + s9 + 0 :|: s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + z08 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + 0 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', app(z', 1 + z09 + 0)) :|: z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {app}, {null}, {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]

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

(51) 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'

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

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

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + z0'' + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + 0 + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 1 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', app(z', 1 + z01 + 0)) + 1 :|: z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + s2 + s :|: s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + s3 + 0 :|: s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + s4 + 0 :|: s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + s5 + 0 :|: s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + s6 + 0 :|: s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + s10 + 0 :|: s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + s7 + 0 :|: s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + s8 + 0 :|: s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', app(z', 1 + 0 + 0)) + s9 + 0 :|: s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + z08 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + 0 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', app(z', 1 + z09 + 0)) :|: z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {app}, {null}, {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: ?, size: O(n^1) [z + z']

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: app
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + z0'' + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', app(z', 1 + 0 + 0)) + 1 :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 1 :|: z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + 0 :|: z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', app(z', 1 + z01 + 0)) + 1 :|: z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', app(z', 1 + 0 + 0)) + s2 + s :|: s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + s3 + 0 :|: s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + z02 + 0)) + s4 + 0 :|: s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + s5 + 0 :|: s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', app(z', 1 + 0 + 0)) + s6 + 0 :|: s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', app(z', 1 + 0 + 0)) + s10 + 0 :|: s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + s7 + 0 :|: s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', app(z', 1 + z03 + 0)) + s8 + 0 :|: s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', app(z', 1 + 0 + 0)) + s9 + 0 :|: s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + app(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> app(0, 1 + z0 + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(0, 1 + (z - 1) + 0) :|: z - 1 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + z08 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', app(z', 1 + 0 + 0)) :|: z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', app(z', 1 + 0 + 0)) :|: z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', app(z', 1 + z09 + 0)) :|: z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {null}, {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']

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

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

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

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', s19) + 0 :|: s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s20) + 1 :|: s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s21) + 1 :|: s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s23) + 1 :|: s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s24) + 0 :|: s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', s22) + 1 :|: s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', s25) + s2 + s :|: s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s26) + s3 + 0 :|: s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s27) + s4 + 0 :|: s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s28) + s5 + 0 :|: s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s29) + s6 + 0 :|: s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', s33) + s10 + 0 :|: s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s30) + s7 + 0 :|: s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s31) + s8 + 0 :|: s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', s32) + s9 + 0 :|: s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', s14) :|: s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s15) :|: s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s16) :|: s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', s18) :|: s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', s17) :|: s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {null}, {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']

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

(57) 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: 2

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

(58)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', s19) + 0 :|: s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s20) + 1 :|: s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s21) + 1 :|: s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s23) + 1 :|: s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s24) + 0 :|: s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', s22) + 1 :|: s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', s25) + s2 + s :|: s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s26) + s3 + 0 :|: s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s27) + s4 + 0 :|: s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s28) + s5 + 0 :|: s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s29) + s6 + 0 :|: s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', s33) + s10 + 0 :|: s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s30) + s7 + 0 :|: s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s31) + s8 + 0 :|: s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', s32) + s9 + 0 :|: s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', s14) :|: s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s15) :|: s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s16) :|: s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', s18) :|: s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', s17) :|: s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {null}, {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: ?, size: O(1) [2]

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

(59) 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: 0

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

(60)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', s19) + 0 :|: s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s20) + 1 :|: s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s21) + 1 :|: s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s23) + 1 :|: s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s24) + 0 :|: s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', s22) + 1 :|: s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', s25) + s2 + s :|: s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s26) + s3 + 0 :|: s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s27) + s4 + 0 :|: s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s28) + s5 + 0 :|: s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s29) + s6 + 0 :|: s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', s33) + s10 + 0 :|: s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s30) + s7 + 0 :|: s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s31) + s8 + 0 :|: s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', s32) + s9 + 0 :|: s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', s14) :|: s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s15) :|: s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s16) :|: s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', s18) :|: s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', s17) :|: s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: O(1) [0], size: O(1) [2]

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

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

(62)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', s19) + 0 :|: s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s20) + 1 :|: s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s21) + 1 :|: s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s23) + 1 :|: s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s24) + 0 :|: s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', s22) + 1 :|: s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', s25) + s2 + s :|: s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s26) + s3 + 0 :|: s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s27) + s4 + 0 :|: s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s28) + s5 + 0 :|: s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s29) + s6 + 0 :|: s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', s33) + s10 + 0 :|: s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s30) + s7 + 0 :|: s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s31) + s8 + 0 :|: s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', s32) + s9 + 0 :|: s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', s14) :|: s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s15) :|: s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s16) :|: s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', s18) :|: s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', s17) :|: s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: O(1) [0], size: O(1) [2]

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(64)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', s19) + 0 :|: s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s20) + 1 :|: s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s21) + 1 :|: s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s23) + 1 :|: s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s24) + 0 :|: s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', s22) + 1 :|: s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', s25) + s2 + s :|: s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s26) + s3 + 0 :|: s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s27) + s4 + 0 :|: s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s28) + s5 + 0 :|: s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s29) + s6 + 0 :|: s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', s33) + s10 + 0 :|: s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s30) + s7 + 0 :|: s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s31) + s8 + 0 :|: s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', s32) + s9 + 0 :|: s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', s14) :|: s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s15) :|: s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s16) :|: s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', s18) :|: s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', s17) :|: s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {TAIL}, {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: O(1) [0], size: O(1) [2]
  TAIL: runtime: ?, size: O(1) [1]

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

(65) 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

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

(66)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', s19) + 0 :|: s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s20) + 1 :|: s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s21) + 1 :|: s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s23) + 1 :|: s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s24) + 0 :|: s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', s22) + 1 :|: s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', s25) + s2 + s :|: s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s26) + s3 + 0 :|: s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s27) + s4 + 0 :|: s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s28) + s5 + 0 :|: s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s29) + s6 + 0 :|: s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', s33) + s10 + 0 :|: s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s30) + s7 + 0 :|: s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s31) + s8 + 0 :|: s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', s32) + s9 + 0 :|: s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', s14) :|: s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s15) :|: s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s16) :|: s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', s18) :|: s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', s17) :|: s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: O(1) [0], size: O(1) [2]
  TAIL: runtime: O(1) [1], size: O(1) [1]

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

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

(68)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', s19) + 0 :|: s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s20) + 1 :|: s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s21) + 1 :|: s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s23) + 1 :|: s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s24) + 0 :|: s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', s22) + 1 :|: s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', s25) + s2 + s :|: s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s26) + s3 + 0 :|: s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s27) + s4 + 0 :|: s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s28) + s5 + 0 :|: s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s29) + s6 + 0 :|: s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', s33) + s10 + 0 :|: s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s30) + s7 + 0 :|: s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s31) + s8 + 0 :|: s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', s32) + s9 + 0 :|: s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', s14) :|: s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s15) :|: s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s16) :|: s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', s18) :|: s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', s17) :|: s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: O(1) [0], size: O(1) [2]
  TAIL: runtime: O(1) [1], size: O(1) [1]

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

(69) 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

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

(70)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', s19) + 0 :|: s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s20) + 1 :|: s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s21) + 1 :|: s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s23) + 1 :|: s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s24) + 0 :|: s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', s22) + 1 :|: s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', s25) + s2 + s :|: s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s26) + s3 + 0 :|: s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s27) + s4 + 0 :|: s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s28) + s5 + 0 :|: s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s29) + s6 + 0 :|: s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', s33) + s10 + 0 :|: s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s30) + s7 + 0 :|: s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s31) + s8 + 0 :|: s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', s32) + s9 + 0 :|: s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', s14) :|: s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s15) :|: s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s16) :|: s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', s18) :|: s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', s17) :|: s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {reverse}, {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: O(1) [0], size: O(1) [2]
  TAIL: runtime: O(1) [1], size: O(1) [1]
  reverse: runtime: ?, size: O(n^1) [z]

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

(71) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: reverse
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(72)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(z15) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(z15), z3) + REVERSE(0) + 0 :|: z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(z16) + 0 :|: z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 1 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(reverse(0), z3) + REVERSE(0) + 0 :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 1 }-> 1 + APP(app(reverse(z1'), 1 + z0' + 0), 1 + z0 + 0) + REVERSE(1 + z0' + z1') :|: z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', s19) + 0 :|: s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s20) + 1 :|: s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s21) + 1 :|: s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s23) + 1 :|: s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s24) + 0 :|: s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', s22) + 1 :|: s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', s25) + s2 + s :|: s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s26) + s3 + 0 :|: s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s27) + s4 + 0 :|: s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s28) + s5 + 0 :|: s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s29) + s6 + 0 :|: s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', s33) + s10 + 0 :|: s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s30) + s7 + 0 :|: s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s31) + s8 + 0 :|: s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', s32) + s9 + 0 :|: s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(z110), z3) :|: z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(reverse(0), z3) :|: z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> app(app(reverse(z17), 1 + z07 + 0), 1 + z0 + 0) :|: z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', s14) :|: s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s15) :|: s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s16) :|: s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', s18) :|: s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', s17) :|: s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: O(1) [0], size: O(1) [2]
  TAIL: runtime: O(1) [1], size: O(1) [1]
  reverse: runtime: O(1) [0], size: O(n^1) [z]

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

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

(74)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s43, z3) + REVERSE(z15) + 0 :|: s43 >= 0, s43 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s44, z3) + REVERSE(0) + 0 :|: s44 >= 0, s44 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s45, z3) + REVERSE(0) + 1 :|: s45 >= 0, s45 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s46, z3) + REVERSE(z16) + 0 :|: s46 >= 0, s46 <= 0, z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s47, z3) + REVERSE(0) + 0 :|: s47 >= 0, s47 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s48, z3) + REVERSE(0) + 1 :|: s48 >= 0, s48 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 2 + s35 }-> 1 + s36 + REVERSE(1 + z0' + z1') :|: s34 >= 0, s34 <= z1', s35 >= 0, s35 <= s34 + (1 + z0' + 0), s36 >= 0, s36 <= s35, z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', s19) + 0 :|: s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s20) + 1 :|: s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s21) + 1 :|: s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s23) + 1 :|: s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s24) + 0 :|: s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', s22) + 1 :|: s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', s25) + s2 + s :|: s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s26) + s3 + 0 :|: s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s27) + s4 + 0 :|: s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s28) + s5 + 0 :|: s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s29) + s6 + 0 :|: s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', s33) + s10 + 0 :|: s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s30) + s7 + 0 :|: s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s31) + s8 + 0 :|: s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', s32) + s9 + 0 :|: s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(s40, z3) :|: s40 >= 0, s40 <= z110, z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(s41, z3) :|: s41 >= 0, s41 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(s42, z3) :|: s42 >= 0, s42 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> s39 :|: s37 >= 0, s37 <= z17, s38 >= 0, s38 <= s37 + (1 + z07 + 0), s39 >= 0, s39 <= s38 + (1 + z0 + 0), z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', s14) :|: s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s15) :|: s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s16) :|: s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', s18) :|: s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', s17) :|: s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: O(1) [0], size: O(1) [2]
  TAIL: runtime: O(1) [1], size: O(1) [1]
  reverse: runtime: O(1) [0], size: O(n^1) [z]

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

(75) 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 + 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' + z'^2 + z'' + 3*z3

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

(76)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s43, z3) + REVERSE(z15) + 0 :|: s43 >= 0, s43 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s44, z3) + REVERSE(0) + 0 :|: s44 >= 0, s44 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s45, z3) + REVERSE(0) + 1 :|: s45 >= 0, s45 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s46, z3) + REVERSE(z16) + 0 :|: s46 >= 0, s46 <= 0, z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s47, z3) + REVERSE(0) + 0 :|: s47 >= 0, s47 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s48, z3) + REVERSE(0) + 1 :|: s48 >= 0, s48 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 2 + s35 }-> 1 + s36 + REVERSE(1 + z0' + z1') :|: s34 >= 0, s34 <= z1', s35 >= 0, s35 <= s34 + (1 + z0' + 0), s36 >= 0, s36 <= s35, z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', s19) + 0 :|: s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s20) + 1 :|: s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s21) + 1 :|: s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s23) + 1 :|: s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s24) + 0 :|: s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', s22) + 1 :|: s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', s25) + s2 + s :|: s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s26) + s3 + 0 :|: s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s27) + s4 + 0 :|: s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s28) + s5 + 0 :|: s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s29) + s6 + 0 :|: s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', s33) + s10 + 0 :|: s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s30) + s7 + 0 :|: s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s31) + s8 + 0 :|: s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', s32) + s9 + 0 :|: s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(s40, z3) :|: s40 >= 0, s40 <= z110, z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(s41, z3) :|: s41 >= 0, s41 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(s42, z3) :|: s42 >= 0, s42 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> s39 :|: s37 >= 0, s37 <= z17, s38 >= 0, s38 <= s37 + (1 + z07 + 0), s39 >= 0, s39 <= s38 + (1 + z0 + 0), z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', s14) :|: s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s15) :|: s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s16) :|: s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', s18) :|: s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', s17) :|: s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {shuff,if}, {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: O(1) [0], size: O(1) [2]
  TAIL: runtime: O(1) [1], size: O(1) [1]
  reverse: runtime: O(1) [0], size: O(n^1) [z]
  shuff: runtime: ?, size: O(n^2) [z + z^2 + z']
  if: runtime: ?, size: O(n^2) [z' + z'^2 + z'' + 3*z3]

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

(77) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: shuff
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed RUNTIME bound using CoFloCo for: if
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(78)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s43, z3) + REVERSE(z15) + 0 :|: s43 >= 0, s43 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s44, z3) + REVERSE(0) + 0 :|: s44 >= 0, s44 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s45, z3) + REVERSE(0) + 1 :|: s45 >= 0, s45 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s46, z3) + REVERSE(z16) + 0 :|: s46 >= 0, s46 <= 0, z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s47, z3) + REVERSE(0) + 0 :|: s47 >= 0, s47 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s48, z3) + REVERSE(0) + 1 :|: s48 >= 0, s48 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 2 + s35 }-> 1 + s36 + REVERSE(1 + z0' + z1') :|: s34 >= 0, s34 <= z1', s35 >= 0, s35 <= s34 + (1 + z0' + 0), s36 >= 0, s36 <= s35, z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', s19) + 0 :|: s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s20) + 1 :|: s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s21) + 1 :|: s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s23) + 1 :|: s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s24) + 0 :|: s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', s22) + 1 :|: s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', s25) + s2 + s :|: s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s26) + s3 + 0 :|: s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s27) + s4 + 0 :|: s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s28) + s5 + 0 :|: s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s29) + s6 + 0 :|: s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', s33) + s10 + 0 :|: s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s30) + s7 + 0 :|: s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s31) + s8 + 0 :|: s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', s32) + s9 + 0 :|: s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(s40, z3) :|: s40 >= 0, s40 <= z110, z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> shuff(s41, z3) :|: s41 >= 0, s41 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> shuff(s42, z3) :|: s42 >= 0, s42 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> s39 :|: s37 >= 0, s37 <= z17, s38 >= 0, s38 <= s37 + (1 + z07 + 0), s39 >= 0, s39 <= s38 + (1 + z0 + 0), z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> if(2, 0, z', s14) :|: s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s15) :|: s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(1, 1 + z08 + z18, z', s16) :|: s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> if(0, z, z', s18) :|: s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> if(0, 1 + z09 + z19, z', s17) :|: s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> shuff(z, 0) :|: z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: O(1) [0], size: O(1) [2]
  TAIL: runtime: O(1) [1], size: O(1) [1]
  reverse: runtime: O(1) [0], size: O(n^1) [z]
  shuff: runtime: O(1) [0], size: O(n^2) [z + z^2 + z']
  if: runtime: O(1) [0], size: O(n^2) [z' + z'^2 + z'' + 3*z3]

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

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

(80)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s43, z3) + REVERSE(z15) + 0 :|: s43 >= 0, s43 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s44, z3) + REVERSE(0) + 0 :|: s44 >= 0, s44 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s45, z3) + REVERSE(0) + 1 :|: s45 >= 0, s45 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s46, z3) + REVERSE(z16) + 0 :|: s46 >= 0, s46 <= 0, z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s47, z3) + REVERSE(0) + 0 :|: s47 >= 0, s47 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s48, z3) + REVERSE(0) + 1 :|: s48 >= 0, s48 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 2 + s35 }-> 1 + s36 + REVERSE(1 + z0' + z1') :|: s34 >= 0, s34 <= z1', s35 >= 0, s35 <= s34 + (1 + z0' + 0), s36 >= 0, s36 <= s35, z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', s19) + 0 :|: s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s20) + 1 :|: s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s21) + 1 :|: s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s23) + 1 :|: s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s24) + 0 :|: s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', s22) + 1 :|: s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', s25) + s2 + s :|: s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s26) + s3 + 0 :|: s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s27) + s4 + 0 :|: s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s28) + s5 + 0 :|: s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s29) + s6 + 0 :|: s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', s33) + s10 + 0 :|: s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s30) + s7 + 0 :|: s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s31) + s8 + 0 :|: s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', s32) + s9 + 0 :|: s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> s55 :|: s55 >= 0, s55 <= s40 + s40 * s40 + z3, s40 >= 0, s40 <= z110, z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> s56 :|: s56 >= 0, s56 <= s41 + s41 * s41 + z3, s41 >= 0, s41 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> s57 :|: s57 >= 0, s57 <= s42 + s42 * s42 + z3, s42 >= 0, s42 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> s39 :|: s37 >= 0, s37 <= z17, s38 >= 0, s38 <= s37 + (1 + z07 + 0), s39 >= 0, s39 <= s38 + (1 + z0 + 0), z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> s50 :|: s50 >= 0, s50 <= 3 * s14 + 0 + 0 * 0 + z', s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> s51 :|: s51 >= 0, s51 <= 3 * s15 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s52 :|: s52 >= 0, s52 <= 3 * s16 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s53 :|: s53 >= 0, s53 <= 3 * s17 + (1 + z09 + z19) + (1 + z09 + z19) * (1 + z09 + z19) + z', s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 3 * s18 + z + z * z + z', s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> s49 :|: s49 >= 0, s49 <= z + z * z + 0, z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: O(1) [0], size: O(1) [2]
  TAIL: runtime: O(1) [1], size: O(1) [1]
  reverse: runtime: O(1) [0], size: O(n^1) [z]
  shuff: runtime: O(1) [0], size: O(n^2) [z + z^2 + z']
  if: runtime: O(1) [0], size: O(n^2) [z' + z'^2 + z'' + 3*z3]

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

(81) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(82)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s43, z3) + REVERSE(z15) + 0 :|: s43 >= 0, s43 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s44, z3) + REVERSE(0) + 0 :|: s44 >= 0, s44 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s45, z3) + REVERSE(0) + 1 :|: s45 >= 0, s45 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s46, z3) + REVERSE(z16) + 0 :|: s46 >= 0, s46 <= 0, z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s47, z3) + REVERSE(0) + 0 :|: s47 >= 0, s47 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s48, z3) + REVERSE(0) + 1 :|: s48 >= 0, s48 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 2 + s35 }-> 1 + s36 + REVERSE(1 + z0' + z1') :|: s34 >= 0, s34 <= z1', s35 >= 0, s35 <= s34 + (1 + z0' + 0), s36 >= 0, s36 <= s35, z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', s19) + 0 :|: s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s20) + 1 :|: s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s21) + 1 :|: s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s23) + 1 :|: s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s24) + 0 :|: s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', s22) + 1 :|: s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', s25) + s2 + s :|: s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s26) + s3 + 0 :|: s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s27) + s4 + 0 :|: s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s28) + s5 + 0 :|: s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s29) + s6 + 0 :|: s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', s33) + s10 + 0 :|: s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s30) + s7 + 0 :|: s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s31) + s8 + 0 :|: s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', s32) + s9 + 0 :|: s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> s55 :|: s55 >= 0, s55 <= s40 + s40 * s40 + z3, s40 >= 0, s40 <= z110, z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> s56 :|: s56 >= 0, s56 <= s41 + s41 * s41 + z3, s41 >= 0, s41 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> s57 :|: s57 >= 0, s57 <= s42 + s42 * s42 + z3, s42 >= 0, s42 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> s39 :|: s37 >= 0, s37 <= z17, s38 >= 0, s38 <= s37 + (1 + z07 + 0), s39 >= 0, s39 <= s38 + (1 + z0 + 0), z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> s50 :|: s50 >= 0, s50 <= 3 * s14 + 0 + 0 * 0 + z', s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> s51 :|: s51 >= 0, s51 <= 3 * s15 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s52 :|: s52 >= 0, s52 <= 3 * s16 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s53 :|: s53 >= 0, s53 <= 3 * s17 + (1 + z09 + z19) + (1 + z09 + z19) * (1 + z09 + z19) + z', s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 3 * s18 + z + z * z + z', s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> s49 :|: s49 >= 0, s49 <= z + z * z + 0, z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {REVERSE}, {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: O(1) [0], size: O(1) [2]
  TAIL: runtime: O(1) [1], size: O(1) [1]
  reverse: runtime: O(1) [0], size: O(n^1) [z]
  shuff: runtime: O(1) [0], size: O(n^2) [z + z^2 + z']
  if: runtime: O(1) [0], size: O(n^2) [z' + z'^2 + z'' + 3*z3]
  REVERSE: runtime: ?, size: O(n^2) [z + z^2]

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

(83) 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: 1 + 2*z + z^2

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

(84)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s43, z3) + REVERSE(z15) + 0 :|: s43 >= 0, s43 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s44, z3) + REVERSE(0) + 0 :|: s44 >= 0, s44 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s45, z3) + REVERSE(0) + 1 :|: s45 >= 0, s45 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s46, z3) + REVERSE(z16) + 0 :|: s46 >= 0, s46 <= 0, z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s47, z3) + REVERSE(0) + 0 :|: s47 >= 0, s47 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   IF(z, z', z'', z3) -{ 2 }-> 1 + SHUFF(s48, z3) + REVERSE(0) + 1 :|: s48 >= 0, s48 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 2 }-> 1 + s'' + REVERSE(0) :|: s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 2 }-> 1 + s1 + REVERSE(z1) :|: s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 2 + s35 }-> 1 + s36 + REVERSE(1 + z0' + z1') :|: s34 >= 0, s34 <= z1', s35 >= 0, s35 <= s34 + (1 + z0' + 0), s36 >= 0, s36 <= s35, z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', s19) + 0 :|: s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s20) + 1 :|: s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s21) + 1 :|: s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s23) + 1 :|: s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s24) + 0 :|: s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', s22) + 1 :|: s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', s25) + s2 + s :|: s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s26) + s3 + 0 :|: s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s27) + s4 + 0 :|: s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s28) + s5 + 0 :|: s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s29) + s6 + 0 :|: s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', s33) + s10 + 0 :|: s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s30) + s7 + 0 :|: s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s31) + s8 + 0 :|: s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', s32) + s9 + 0 :|: s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> s55 :|: s55 >= 0, s55 <= s40 + s40 * s40 + z3, s40 >= 0, s40 <= z110, z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> s56 :|: s56 >= 0, s56 <= s41 + s41 * s41 + z3, s41 >= 0, s41 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> s57 :|: s57 >= 0, s57 <= s42 + s42 * s42 + z3, s42 >= 0, s42 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> s39 :|: s37 >= 0, s37 <= z17, s38 >= 0, s38 <= s37 + (1 + z07 + 0), s39 >= 0, s39 <= s38 + (1 + z0 + 0), z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> s50 :|: s50 >= 0, s50 <= 3 * s14 + 0 + 0 * 0 + z', s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> s51 :|: s51 >= 0, s51 <= 3 * s15 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s52 :|: s52 >= 0, s52 <= 3 * s16 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s53 :|: s53 >= 0, s53 <= 3 * s17 + (1 + z09 + z19) + (1 + z09 + z19) * (1 + z09 + z19) + z', s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 3 * s18 + z + z * z + z', s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> s49 :|: s49 >= 0, s49 <= z + z * z + 0, z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: O(1) [0], size: O(1) [2]
  TAIL: runtime: O(1) [1], size: O(1) [1]
  reverse: runtime: O(1) [0], size: O(n^1) [z]
  shuff: runtime: O(1) [0], size: O(n^2) [z + z^2 + z']
  if: runtime: O(1) [0], size: O(n^2) [z' + z'^2 + z'' + 3*z3]
  REVERSE: runtime: O(n^2) [1 + 2*z + z^2], size: O(n^2) [z + z^2]

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

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

(86)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 3 + 2*z15 + z15^2 }-> 1 + SHUFF(s43, z3) + s61 + 0 :|: s61 >= 0, s61 <= z15 + z15 * z15, s43 >= 0, s43 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s44, z3) + s62 + 0 :|: s62 >= 0, s62 <= 0 + 0 * 0, s44 >= 0, s44 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s45, z3) + s63 + 1 :|: s63 >= 0, s63 <= 0 + 0 * 0, s45 >= 0, s45 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 3 + 2*z16 + z16^2 }-> 1 + SHUFF(s46, z3) + s64 + 0 :|: s64 >= 0, s64 <= z16 + z16 * z16, s46 >= 0, s46 <= 0, z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s47, z3) + s65 + 0 :|: s65 >= 0, s65 <= 0 + 0 * 0, s47 >= 0, s47 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s48, z3) + s66 + 1 :|: s66 >= 0, s66 <= 0 + 0 * 0, s48 >= 0, s48 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 3 }-> 1 + s'' + s58 :|: s58 >= 0, s58 <= 0 + 0 * 0, s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 3 + 2*z1 + z1^2 }-> 1 + s1 + s60 :|: s60 >= 0, s60 <= z1 + z1 * z1, s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 6 + s35 + 4*z0' + 2*z0'*z1' + z0'^2 + 4*z1' + z1'^2 }-> 1 + s36 + s59 :|: s59 >= 0, s59 <= 1 + z0' + z1' + (1 + z0' + z1') * (1 + z0' + z1'), s34 >= 0, s34 <= z1', s35 >= 0, s35 <= s34 + (1 + z0' + 0), s36 >= 0, s36 <= s35, z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', s19) + 0 :|: s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s20) + 1 :|: s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s21) + 1 :|: s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s23) + 1 :|: s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s24) + 0 :|: s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', s22) + 1 :|: s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', s25) + s2 + s :|: s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s26) + s3 + 0 :|: s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s27) + s4 + 0 :|: s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s28) + s5 + 0 :|: s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s29) + s6 + 0 :|: s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', s33) + s10 + 0 :|: s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s30) + s7 + 0 :|: s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s31) + s8 + 0 :|: s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', s32) + s9 + 0 :|: s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> s55 :|: s55 >= 0, s55 <= s40 + s40 * s40 + z3, s40 >= 0, s40 <= z110, z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> s56 :|: s56 >= 0, s56 <= s41 + s41 * s41 + z3, s41 >= 0, s41 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> s57 :|: s57 >= 0, s57 <= s42 + s42 * s42 + z3, s42 >= 0, s42 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> s39 :|: s37 >= 0, s37 <= z17, s38 >= 0, s38 <= s37 + (1 + z07 + 0), s39 >= 0, s39 <= s38 + (1 + z0 + 0), z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> s50 :|: s50 >= 0, s50 <= 3 * s14 + 0 + 0 * 0 + z', s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> s51 :|: s51 >= 0, s51 <= 3 * s15 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s52 :|: s52 >= 0, s52 <= 3 * s16 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s53 :|: s53 >= 0, s53 <= 3 * s17 + (1 + z09 + z19) + (1 + z09 + z19) * (1 + z09 + z19) + z', s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 3 * s18 + z + z * z + z', s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> s49 :|: s49 >= 0, s49 <= z + z * z + 0, z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: O(1) [0], size: O(1) [2]
  TAIL: runtime: O(1) [1], size: O(1) [1]
  reverse: runtime: O(1) [0], size: O(n^1) [z]
  shuff: runtime: O(1) [0], size: O(n^2) [z + z^2 + z']
  if: runtime: O(1) [0], size: O(n^2) [z' + z'^2 + z'' + 3*z3]
  REVERSE: runtime: O(n^2) [1 + 2*z + z^2], size: O(n^2) [z + z^2]

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

(87) 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 + z^2

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

(88)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 3 + 2*z15 + z15^2 }-> 1 + SHUFF(s43, z3) + s61 + 0 :|: s61 >= 0, s61 <= z15 + z15 * z15, s43 >= 0, s43 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s44, z3) + s62 + 0 :|: s62 >= 0, s62 <= 0 + 0 * 0, s44 >= 0, s44 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s45, z3) + s63 + 1 :|: s63 >= 0, s63 <= 0 + 0 * 0, s45 >= 0, s45 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 3 + 2*z16 + z16^2 }-> 1 + SHUFF(s46, z3) + s64 + 0 :|: s64 >= 0, s64 <= z16 + z16 * z16, s46 >= 0, s46 <= 0, z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s47, z3) + s65 + 0 :|: s65 >= 0, s65 <= 0 + 0 * 0, s47 >= 0, s47 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s48, z3) + s66 + 1 :|: s66 >= 0, s66 <= 0 + 0 * 0, s48 >= 0, s48 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 3 }-> 1 + s'' + s58 :|: s58 >= 0, s58 <= 0 + 0 * 0, s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 3 + 2*z1 + z1^2 }-> 1 + s1 + s60 :|: s60 >= 0, s60 <= z1 + z1 * z1, s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 6 + s35 + 4*z0' + 2*z0'*z1' + z0'^2 + 4*z1' + z1'^2 }-> 1 + s36 + s59 :|: s59 >= 0, s59 <= 1 + z0' + z1' + (1 + z0' + z1') * (1 + z0' + z1'), s34 >= 0, s34 <= z1', s35 >= 0, s35 <= s34 + (1 + z0' + 0), s36 >= 0, s36 <= s35, z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', s19) + 0 :|: s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s20) + 1 :|: s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s21) + 1 :|: s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s23) + 1 :|: s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s24) + 0 :|: s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', s22) + 1 :|: s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', s25) + s2 + s :|: s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s26) + s3 + 0 :|: s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s27) + s4 + 0 :|: s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s28) + s5 + 0 :|: s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s29) + s6 + 0 :|: s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', s33) + s10 + 0 :|: s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s30) + s7 + 0 :|: s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s31) + s8 + 0 :|: s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', s32) + s9 + 0 :|: s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> s55 :|: s55 >= 0, s55 <= s40 + s40 * s40 + z3, s40 >= 0, s40 <= z110, z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> s56 :|: s56 >= 0, s56 <= s41 + s41 * s41 + z3, s41 >= 0, s41 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> s57 :|: s57 >= 0, s57 <= s42 + s42 * s42 + z3, s42 >= 0, s42 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> s39 :|: s37 >= 0, s37 <= z17, s38 >= 0, s38 <= s37 + (1 + z07 + 0), s39 >= 0, s39 <= s38 + (1 + z0 + 0), z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> s50 :|: s50 >= 0, s50 <= 3 * s14 + 0 + 0 * 0 + z', s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> s51 :|: s51 >= 0, s51 <= 3 * s15 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s52 :|: s52 >= 0, s52 <= 3 * s16 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s53 :|: s53 >= 0, s53 <= 3 * s17 + (1 + z09 + z19) + (1 + z09 + z19) * (1 + z09 + z19) + z', s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 3 * s18 + z + z * z + z', s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> s49 :|: s49 >= 0, s49 <= z + z * z + 0, z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {shuffle}, {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: O(1) [0], size: O(1) [2]
  TAIL: runtime: O(1) [1], size: O(1) [1]
  reverse: runtime: O(1) [0], size: O(n^1) [z]
  shuff: runtime: O(1) [0], size: O(n^2) [z + z^2 + z']
  if: runtime: O(1) [0], size: O(n^2) [z' + z'^2 + z'' + 3*z3]
  REVERSE: runtime: O(n^2) [1 + 2*z + z^2], size: O(n^2) [z + z^2]
  shuffle: runtime: ?, size: O(n^2) [z + z^2]

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

(89) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: shuffle
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(90)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 3 + 2*z15 + z15^2 }-> 1 + SHUFF(s43, z3) + s61 + 0 :|: s61 >= 0, s61 <= z15 + z15 * z15, s43 >= 0, s43 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s44, z3) + s62 + 0 :|: s62 >= 0, s62 <= 0 + 0 * 0, s44 >= 0, s44 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s45, z3) + s63 + 1 :|: s63 >= 0, s63 <= 0 + 0 * 0, s45 >= 0, s45 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 3 + 2*z16 + z16^2 }-> 1 + SHUFF(s46, z3) + s64 + 0 :|: s64 >= 0, s64 <= z16 + z16 * z16, s46 >= 0, s46 <= 0, z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s47, z3) + s65 + 0 :|: s65 >= 0, s65 <= 0 + 0 * 0, s47 >= 0, s47 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s48, z3) + s66 + 1 :|: s66 >= 0, s66 <= 0 + 0 * 0, s48 >= 0, s48 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 3 }-> 1 + s'' + s58 :|: s58 >= 0, s58 <= 0 + 0 * 0, s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 3 + 2*z1 + z1^2 }-> 1 + s1 + s60 :|: s60 >= 0, s60 <= z1 + z1 * z1, s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 6 + s35 + 4*z0' + 2*z0'*z1' + z0'^2 + 4*z1' + z1'^2 }-> 1 + s36 + s59 :|: s59 >= 0, s59 <= 1 + z0' + z1' + (1 + z0' + z1') * (1 + z0' + z1'), s34 >= 0, s34 <= z1', s35 >= 0, s35 <= s34 + (1 + z0' + 0), s36 >= 0, s36 <= s35, z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', s19) + 0 :|: s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s20) + 1 :|: s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s21) + 1 :|: s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s23) + 1 :|: s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s24) + 0 :|: s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', s22) + 1 :|: s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', s25) + s2 + s :|: s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s26) + s3 + 0 :|: s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s27) + s4 + 0 :|: s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s28) + s5 + 0 :|: s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s29) + s6 + 0 :|: s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', s33) + s10 + 0 :|: s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s30) + s7 + 0 :|: s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s31) + s8 + 0 :|: s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', s32) + s9 + 0 :|: s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> s55 :|: s55 >= 0, s55 <= s40 + s40 * s40 + z3, s40 >= 0, s40 <= z110, z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> s56 :|: s56 >= 0, s56 <= s41 + s41 * s41 + z3, s41 >= 0, s41 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> s57 :|: s57 >= 0, s57 <= s42 + s42 * s42 + z3, s42 >= 0, s42 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> s39 :|: s37 >= 0, s37 <= z17, s38 >= 0, s38 <= s37 + (1 + z07 + 0), s39 >= 0, s39 <= s38 + (1 + z0 + 0), z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> s50 :|: s50 >= 0, s50 <= 3 * s14 + 0 + 0 * 0 + z', s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> s51 :|: s51 >= 0, s51 <= 3 * s15 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s52 :|: s52 >= 0, s52 <= 3 * s16 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s53 :|: s53 >= 0, s53 <= 3 * s17 + (1 + z09 + z19) + (1 + z09 + z19) * (1 + z09 + z19) + z', s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 3 * s18 + z + z * z + z', s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> s49 :|: s49 >= 0, s49 <= z + z * z + 0, z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: O(1) [0], size: O(1) [2]
  TAIL: runtime: O(1) [1], size: O(1) [1]
  reverse: runtime: O(1) [0], size: O(n^1) [z]
  shuff: runtime: O(1) [0], size: O(n^2) [z + z^2 + z']
  if: runtime: O(1) [0], size: O(n^2) [z' + z'^2 + z'' + 3*z3]
  REVERSE: runtime: O(n^2) [1 + 2*z + z^2], size: O(n^2) [z + z^2]
  shuffle: runtime: O(1) [0], size: O(n^2) [z + z^2]

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

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

(92)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 3 + 2*z15 + z15^2 }-> 1 + SHUFF(s43, z3) + s61 + 0 :|: s61 >= 0, s61 <= z15 + z15 * z15, s43 >= 0, s43 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s44, z3) + s62 + 0 :|: s62 >= 0, s62 <= 0 + 0 * 0, s44 >= 0, s44 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s45, z3) + s63 + 1 :|: s63 >= 0, s63 <= 0 + 0 * 0, s45 >= 0, s45 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 3 + 2*z16 + z16^2 }-> 1 + SHUFF(s46, z3) + s64 + 0 :|: s64 >= 0, s64 <= z16 + z16 * z16, s46 >= 0, s46 <= 0, z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s47, z3) + s65 + 0 :|: s65 >= 0, s65 <= 0 + 0 * 0, s47 >= 0, s47 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s48, z3) + s66 + 1 :|: s66 >= 0, s66 <= 0 + 0 * 0, s48 >= 0, s48 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 3 }-> 1 + s'' + s58 :|: s58 >= 0, s58 <= 0 + 0 * 0, s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 3 + 2*z1 + z1^2 }-> 1 + s1 + s60 :|: s60 >= 0, s60 <= z1 + z1 * z1, s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 6 + s35 + 4*z0' + 2*z0'*z1' + z0'^2 + 4*z1' + z1'^2 }-> 1 + s36 + s59 :|: s59 >= 0, s59 <= 1 + z0' + z1' + (1 + z0' + z1') * (1 + z0' + z1'), s34 >= 0, s34 <= z1', s35 >= 0, s35 <= s34 + (1 + z0' + 0), s36 >= 0, s36 <= s35, z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', s19) + 0 :|: s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s20) + 1 :|: s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s21) + 1 :|: s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s23) + 1 :|: s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s24) + 0 :|: s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', s22) + 1 :|: s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', s25) + s2 + s :|: s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s26) + s3 + 0 :|: s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s27) + s4 + 0 :|: s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s28) + s5 + 0 :|: s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s29) + s6 + 0 :|: s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', s33) + s10 + 0 :|: s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s30) + s7 + 0 :|: s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s31) + s8 + 0 :|: s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', s32) + s9 + 0 :|: s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> s55 :|: s55 >= 0, s55 <= s40 + s40 * s40 + z3, s40 >= 0, s40 <= z110, z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> s56 :|: s56 >= 0, s56 <= s41 + s41 * s41 + z3, s41 >= 0, s41 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> s57 :|: s57 >= 0, s57 <= s42 + s42 * s42 + z3, s42 >= 0, s42 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> s39 :|: s37 >= 0, s37 <= z17, s38 >= 0, s38 <= s37 + (1 + z07 + 0), s39 >= 0, s39 <= s38 + (1 + z0 + 0), z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> s50 :|: s50 >= 0, s50 <= 3 * s14 + 0 + 0 * 0 + z', s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> s51 :|: s51 >= 0, s51 <= 3 * s15 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s52 :|: s52 >= 0, s52 <= 3 * s16 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s53 :|: s53 >= 0, s53 <= 3 * s17 + (1 + z09 + z19) + (1 + z09 + z19) * (1 + z09 + z19) + z', s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 3 * s18 + z + z * z + z', s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> s49 :|: s49 >= 0, s49 <= z + z * z + 0, z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: O(1) [0], size: O(1) [2]
  TAIL: runtime: O(1) [1], size: O(1) [1]
  reverse: runtime: O(1) [0], size: O(n^1) [z]
  shuff: runtime: O(1) [0], size: O(n^2) [z + z^2 + z']
  if: runtime: O(1) [0], size: O(n^2) [z' + z'^2 + z'' + 3*z3]
  REVERSE: runtime: O(n^2) [1 + 2*z + z^2], size: O(n^2) [z + z^2]
  shuffle: runtime: O(1) [0], size: O(n^2) [z + z^2]

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

(93) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

(94)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 3 + 2*z15 + z15^2 }-> 1 + SHUFF(s43, z3) + s61 + 0 :|: s61 >= 0, s61 <= z15 + z15 * z15, s43 >= 0, s43 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s44, z3) + s62 + 0 :|: s62 >= 0, s62 <= 0 + 0 * 0, s44 >= 0, s44 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s45, z3) + s63 + 1 :|: s63 >= 0, s63 <= 0 + 0 * 0, s45 >= 0, s45 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 3 + 2*z16 + z16^2 }-> 1 + SHUFF(s46, z3) + s64 + 0 :|: s64 >= 0, s64 <= z16 + z16 * z16, s46 >= 0, s46 <= 0, z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s47, z3) + s65 + 0 :|: s65 >= 0, s65 <= 0 + 0 * 0, s47 >= 0, s47 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s48, z3) + s66 + 1 :|: s66 >= 0, s66 <= 0 + 0 * 0, s48 >= 0, s48 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 3 }-> 1 + s'' + s58 :|: s58 >= 0, s58 <= 0 + 0 * 0, s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 3 + 2*z1 + z1^2 }-> 1 + s1 + s60 :|: s60 >= 0, s60 <= z1 + z1 * z1, s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 6 + s35 + 4*z0' + 2*z0'*z1' + z0'^2 + 4*z1' + z1'^2 }-> 1 + s36 + s59 :|: s59 >= 0, s59 <= 1 + z0' + z1' + (1 + z0' + z1') * (1 + z0' + z1'), s34 >= 0, s34 <= z1', s35 >= 0, s35 <= s34 + (1 + z0' + 0), s36 >= 0, s36 <= s35, z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', s19) + 0 :|: s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s20) + 1 :|: s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s21) + 1 :|: s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s23) + 1 :|: s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s24) + 0 :|: s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', s22) + 1 :|: s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', s25) + s2 + s :|: s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s26) + s3 + 0 :|: s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s27) + s4 + 0 :|: s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s28) + s5 + 0 :|: s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s29) + s6 + 0 :|: s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', s33) + s10 + 0 :|: s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s30) + s7 + 0 :|: s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s31) + s8 + 0 :|: s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', s32) + s9 + 0 :|: s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> s55 :|: s55 >= 0, s55 <= s40 + s40 * s40 + z3, s40 >= 0, s40 <= z110, z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> s56 :|: s56 >= 0, s56 <= s41 + s41 * s41 + z3, s41 >= 0, s41 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> s57 :|: s57 >= 0, s57 <= s42 + s42 * s42 + z3, s42 >= 0, s42 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> s39 :|: s37 >= 0, s37 <= z17, s38 >= 0, s38 <= s37 + (1 + z07 + 0), s39 >= 0, s39 <= s38 + (1 + z0 + 0), z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> s50 :|: s50 >= 0, s50 <= 3 * s14 + 0 + 0 * 0 + z', s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> s51 :|: s51 >= 0, s51 <= 3 * s15 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s52 :|: s52 >= 0, s52 <= 3 * s16 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s53 :|: s53 >= 0, s53 <= 3 * s17 + (1 + z09 + z19) + (1 + z09 + z19) * (1 + z09 + z19) + z', s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 3 * s18 + z + z * z + z', s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> s49 :|: s49 >= 0, s49 <= z + z * z + 0, z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {IF,SHUFF}, {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: O(1) [0], size: O(1) [2]
  TAIL: runtime: O(1) [1], size: O(1) [1]
  reverse: runtime: O(1) [0], size: O(n^1) [z]
  shuff: runtime: O(1) [0], size: O(n^2) [z + z^2 + z']
  if: runtime: O(1) [0], size: O(n^2) [z' + z'^2 + z'' + 3*z3]
  REVERSE: runtime: O(n^2) [1 + 2*z + z^2], size: O(n^2) [z + z^2]
  shuffle: runtime: O(1) [0], size: O(n^2) [z + z^2]
  IF: runtime: ?, size: INF
  SHUFF: runtime: ?, size: INF

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

(95) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: IF
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 257 + 382*z' + 20*z'*z3 + 278*z'^2 + 122*z'^3 + 22*z3

Computed RUNTIME bound using KoAT for: SHUFF
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 12244 + 27938*z + 460*z*z' + 26834*z^2 + 10126*z^3 + 539*z'

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

(96)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 3 + 2*z15 + z15^2 }-> 1 + SHUFF(s43, z3) + s61 + 0 :|: s61 >= 0, s61 <= z15 + z15 * z15, s43 >= 0, s43 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s44, z3) + s62 + 0 :|: s62 >= 0, s62 <= 0 + 0 * 0, s44 >= 0, s44 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s45, z3) + s63 + 1 :|: s63 >= 0, s63 <= 0 + 0 * 0, s45 >= 0, s45 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 3 + 2*z16 + z16^2 }-> 1 + SHUFF(s46, z3) + s64 + 0 :|: s64 >= 0, s64 <= z16 + z16 * z16, s46 >= 0, s46 <= 0, z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s47, z3) + s65 + 0 :|: s65 >= 0, s65 <= 0 + 0 * 0, s47 >= 0, s47 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   IF(z, z', z'', z3) -{ 3 }-> 1 + SHUFF(s48, z3) + s66 + 1 :|: s66 >= 0, s66 <= 0 + 0 * 0, s48 >= 0, s48 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 3 }-> 1 + s'' + s58 :|: s58 >= 0, s58 <= 0 + 0 * 0, s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 3 + 2*z1 + z1^2 }-> 1 + s1 + s60 :|: s60 >= 0, s60 <= z1 + z1 * z1, s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 6 + s35 + 4*z0' + 2*z0'*z1' + z0'^2 + 4*z1' + z1'^2 }-> 1 + s36 + s59 :|: s59 >= 0, s59 <= 1 + z0' + z1' + (1 + z0' + z1') * (1 + z0' + z1'), s34 >= 0, s34 <= z1', s35 >= 0, s35 <= s34 + (1 + z0' + 0), s36 >= 0, s36 <= s35, z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(2, 0, z', s19) + 0 :|: s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s20) + 1 :|: s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(1, 1 + z0'' + z1'', z', s21) + 1 :|: s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s23) + 1 :|: s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, z, z', s24) + 0 :|: s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 2 }-> 1 + IF(0, 1 + z01 + z11, z', s22) + 1 :|: s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(2, 0, z', s25) + s2 + s :|: s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s26) + s3 + 0 :|: s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s27) + s4 + 0 :|: s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s28) + s5 + 0 :|: s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(1, 1 + z02 + z12, z', s29) + s6 + 0 :|: s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, z, z', s33) + s10 + 0 :|: s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s30) + s7 + 0 :|: s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z03 + z13, z', s31) + s8 + 0 :|: s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 3 + z' }-> 1 + IF(0, 1 + z04 + z14, z', s32) + s9 + 0 :|: s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFFLE(z) -{ 1 }-> 1 + SHUFF(z, 0) :|: z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> s55 :|: s55 >= 0, s55 <= s40 + s40 * s40 + z3, s40 >= 0, s40 <= z110, z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> s56 :|: s56 >= 0, s56 <= s41 + s41 * s41 + z3, s41 >= 0, s41 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> s57 :|: s57 >= 0, s57 <= s42 + s42 * s42 + z3, s42 >= 0, s42 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> s39 :|: s37 >= 0, s37 <= z17, s38 >= 0, s38 <= s37 + (1 + z07 + 0), s39 >= 0, s39 <= s38 + (1 + z0 + 0), z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> s50 :|: s50 >= 0, s50 <= 3 * s14 + 0 + 0 * 0 + z', s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> s51 :|: s51 >= 0, s51 <= 3 * s15 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s52 :|: s52 >= 0, s52 <= 3 * s16 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s53 :|: s53 >= 0, s53 <= 3 * s17 + (1 + z09 + z19) + (1 + z09 + z19) * (1 + z09 + z19) + z', s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 3 * s18 + z + z * z + z', s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> s49 :|: s49 >= 0, s49 <= z + z * z + 0, z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: O(1) [0], size: O(1) [2]
  TAIL: runtime: O(1) [1], size: O(1) [1]
  reverse: runtime: O(1) [0], size: O(n^1) [z]
  shuff: runtime: O(1) [0], size: O(n^2) [z + z^2 + z']
  if: runtime: O(1) [0], size: O(n^2) [z' + z'^2 + z'' + 3*z3]
  REVERSE: runtime: O(n^2) [1 + 2*z + z^2], size: O(n^2) [z + z^2]
  shuffle: runtime: O(1) [0], size: O(n^2) [z + z^2]
  IF: runtime: O(n^3) [257 + 382*z' + 20*z'*z3 + 278*z'^2 + 122*z'^3 + 22*z3], size: INF
  SHUFF: runtime: O(n^3) [12244 + 27938*z + 460*z*z' + 26834*z^2 + 10126*z^3 + 539*z'], size: INF

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

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

(98)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 12247 + 27938*s43 + 460*s43*z3 + 26834*s43^2 + 10126*s43^3 + 2*z15 + z15^2 + 539*z3 }-> 1 + s74 + s61 + 0 :|: s74 >= 0, s74 <= inf5, s61 >= 0, s61 <= z15 + z15 * z15, s43 >= 0, s43 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 12247 + 27938*s44 + 460*s44*z3 + 26834*s44^2 + 10126*s44^3 + 539*z3 }-> 1 + s75 + s62 + 0 :|: s75 >= 0, s75 <= inf6, s62 >= 0, s62 <= 0 + 0 * 0, s44 >= 0, s44 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 12247 + 27938*s45 + 460*s45*z3 + 26834*s45^2 + 10126*s45^3 + 539*z3 }-> 1 + s76 + s63 + 1 :|: s76 >= 0, s76 <= inf7, s63 >= 0, s63 <= 0 + 0 * 0, s45 >= 0, s45 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 12247 + 27938*s46 + 460*s46*z3 + 26834*s46^2 + 10126*s46^3 + 2*z16 + z16^2 + 539*z3 }-> 1 + s77 + s64 + 0 :|: s77 >= 0, s77 <= inf8, s64 >= 0, s64 <= z16 + z16 * z16, s46 >= 0, s46 <= 0, z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 12247 + 27938*s47 + 460*s47*z3 + 26834*s47^2 + 10126*s47^3 + 539*z3 }-> 1 + s78 + s65 + 0 :|: s78 >= 0, s78 <= inf9, s65 >= 0, s65 <= 0 + 0 * 0, s47 >= 0, s47 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   IF(z, z', z'', z3) -{ 12247 + 27938*s48 + 460*s48*z3 + 26834*s48^2 + 10126*s48^3 + 539*z3 }-> 1 + s79 + s66 + 1 :|: s79 >= 0, s79 <= inf10, s66 >= 0, s66 <= 0 + 0 * 0, s48 >= 0, s48 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 3 }-> 1 + s'' + s58 :|: s58 >= 0, s58 <= 0 + 0 * 0, s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 3 + 2*z1 + z1^2 }-> 1 + s1 + s60 :|: s60 >= 0, s60 <= z1 + z1 * z1, s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 6 + s35 + 4*z0' + 2*z0'*z1' + z0'^2 + 4*z1' + z1'^2 }-> 1 + s36 + s59 :|: s59 >= 0, s59 <= 1 + z0' + z1' + (1 + z0' + z1') * (1 + z0' + z1'), s34 >= 0, s34 <= z1', s35 >= 0, s35 <= s34 + (1 + z0' + 0), s36 >= 0, s36 <= s35, z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 259 + 22*s19 }-> 1 + s68 + 0 :|: s68 >= 0, s68 <= inf', s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 1041 + 42*s20 + 20*s20*z0'' + 20*s20*z1'' + 1304*z0'' + 1288*z0''*z1'' + 366*z0''*z1''^2 + 644*z0''^2 + 366*z0''^2*z1'' + 122*z0''^3 + 1304*z1'' + 644*z1''^2 + 122*z1''^3 }-> 1 + s69 + 1 :|: s69 >= 0, s69 <= inf'', s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 1041 + 42*s21 + 20*s21*z0'' + 20*s21*z1'' + 1304*z0'' + 1288*z0''*z1'' + 366*z0''*z1''^2 + 644*z0''^2 + 366*z0''^2*z1'' + 122*z0''^3 + 1304*z1'' + 644*z1''^2 + 122*z1''^3 }-> 1 + s70 + 1 :|: s70 >= 0, s70 <= inf1, s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 1041 + 42*s22 + 20*s22*z01 + 20*s22*z11 + 1304*z01 + 1288*z01*z11 + 366*z01*z11^2 + 644*z01^2 + 366*z01^2*z11 + 122*z01^3 + 1304*z11 + 644*z11^2 + 122*z11^3 }-> 1 + s71 + 1 :|: s71 >= 0, s71 <= inf2, s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 259 + 22*s23 + 20*s23*z + 382*z + 278*z^2 + 122*z^3 }-> 1 + s72 + 1 :|: s72 >= 0, s72 <= inf3, s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 259 + 22*s24 + 20*s24*z + 382*z + 278*z^2 + 122*z^3 }-> 1 + s73 + 0 :|: s73 >= 0, s73 <= inf4, s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 260 + 22*s25 + z' }-> 1 + s80 + s2 + s :|: s80 >= 0, s80 <= inf11, s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 1042 + 42*s26 + 20*s26*z02 + 20*s26*z12 + z' + 1304*z02 + 1288*z02*z12 + 366*z02*z12^2 + 644*z02^2 + 366*z02^2*z12 + 122*z02^3 + 1304*z12 + 644*z12^2 + 122*z12^3 }-> 1 + s81 + s3 + 0 :|: s81 >= 0, s81 <= inf12, s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1042 + 42*s27 + 20*s27*z02 + 20*s27*z12 + z' + 1304*z02 + 1288*z02*z12 + 366*z02*z12^2 + 644*z02^2 + 366*z02^2*z12 + 122*z02^3 + 1304*z12 + 644*z12^2 + 122*z12^3 }-> 1 + s82 + s4 + 0 :|: s82 >= 0, s82 <= inf13, s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1042 + 42*s28 + 20*s28*z02 + 20*s28*z12 + z' + 1304*z02 + 1288*z02*z12 + 366*z02*z12^2 + 644*z02^2 + 366*z02^2*z12 + 122*z02^3 + 1304*z12 + 644*z12^2 + 122*z12^3 }-> 1 + s83 + s5 + 0 :|: s83 >= 0, s83 <= inf14, s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1042 + 42*s29 + 20*s29*z02 + 20*s29*z12 + z' + 1304*z02 + 1288*z02*z12 + 366*z02*z12^2 + 644*z02^2 + 366*z02^2*z12 + 122*z02^3 + 1304*z12 + 644*z12^2 + 122*z12^3 }-> 1 + s84 + s6 + 0 :|: s84 >= 0, s84 <= inf15, s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1042 + 42*s30 + 20*s30*z03 + 20*s30*z13 + z' + 1304*z03 + 1288*z03*z13 + 366*z03*z13^2 + 644*z03^2 + 366*z03^2*z13 + 122*z03^3 + 1304*z13 + 644*z13^2 + 122*z13^3 }-> 1 + s85 + s7 + 0 :|: s85 >= 0, s85 <= inf16, s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1042 + 42*s31 + 20*s31*z03 + 20*s31*z13 + z' + 1304*z03 + 1288*z03*z13 + 366*z03*z13^2 + 644*z03^2 + 366*z03^2*z13 + 122*z03^3 + 1304*z13 + 644*z13^2 + 122*z13^3 }-> 1 + s86 + s8 + 0 :|: s86 >= 0, s86 <= inf17, s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1042 + 42*s32 + 20*s32*z04 + 20*s32*z14 + z' + 1304*z04 + 1288*z04*z14 + 366*z04*z14^2 + 644*z04^2 + 366*z04^2*z14 + 122*z04^3 + 1304*z14 + 644*z14^2 + 122*z14^3 }-> 1 + s87 + s9 + 0 :|: s87 >= 0, s87 <= inf18, s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFF(z, z') -{ 260 + 22*s33 + 20*s33*z + 382*z + 278*z^2 + 122*z^3 + z' }-> 1 + s88 + s10 + 0 :|: s88 >= 0, s88 <= inf19, s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFFLE(z) -{ 12245 + 27938*z + 26834*z^2 + 10126*z^3 }-> 1 + s67 :|: s67 >= 0, s67 <= inf, z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> s55 :|: s55 >= 0, s55 <= s40 + s40 * s40 + z3, s40 >= 0, s40 <= z110, z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> s56 :|: s56 >= 0, s56 <= s41 + s41 * s41 + z3, s41 >= 0, s41 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> s57 :|: s57 >= 0, s57 <= s42 + s42 * s42 + z3, s42 >= 0, s42 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> s39 :|: s37 >= 0, s37 <= z17, s38 >= 0, s38 <= s37 + (1 + z07 + 0), s39 >= 0, s39 <= s38 + (1 + z0 + 0), z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> s50 :|: s50 >= 0, s50 <= 3 * s14 + 0 + 0 * 0 + z', s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> s51 :|: s51 >= 0, s51 <= 3 * s15 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s52 :|: s52 >= 0, s52 <= 3 * s16 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s53 :|: s53 >= 0, s53 <= 3 * s17 + (1 + z09 + z19) + (1 + z09 + z19) * (1 + z09 + z19) + z', s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 3 * s18 + z + z * z + z', s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> s49 :|: s49 >= 0, s49 <= z + z * z + 0, z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: O(1) [0], size: O(1) [2]
  TAIL: runtime: O(1) [1], size: O(1) [1]
  reverse: runtime: O(1) [0], size: O(n^1) [z]
  shuff: runtime: O(1) [0], size: O(n^2) [z + z^2 + z']
  if: runtime: O(1) [0], size: O(n^2) [z' + z'^2 + z'' + 3*z3]
  REVERSE: runtime: O(n^2) [1 + 2*z + z^2], size: O(n^2) [z + z^2]
  shuffle: runtime: O(1) [0], size: O(n^2) [z + z^2]
  IF: runtime: O(n^3) [257 + 382*z' + 20*z'*z3 + 278*z'^2 + 122*z'^3 + 22*z3], size: INF
  SHUFF: runtime: O(n^3) [12244 + 27938*z + 460*z*z' + 26834*z^2 + 10126*z^3 + 539*z'], size: INF

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

(99) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(100)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 12247 + 27938*s43 + 460*s43*z3 + 26834*s43^2 + 10126*s43^3 + 2*z15 + z15^2 + 539*z3 }-> 1 + s74 + s61 + 0 :|: s74 >= 0, s74 <= inf5, s61 >= 0, s61 <= z15 + z15 * z15, s43 >= 0, s43 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 12247 + 27938*s44 + 460*s44*z3 + 26834*s44^2 + 10126*s44^3 + 539*z3 }-> 1 + s75 + s62 + 0 :|: s75 >= 0, s75 <= inf6, s62 >= 0, s62 <= 0 + 0 * 0, s44 >= 0, s44 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 12247 + 27938*s45 + 460*s45*z3 + 26834*s45^2 + 10126*s45^3 + 539*z3 }-> 1 + s76 + s63 + 1 :|: s76 >= 0, s76 <= inf7, s63 >= 0, s63 <= 0 + 0 * 0, s45 >= 0, s45 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 12247 + 27938*s46 + 460*s46*z3 + 26834*s46^2 + 10126*s46^3 + 2*z16 + z16^2 + 539*z3 }-> 1 + s77 + s64 + 0 :|: s77 >= 0, s77 <= inf8, s64 >= 0, s64 <= z16 + z16 * z16, s46 >= 0, s46 <= 0, z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 12247 + 27938*s47 + 460*s47*z3 + 26834*s47^2 + 10126*s47^3 + 539*z3 }-> 1 + s78 + s65 + 0 :|: s78 >= 0, s78 <= inf9, s65 >= 0, s65 <= 0 + 0 * 0, s47 >= 0, s47 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   IF(z, z', z'', z3) -{ 12247 + 27938*s48 + 460*s48*z3 + 26834*s48^2 + 10126*s48^3 + 539*z3 }-> 1 + s79 + s66 + 1 :|: s79 >= 0, s79 <= inf10, s66 >= 0, s66 <= 0 + 0 * 0, s48 >= 0, s48 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 3 }-> 1 + s'' + s58 :|: s58 >= 0, s58 <= 0 + 0 * 0, s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 3 + 2*z1 + z1^2 }-> 1 + s1 + s60 :|: s60 >= 0, s60 <= z1 + z1 * z1, s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 6 + s35 + 4*z0' + 2*z0'*z1' + z0'^2 + 4*z1' + z1'^2 }-> 1 + s36 + s59 :|: s59 >= 0, s59 <= 1 + z0' + z1' + (1 + z0' + z1') * (1 + z0' + z1'), s34 >= 0, s34 <= z1', s35 >= 0, s35 <= s34 + (1 + z0' + 0), s36 >= 0, s36 <= s35, z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 259 + 22*s19 }-> 1 + s68 + 0 :|: s68 >= 0, s68 <= inf', s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 1041 + 42*s20 + 20*s20*z0'' + 20*s20*z1'' + 1304*z0'' + 1288*z0''*z1'' + 366*z0''*z1''^2 + 644*z0''^2 + 366*z0''^2*z1'' + 122*z0''^3 + 1304*z1'' + 644*z1''^2 + 122*z1''^3 }-> 1 + s69 + 1 :|: s69 >= 0, s69 <= inf'', s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 1041 + 42*s21 + 20*s21*z0'' + 20*s21*z1'' + 1304*z0'' + 1288*z0''*z1'' + 366*z0''*z1''^2 + 644*z0''^2 + 366*z0''^2*z1'' + 122*z0''^3 + 1304*z1'' + 644*z1''^2 + 122*z1''^3 }-> 1 + s70 + 1 :|: s70 >= 0, s70 <= inf1, s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 1041 + 42*s22 + 20*s22*z01 + 20*s22*z11 + 1304*z01 + 1288*z01*z11 + 366*z01*z11^2 + 644*z01^2 + 366*z01^2*z11 + 122*z01^3 + 1304*z11 + 644*z11^2 + 122*z11^3 }-> 1 + s71 + 1 :|: s71 >= 0, s71 <= inf2, s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 259 + 22*s23 + 20*s23*z + 382*z + 278*z^2 + 122*z^3 }-> 1 + s72 + 1 :|: s72 >= 0, s72 <= inf3, s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 259 + 22*s24 + 20*s24*z + 382*z + 278*z^2 + 122*z^3 }-> 1 + s73 + 0 :|: s73 >= 0, s73 <= inf4, s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 260 + 22*s25 + z' }-> 1 + s80 + s2 + s :|: s80 >= 0, s80 <= inf11, s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 1042 + 42*s26 + 20*s26*z02 + 20*s26*z12 + z' + 1304*z02 + 1288*z02*z12 + 366*z02*z12^2 + 644*z02^2 + 366*z02^2*z12 + 122*z02^3 + 1304*z12 + 644*z12^2 + 122*z12^3 }-> 1 + s81 + s3 + 0 :|: s81 >= 0, s81 <= inf12, s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1042 + 42*s27 + 20*s27*z02 + 20*s27*z12 + z' + 1304*z02 + 1288*z02*z12 + 366*z02*z12^2 + 644*z02^2 + 366*z02^2*z12 + 122*z02^3 + 1304*z12 + 644*z12^2 + 122*z12^3 }-> 1 + s82 + s4 + 0 :|: s82 >= 0, s82 <= inf13, s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1042 + 42*s28 + 20*s28*z02 + 20*s28*z12 + z' + 1304*z02 + 1288*z02*z12 + 366*z02*z12^2 + 644*z02^2 + 366*z02^2*z12 + 122*z02^3 + 1304*z12 + 644*z12^2 + 122*z12^3 }-> 1 + s83 + s5 + 0 :|: s83 >= 0, s83 <= inf14, s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1042 + 42*s29 + 20*s29*z02 + 20*s29*z12 + z' + 1304*z02 + 1288*z02*z12 + 366*z02*z12^2 + 644*z02^2 + 366*z02^2*z12 + 122*z02^3 + 1304*z12 + 644*z12^2 + 122*z12^3 }-> 1 + s84 + s6 + 0 :|: s84 >= 0, s84 <= inf15, s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1042 + 42*s30 + 20*s30*z03 + 20*s30*z13 + z' + 1304*z03 + 1288*z03*z13 + 366*z03*z13^2 + 644*z03^2 + 366*z03^2*z13 + 122*z03^3 + 1304*z13 + 644*z13^2 + 122*z13^3 }-> 1 + s85 + s7 + 0 :|: s85 >= 0, s85 <= inf16, s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1042 + 42*s31 + 20*s31*z03 + 20*s31*z13 + z' + 1304*z03 + 1288*z03*z13 + 366*z03*z13^2 + 644*z03^2 + 366*z03^2*z13 + 122*z03^3 + 1304*z13 + 644*z13^2 + 122*z13^3 }-> 1 + s86 + s8 + 0 :|: s86 >= 0, s86 <= inf17, s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1042 + 42*s32 + 20*s32*z04 + 20*s32*z14 + z' + 1304*z04 + 1288*z04*z14 + 366*z04*z14^2 + 644*z04^2 + 366*z04^2*z14 + 122*z04^3 + 1304*z14 + 644*z14^2 + 122*z14^3 }-> 1 + s87 + s9 + 0 :|: s87 >= 0, s87 <= inf18, s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFF(z, z') -{ 260 + 22*s33 + 20*s33*z + 382*z + 278*z^2 + 122*z^3 + z' }-> 1 + s88 + s10 + 0 :|: s88 >= 0, s88 <= inf19, s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFFLE(z) -{ 12245 + 27938*z + 26834*z^2 + 10126*z^3 }-> 1 + s67 :|: s67 >= 0, s67 <= inf, z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> s55 :|: s55 >= 0, s55 <= s40 + s40 * s40 + z3, s40 >= 0, s40 <= z110, z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> s56 :|: s56 >= 0, s56 <= s41 + s41 * s41 + z3, s41 >= 0, s41 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> s57 :|: s57 >= 0, s57 <= s42 + s42 * s42 + z3, s42 >= 0, s42 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> s39 :|: s37 >= 0, s37 <= z17, s38 >= 0, s38 <= s37 + (1 + z07 + 0), s39 >= 0, s39 <= s38 + (1 + z0 + 0), z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> s50 :|: s50 >= 0, s50 <= 3 * s14 + 0 + 0 * 0 + z', s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> s51 :|: s51 >= 0, s51 <= 3 * s15 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s52 :|: s52 >= 0, s52 <= 3 * s16 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s53 :|: s53 >= 0, s53 <= 3 * s17 + (1 + z09 + z19) + (1 + z09 + z19) * (1 + z09 + z19) + z', s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 3 * s18 + z + z * z + z', s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> s49 :|: s49 >= 0, s49 <= z + z * z + 0, z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {SHUFFLE}
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: O(1) [0], size: O(1) [2]
  TAIL: runtime: O(1) [1], size: O(1) [1]
  reverse: runtime: O(1) [0], size: O(n^1) [z]
  shuff: runtime: O(1) [0], size: O(n^2) [z + z^2 + z']
  if: runtime: O(1) [0], size: O(n^2) [z' + z'^2 + z'' + 3*z3]
  REVERSE: runtime: O(n^2) [1 + 2*z + z^2], size: O(n^2) [z + z^2]
  shuffle: runtime: O(1) [0], size: O(n^2) [z + z^2]
  IF: runtime: O(n^3) [257 + 382*z' + 20*z'*z3 + 278*z'^2 + 122*z'^3 + 22*z3], size: INF
  SHUFF: runtime: O(n^3) [12244 + 27938*z + 460*z*z' + 26834*z^2 + 10126*z^3 + 539*z'], size: INF
  SHUFFLE: runtime: ?, size: INF

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

(101) 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: 12245 + 27938*z + 26834*z^2 + 10126*z^3

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

(102)
Obligation:
Complexity RNTS consisting of the following rules:

   APP(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   APP(z, z') -{ 2 + z1 }-> 1 + s' :|: s' >= 0, s' <= z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   HEAD(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   IF(z, z', z'', z3) -{ 1 }-> 0 :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   IF(z, z', z'', z3) -{ 12247 + 27938*s43 + 460*s43*z3 + 26834*s43^2 + 10126*s43^3 + 2*z15 + z15^2 + 539*z3 }-> 1 + s74 + s61 + 0 :|: s74 >= 0, s74 <= inf5, s61 >= 0, s61 <= z15 + z15 * z15, s43 >= 0, s43 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 12247 + 27938*s44 + 460*s44*z3 + 26834*s44^2 + 10126*s44^3 + 539*z3 }-> 1 + s75 + s62 + 0 :|: s75 >= 0, s75 <= inf6, s62 >= 0, s62 <= 0 + 0 * 0, s44 >= 0, s44 <= z15, z15 >= 0, z'' >= 0, z = 1, z' = 1 + z05 + z15, z05 >= 0, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z05 + z15 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 12247 + 27938*s45 + 460*s45*z3 + 26834*s45^2 + 10126*s45^3 + 539*z3 }-> 1 + s76 + s63 + 1 :|: s76 >= 0, s76 <= inf7, s63 >= 0, s63 <= 0 + 0 * 0, s45 >= 0, s45 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0, 0 = 0
   IF(z, z', z'', z3) -{ 12247 + 27938*s46 + 460*s46*z3 + 26834*s46^2 + 10126*s46^3 + 2*z16 + z16^2 + 539*z3 }-> 1 + s77 + s64 + 0 :|: s77 >= 0, s77 <= inf8, s64 >= 0, s64 <= z16 + z16 * z16, s46 >= 0, s46 <= 0, z'' >= 0, z = 1, z06 >= 0, z16 >= 0, z' = 1 + z06 + z16, z3 >= 0, z1' >= 0, z0 >= 0, 1 + z06 + z16 = 1 + z0 + z1'
   IF(z, z', z'', z3) -{ 12247 + 27938*s47 + 460*s47*z3 + 26834*s47^2 + 10126*s47^3 + 539*z3 }-> 1 + s78 + s65 + 0 :|: s78 >= 0, s78 <= inf9, s65 >= 0, s65 <= 0 + 0 * 0, s47 >= 0, s47 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z1' >= 0, z0' >= 0, z' = 1 + z0' + z1'
   IF(z, z', z'', z3) -{ 12247 + 27938*s48 + 460*s48*z3 + 26834*s48^2 + 10126*s48^3 + 539*z3 }-> 1 + s79 + s66 + 1 :|: s79 >= 0, s79 <= inf10, s66 >= 0, s66 <= 0 + 0 * 0, s48 >= 0, s48 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0, z' = 0
   NULL(z) -{ 1 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   NULL(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 1 }-> 0 :|: z = 0
   REVERSE(z) -{ 3 }-> 1 + s'' + s58 :|: s58 >= 0, s58 <= 0 + 0 * 0, s'' >= 0, s'' <= 0, z - 1 >= 0
   REVERSE(z) -{ 3 + 2*z1 + z1^2 }-> 1 + s1 + s60 :|: s60 >= 0, s60 <= z1 + z1 * z1, s1 >= 0, s1 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   REVERSE(z) -{ 6 + s35 + 4*z0' + 2*z0'*z1' + z0'^2 + 4*z1' + z1'^2 }-> 1 + s36 + s59 :|: s59 >= 0, s59 <= 1 + z0' + z1' + (1 + z0' + z1') * (1 + z0' + z1'), s34 >= 0, s34 <= z1', s35 >= 0, s35 <= s34 + (1 + z0' + 0), s36 >= 0, s36 <= s35, z = 1 + z0 + (1 + z0' + z1'), z0' >= 0, z1' >= 0, z0 >= 0
   SHUFF(z, z') -{ 259 + 22*s19 }-> 1 + s68 + 0 :|: s68 >= 0, s68 <= inf', s19 >= 0, s19 <= z' + (1 + 0 + 0), z' >= 0, z = 0, 0 = 0
   SHUFF(z, z') -{ 1041 + 42*s20 + 20*s20*z0'' + 20*s20*z1'' + 1304*z0'' + 1288*z0''*z1'' + 366*z0''*z1''^2 + 644*z0''^2 + 366*z0''^2*z1'' + 122*z0''^3 + 1304*z1'' + 644*z1''^2 + 122*z1''^3 }-> 1 + s69 + 1 :|: s69 >= 0, s69 <= inf'', s20 >= 0, s20 <= z' + (1 + z0'' + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 1041 + 42*s21 + 20*s21*z0'' + 20*s21*z1'' + 1304*z0'' + 1288*z0''*z1'' + 366*z0''*z1''^2 + 644*z0''^2 + 366*z0''^2*z1'' + 122*z0''^3 + 1304*z1'' + 644*z1''^2 + 122*z1''^3 }-> 1 + s70 + 1 :|: s70 >= 0, s70 <= inf1, s21 >= 0, s21 <= z' + (1 + 0 + 0), z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z1'' >= 0, z1' >= 0, z0 >= 0, 1 + z0'' + z1'' = 1 + z0 + z1'
   SHUFF(z, z') -{ 1041 + 42*s22 + 20*s22*z01 + 20*s22*z11 + 1304*z01 + 1288*z01*z11 + 366*z01*z11^2 + 644*z01^2 + 366*z01^2*z11 + 122*z01^3 + 1304*z11 + 644*z11^2 + 122*z11^3 }-> 1 + s71 + 1 :|: s71 >= 0, s71 <= inf2, s22 >= 0, s22 <= z' + (1 + z01 + 0), z11 >= 0, z' >= 0, z01 >= 0, z = 1 + z01 + z11, z1' >= 0, z0 >= 0, 1 + z01 + z11 = 1 + z0 + z1'
   SHUFF(z, z') -{ 259 + 22*s23 + 20*s23*z + 382*z + 278*z^2 + 122*z^3 }-> 1 + s72 + 1 :|: s72 >= 0, s72 <= inf3, s23 >= 0, s23 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFF(z, z') -{ 259 + 22*s24 + 20*s24*z + 382*z + 278*z^2 + 122*z^3 }-> 1 + s73 + 0 :|: s73 >= 0, s73 <= inf4, s24 >= 0, s24 <= z' + (1 + 0 + 0), z' >= 0, z >= 0, z = 0
   SHUFF(z, z') -{ 260 + 22*s25 + z' }-> 1 + s80 + s2 + s :|: s80 >= 0, s80 <= inf11, s25 >= 0, s25 <= z' + (1 + 0 + 0), s2 >= 0, s2 <= z', s >= 0, s <= 0, z' >= 0, z = 0
   SHUFF(z, z') -{ 1042 + 42*s26 + 20*s26*z02 + 20*s26*z12 + z' + 1304*z02 + 1288*z02*z12 + 366*z02*z12^2 + 644*z02^2 + 366*z02^2*z12 + 122*z02^3 + 1304*z12 + 644*z12^2 + 122*z12^3 }-> 1 + s81 + s3 + 0 :|: s81 >= 0, s81 <= inf12, s26 >= 0, s26 <= z' + (1 + z02 + 0), s3 >= 0, s3 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1042 + 42*s27 + 20*s27*z02 + 20*s27*z12 + z' + 1304*z02 + 1288*z02*z12 + 366*z02*z12^2 + 644*z02^2 + 366*z02^2*z12 + 122*z02^3 + 1304*z12 + 644*z12^2 + 122*z12^3 }-> 1 + s82 + s4 + 0 :|: s82 >= 0, s82 <= inf13, s27 >= 0, s27 <= z' + (1 + z02 + 0), s4 >= 0, s4 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1042 + 42*s28 + 20*s28*z02 + 20*s28*z12 + z' + 1304*z02 + 1288*z02*z12 + 366*z02*z12^2 + 644*z02^2 + 366*z02^2*z12 + 122*z02^3 + 1304*z12 + 644*z12^2 + 122*z12^3 }-> 1 + s83 + s5 + 0 :|: s83 >= 0, s83 <= inf14, s28 >= 0, s28 <= z' + (1 + 0 + 0), s5 >= 0, s5 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1042 + 42*s29 + 20*s29*z02 + 20*s29*z12 + z' + 1304*z02 + 1288*z02*z12 + 366*z02*z12^2 + 644*z02^2 + 366*z02^2*z12 + 122*z02^3 + 1304*z12 + 644*z12^2 + 122*z12^3 }-> 1 + s84 + s6 + 0 :|: s84 >= 0, s84 <= inf15, s29 >= 0, s29 <= z' + (1 + 0 + 0), s6 >= 0, s6 <= z', z = 1 + z02 + z12, z' >= 0, z02 >= 0, z12 >= 0, z1' >= 0, z0 >= 0, 1 + z02 + z12 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1042 + 42*s30 + 20*s30*z03 + 20*s30*z13 + z' + 1304*z03 + 1288*z03*z13 + 366*z03*z13^2 + 644*z03^2 + 366*z03^2*z13 + 122*z03^3 + 1304*z13 + 644*z13^2 + 122*z13^3 }-> 1 + s85 + s7 + 0 :|: s85 >= 0, s85 <= inf16, s30 >= 0, s30 <= z' + (1 + z03 + 0), s7 >= 0, s7 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1042 + 42*s31 + 20*s31*z03 + 20*s31*z13 + z' + 1304*z03 + 1288*z03*z13 + 366*z03*z13^2 + 644*z03^2 + 366*z03^2*z13 + 122*z03^3 + 1304*z13 + 644*z13^2 + 122*z13^3 }-> 1 + s86 + s8 + 0 :|: s86 >= 0, s86 <= inf17, s31 >= 0, s31 <= z' + (1 + z03 + 0), s8 >= 0, s8 <= z', z = 1 + z03 + z13, z' >= 0, z03 >= 0, z13 >= 0, z1' >= 0, z0 >= 0, 1 + z03 + z13 = 1 + z0 + z1'
   SHUFF(z, z') -{ 1042 + 42*s32 + 20*s32*z04 + 20*s32*z14 + z' + 1304*z04 + 1288*z04*z14 + 366*z04*z14^2 + 644*z04^2 + 366*z04^2*z14 + 122*z04^3 + 1304*z14 + 644*z14^2 + 122*z14^3 }-> 1 + s87 + s9 + 0 :|: s87 >= 0, s87 <= inf18, s32 >= 0, s32 <= z' + (1 + 0 + 0), s9 >= 0, s9 <= z', z04 >= 0, z = 1 + z04 + z14, z' >= 0, z14 >= 0, z1' >= 0, z0 >= 0, 1 + z04 + z14 = 1 + z0 + z1'
   SHUFF(z, z') -{ 260 + 22*s33 + 20*s33*z + 382*z + 278*z^2 + 122*z^3 + z' }-> 1 + s88 + s10 + 0 :|: s88 >= 0, s88 <= inf19, s33 >= 0, s33 <= z' + (1 + 0 + 0), s10 >= 0, s10 <= z', z' >= 0, z >= 0, z1' >= 0, z0' >= 0, z = 1 + z0' + z1'
   SHUFFLE(z) -{ 12245 + 27938*z + 26834*z^2 + 10126*z^3 }-> 1 + s67 :|: s67 >= 0, s67 <= inf, z >= 0
   TAIL(z) -{ 1 }-> 1 :|: z = 0
   TAIL(z) -{ 1 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   app(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0
   app(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   app(z, z') -{ 0 }-> 1 + z0 + s11 :|: s11 >= 0, s11 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   head(z) -{ 0 }-> z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   head(z) -{ 0 }-> 0 :|: z >= 0
   if(z, z', z'', z3) -{ 0 }-> s55 :|: s55 >= 0, s55 <= s40 + s40 * s40 + z3, s40 >= 0, s40 <= z110, z110 >= 0, z'' >= 0, z = 1, z' = 1 + z010 + z110, z010 >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> s56 :|: s56 >= 0, s56 <= s41 + s41 * s41 + z3, s41 >= 0, s41 <= 0, z'' >= 0, z = 1, z3 >= 0, z' = 0
   if(z, z', z'', z3) -{ 0 }-> s57 :|: s57 >= 0, s57 <= s42 + s42 * s42 + z3, s42 >= 0, s42 <= 0, z'' >= 0, z = 1, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> z'' :|: z = 2, z'' >= 0, z' >= 0, z3 >= 0
   if(z, z', z'', z3) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z3 >= 0
   null(z) -{ 0 }-> 2 :|: z = 0
   null(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   null(z) -{ 0 }-> 0 :|: z >= 0
   reverse(z) -{ 0 }-> s12 :|: s12 >= 0, s12 <= 0 + (1 + (z - 1) + 0), z - 1 >= 0
   reverse(z) -{ 0 }-> s13 :|: s13 >= 0, s13 <= 0 + (1 + z0 + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   reverse(z) -{ 0 }-> s39 :|: s37 >= 0, s37 <= z17, s38 >= 0, s38 <= s37 + (1 + z07 + 0), s39 >= 0, s39 <= s38 + (1 + z0 + 0), z = 1 + z0 + (1 + z07 + z17), z07 >= 0, z17 >= 0, z0 >= 0
   reverse(z) -{ 0 }-> 0 :|: z = 0
   reverse(z) -{ 0 }-> 0 :|: z >= 0
   shuff(z, z') -{ 0 }-> s50 :|: s50 >= 0, s50 <= 3 * s14 + 0 + 0 * 0 + z', s14 >= 0, s14 <= z' + (1 + 0 + 0), z' >= 0, z = 0
   shuff(z, z') -{ 0 }-> s51 :|: s51 >= 0, s51 <= 3 * s15 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s15 >= 0, s15 <= z' + (1 + z08 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s52 :|: s52 >= 0, s52 <= 3 * s16 + (1 + z08 + z18) + (1 + z08 + z18) * (1 + z08 + z18) + z', s16 >= 0, s16 <= z' + (1 + 0 + 0), z = 1 + z08 + z18, z08 >= 0, z18 >= 0, z' >= 0
   shuff(z, z') -{ 0 }-> s53 :|: s53 >= 0, s53 <= 3 * s17 + (1 + z09 + z19) + (1 + z09 + z19) * (1 + z09 + z19) + z', s17 >= 0, s17 <= z' + (1 + z09 + 0), z = 1 + z09 + z19, z' >= 0, z19 >= 0, z09 >= 0
   shuff(z, z') -{ 0 }-> s54 :|: s54 >= 0, s54 <= 3 * s18 + z + z * z + z', s18 >= 0, s18 <= z' + (1 + 0 + 0), z' >= 0, z >= 0
   shuff(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   shuffle(z) -{ 0 }-> s49 :|: s49 >= 0, s49 <= z + z * z + 0, z >= 0
   shuffle(z) -{ 0 }-> 0 :|: z >= 0
   tail(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tail(z) -{ 0 }-> 0 :|: z = 0
   tail(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  HEAD: runtime: O(1) [1], size: O(1) [0]
  tail: runtime: O(1) [0], size: O(n^1) [z]
  APP: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  head: runtime: O(1) [0], size: O(n^1) [z]
  NULL: runtime: O(1) [1], size: O(1) [1]
  app: runtime: O(1) [0], size: O(n^1) [z + z']
  null: runtime: O(1) [0], size: O(1) [2]
  TAIL: runtime: O(1) [1], size: O(1) [1]
  reverse: runtime: O(1) [0], size: O(n^1) [z]
  shuff: runtime: O(1) [0], size: O(n^2) [z + z^2 + z']
  if: runtime: O(1) [0], size: O(n^2) [z' + z'^2 + z'' + 3*z3]
  REVERSE: runtime: O(n^2) [1 + 2*z + z^2], size: O(n^2) [z + z^2]
  shuffle: runtime: O(1) [0], size: O(n^2) [z + z^2]
  IF: runtime: O(n^3) [257 + 382*z' + 20*z'*z3 + 278*z'^2 + 122*z'^3 + 22*z3], size: INF
  SHUFF: runtime: O(n^3) [12244 + 27938*z + 460*z*z' + 26834*z^2 + 10126*z^3 + 539*z'], size: INF
  SHUFFLE: runtime: O(n^3) [12245 + 27938*z + 26834*z^2 + 10126*z^3], size: INF

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

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

(104)
BOUNDS(1, n^3)

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

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

(106)
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
----------------------------------------

(107) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
IF/2

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

(108)
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, app(z1, add(head(z0), nil))), NULL(z0))
   SHUFF(z0, z1) -> c11(IF(null(z0), z0, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
   IF(true, z0, z2) -> c12
   IF(false, z0, 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
----------------------------------------

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

(110)
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, app(z1, add(head(z0), nil))), NULL(z0))
SHUFF(z0, z1) -> c11(IF(null(z0), z0, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
IF(true, z0, z2) -> c12
IF(false, z0, 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 -> 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

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

(111) 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

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

(112)
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, app(z1, add(head(z0), nil))), NULL(z0))
SHUFF(z0, z1) -> c11(IF(null(z0), z0, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
IF(true, z0, z2) -> c12
IF(false, z0, 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 -> 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

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

(113) 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).
----------------------------------------

(114)
Complex Obligation (BEST)

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

(115)
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, app(z1, add(head(z0), nil))), NULL(z0))
SHUFF(z0, z1) -> c11(IF(null(z0), z0, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
IF(true, z0, z2) -> c12
IF(false, z0, 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 -> 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

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

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

(117)
BOUNDS(n^1, INF)

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

(118)
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, app(z1, add(head(z0), nil))), NULL(z0))
SHUFF(z0, z1) -> c11(IF(null(z0), z0, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
IF(true, z0, z2) -> c12
IF(false, z0, 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 -> 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

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

(119) 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).
----------------------------------------

(120)
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, app(z1, add(head(z0), nil))), NULL(z0))
SHUFF(z0, z1) -> c11(IF(null(z0), z0, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
IF(true, z0, z2) -> c12
IF(false, z0, 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 -> 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

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

(121) 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).
----------------------------------------

(122)
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, app(z1, add(head(z0), nil))), NULL(z0))
SHUFF(z0, z1) -> c11(IF(null(z0), z0, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
IF(true, z0, z2) -> c12
IF(false, z0, 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 -> 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

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

(123) 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).
----------------------------------------

(124)
Complex Obligation (BEST)

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

(125)
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, app(z1, add(head(z0), nil))), NULL(z0))
SHUFF(z0, z1) -> c11(IF(null(z0), z0, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
IF(true, z0, z2) -> c12
IF(false, z0, 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 -> 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

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

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

(127)
BOUNDS(n^2, INF)

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

(128)
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, app(z1, add(head(z0), nil))), NULL(z0))
SHUFF(z0, z1) -> c11(IF(null(z0), z0, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
IF(true, z0, z2) -> c12
IF(false, z0, 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 -> 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
----------------------------------------

(129) 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)), 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)), 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)), 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(+(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).
----------------------------------------

(130)
Complex Obligation (BEST)

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

(131)
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, app(z1, add(head(z0), nil))), NULL(z0))
SHUFF(z0, z1) -> c11(IF(null(z0), z0, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
IF(true, z0, z2) -> c12
IF(false, z0, 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 -> 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
----------------------------------------

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

(133)
BOUNDS(n^3, INF)

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

(134)
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, app(z1, add(head(z0), nil))), NULL(z0))
SHUFF(z0, z1) -> c11(IF(null(z0), z0, app(z1, add(head(z0), nil))), APP(z1, add(head(z0), nil)), HEAD(z0))
IF(true, z0, z2) -> c12
IF(false, z0, 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 -> 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
----------------------------------------

(135) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
shuff(gen_nil:add12_14(n16398_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(+(n16398_14, 1)), gen_nil:add12_14(b)) ->_R^Omega(0)
if(null(gen_nil:add12_14(+(n16398_14, 1))), gen_nil:add12_14(+(n16398_14, 1)), gen_nil:add12_14(b), app(gen_nil:add12_14(b), add(head(gen_nil:add12_14(+(n16398_14, 1))), nil))) ->_R^Omega(0)
if(false, gen_nil:add12_14(+(1, n16398_14)), gen_nil:add12_14(b), app(gen_nil:add12_14(b), add(head(gen_nil:add12_14(+(1, n16398_14))), nil))) ->_R^Omega(0)
if(false, gen_nil:add12_14(+(1, n16398_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, n16398_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, n16398_14)))), gen_nil:add12_14(+(1, b))) ->_R^Omega(0)
shuff(reverse(gen_nil:add12_14(n16398_14)), gen_nil:add12_14(+(1, b))) ->_L^Omega(0)
shuff(gen_nil:add12_14(n16398_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).
----------------------------------------

(136)
BOUNDS(1, INF)