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


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WORST_CASE(Omega(n^3), O(n^3))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 5 ms]
(2) CpxTRS
    (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), 1 ms]
    (12) CpxRNTS
    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 1 ms]
    (16) CpxRNTS
    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 404 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 148 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 420 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 114 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 352 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 105 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 583 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 260 ms]
    (40) CpxRNTS
    (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 896 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 559 ms]
    (46) CpxRNTS
    (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 410 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 135 ms]
    (52) CpxRNTS
    (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (54) CpxRNTS
    (55) IntTrsBoundProof [UPPER BOUND(ID), 341 ms]
    (56) CpxRNTS
    (57) IntTrsBoundProof [UPPER BOUND(ID), 13 ms]
    (58) CpxRNTS
    (59) FinalProof [FINISHED, 0 ms]
    (60) BOUNDS(1, n^3)
(61) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(62) CpxTRS
    (63) SlicingProof [LOWER BOUND(ID), 0 ms]
    (64) CpxTRS
    (65) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) typed CpxTrs
    (67) OrderProof [LOWER BOUND(ID), 0 ms]
    (68) typed CpxTrs
    (69) RewriteLemmaProof [LOWER BOUND(ID), 278 ms]
    (70) BEST
        (71) proven lower bound
            (72) LowerBoundPropagationProof [FINISHED, 0 ms]
            (73) BOUNDS(n^1, INF)
        (74) typed CpxTrs
            (75) RewriteLemmaProof [LOWER BOUND(ID), 41 ms]
            (76) typed CpxTrs
            (77) RewriteLemmaProof [LOWER BOUND(ID), 9 ms]
            (78) BEST
                (79) proven lower bound
                    (80) LowerBoundPropagationProof [FINISHED, 0 ms]
                    (81) BOUNDS(n^2, INF)
                (82) typed CpxTrs
                    (83) RewriteLemmaProof [LOWER BOUND(ID), 22 ms]
                    (84) BEST
                        (85) proven lower bound
                            (86) LowerBoundPropagationProof [FINISHED, 0 ms]
                            (87) BOUNDS(n^3, INF)
                        (88) typed CpxTrs
                            (89) RewriteLemmaProof [LOWER BOUND(ID), 42 ms]
                            (90) typed CpxTrs
                            (91) RewriteLemmaProof [LOWER BOUND(ID), 53 ms]
                            (92) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   reverse(nil) -> nil
   reverse(add(n, x)) -> app(reverse(x), add(n, nil))
   shuffle(nil) -> nil
   shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))
   less_leaves(x, leaf) -> false
   less_leaves(leaf, cons(w, z)) -> true
   less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

The duplicating contexts are:
quot(s(x), s([]))


The defined contexts are:
quot([], s(x1))
shuffle([])
less_leaves([], x1)
less_leaves(x0, [])
app([], add(x1, nil))
minus([], x1)
app([], x1)
reverse([])
concat([], x1)
concat(x0, [])
app(x0, add([], nil))
app(x0, [])


[] just represents basic- or constructor-terms in the following defined contexts:
quot([], s(x1))


As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   reverse(nil) -> nil
   reverse(add(n, x)) -> app(reverse(x), add(n, nil))
   shuffle(nil) -> nil
   shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))
   less_leaves(x, leaf) -> false
   less_leaves(leaf, cons(w, z)) -> true
   less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

S is empty.
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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   app(nil, y) -> y [1]
   app(add(n, x), y) -> add(n, app(x, y)) [1]
   reverse(nil) -> nil [1]
   reverse(add(n, x)) -> app(reverse(x), add(n, nil)) [1]
   shuffle(nil) -> nil [1]
   shuffle(add(n, x)) -> add(n, shuffle(reverse(x))) [1]
   concat(leaf, y) -> y [1]
   concat(cons(u, v), y) -> cons(u, concat(v, y)) [1]
   less_leaves(x, leaf) -> false [1]
   less_leaves(leaf, cons(w, z)) -> true [1]
   less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z)) [1]

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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   app(nil, y) -> y [1]
   app(add(n, x), y) -> add(n, app(x, y)) [1]
   reverse(nil) -> nil [1]
   reverse(add(n, x)) -> app(reverse(x), add(n, nil)) [1]
   shuffle(nil) -> nil [1]
   shuffle(add(n, x)) -> add(n, shuffle(reverse(x))) [1]
   concat(leaf, y) -> y [1]
   concat(cons(u, v), y) -> cons(u, concat(v, y)) [1]
   less_leaves(x, leaf) -> false [1]
   less_leaves(leaf, cons(w, z)) -> true [1]
   less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z)) [1]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: a -> nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true

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:

   quot_2
   shuffle_1
   less_leaves_2

(c) The following functions are completely defined:

   minus_2
   reverse_1
   concat_2
   app_2

Due to the following rules being added:

   minus(v0, v1) -> 0 [0]

And the following fresh constants:    const

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

(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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y))) [1]
   app(nil, y) -> y [1]
   app(add(n, x), y) -> add(n, app(x, y)) [1]
   reverse(nil) -> nil [1]
   reverse(add(n, x)) -> app(reverse(x), add(n, nil)) [1]
   shuffle(nil) -> nil [1]
   shuffle(add(n, x)) -> add(n, shuffle(reverse(x))) [1]
   concat(leaf, y) -> y [1]
   concat(cons(u, v), y) -> cons(u, concat(v, y)) [1]
   less_leaves(x, leaf) -> false [1]
   less_leaves(leaf, cons(w, z)) -> true [1]
   less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z)) [1]
   minus(v0, v1) -> 0 [0]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: a -> nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
const :: a

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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   quot(0, s(y)) -> 0 [1]
   quot(s(x), s(0)) -> s(quot(x, s(0))) [2]
   quot(s(s(x')), s(s(y'))) -> s(quot(minus(x', y'), s(s(y')))) [2]
   quot(s(x), s(y)) -> s(quot(0, s(y))) [1]
   app(nil, y) -> y [1]
   app(add(n, x), y) -> add(n, app(x, y)) [1]
   reverse(nil) -> nil [1]
   reverse(add(n, nil)) -> app(nil, add(n, nil)) [2]
   reverse(add(n, add(n', x''))) -> app(app(reverse(x''), add(n', nil)), add(n, nil)) [2]
   shuffle(nil) -> nil [1]
   shuffle(add(n, nil)) -> add(n, shuffle(nil)) [2]
   shuffle(add(n, add(n'', x1))) -> add(n, shuffle(app(reverse(x1), add(n'', nil)))) [2]
   concat(leaf, y) -> y [1]
   concat(cons(u, v), y) -> cons(u, concat(v, y)) [1]
   less_leaves(x, leaf) -> false [1]
   less_leaves(leaf, cons(w, z)) -> true [1]
   less_leaves(cons(leaf, v), cons(leaf, z)) -> less_leaves(v, z) [3]
   less_leaves(cons(leaf, v), cons(cons(u'', v''), z)) -> less_leaves(v, cons(u'', concat(v'', z))) [3]
   less_leaves(cons(cons(u', v'), v), cons(leaf, z)) -> less_leaves(cons(u', concat(v', v)), z) [3]
   less_leaves(cons(cons(u', v'), v), cons(cons(u1, v2), z)) -> less_leaves(cons(u', concat(v', v)), cons(u1, concat(v2, z))) [3]
   minus(v0, v1) -> 0 [0]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: a -> nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
const :: a

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:

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

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

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

   app(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, y) :|: n >= 0, z'' = y, z' = 1 + n + x, x >= 0, y >= 0
   concat(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   concat(z', z'') -{ 1 }-> 1 + u + concat(v, y) :|: v >= 0, z' = 1 + u + v, z'' = y, y >= 0, u >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(v, z) :|: v >= 0, z >= 0, z' = 1 + 0 + v, z'' = 1 + 0 + z
   less_leaves(z', z'') -{ 3 }-> less_leaves(v, 1 + u'' + concat(v'', z)) :|: v >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, z' = 1 + 0 + v, v'' >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(1 + u' + concat(v', v), z) :|: v >= 0, z >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + 0 + z
   less_leaves(z', z'') -{ 3 }-> less_leaves(1 + u' + concat(v', v), 1 + u1 + concat(v2, z)) :|: v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = x, x >= 0
   minus(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0
   minus(z', z'') -{ 1 }-> minus(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   minus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   quot(z', z'') -{ 1 }-> 0 :|: y >= 0, z'' = 1 + y, z' = 0
   quot(z', z'') -{ 2 }-> 1 + quot(x, 1 + 0) :|: z' = 1 + x, x >= 0, z'' = 1 + 0
   quot(z', z'') -{ 2 }-> 1 + quot(minus(x', y'), 1 + (1 + y')) :|: z' = 1 + (1 + x'), x' >= 0, z'' = 1 + (1 + y'), y' >= 0
   quot(z', z'') -{ 1 }-> 1 + quot(0, 1 + y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   reverse(z') -{ 2 }-> app(app(reverse(x''), 1 + n' + 0), 1 + n + 0) :|: n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 2 }-> app(0, 1 + n + 0) :|: n >= 0, z' = 1 + n + 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 2 }-> 1 + n + shuffle(app(reverse(x1), 1 + n'' + 0)) :|: n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + n + shuffle(0) :|: n >= 0, z' = 1 + n + 0


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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 1 }-> 1 + u + concat(v, z'') :|: v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, 1 + u'' + concat(v'', z)) :|: z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(1 + u' + concat(v', v), z'' - 1) :|: v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 3 }-> less_leaves(1 + u' + concat(v', v), 1 + u1 + concat(v2, z)) :|: v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 2 }-> 1 + quot(minus(z' - 2, z'' - 2), 1 + (1 + (z'' - 2))) :|: z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 1 }-> 1 + quot(0, 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'') -{ 2 }-> 1 + quot(z' - 1, 1 + 0) :|: z' - 1 >= 0, z'' = 1 + 0
   reverse(z') -{ 2 }-> app(app(reverse(x''), 1 + n' + 0), 1 + n + 0) :|: n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 2 }-> app(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 2 }-> 1 + n + shuffle(app(reverse(x1), 1 + n'' + 0)) :|: n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0


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

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

   { minus }
   { concat }
   { app }
   { quot }
   { less_leaves }
   { reverse }
   { shuffle }

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 1 }-> 1 + u + concat(v, z'') :|: v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, 1 + u'' + concat(v'', z)) :|: z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(1 + u' + concat(v', v), z'' - 1) :|: v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 3 }-> less_leaves(1 + u' + concat(v', v), 1 + u1 + concat(v2, z)) :|: v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 2 }-> 1 + quot(minus(z' - 2, z'' - 2), 1 + (1 + (z'' - 2))) :|: z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 1 }-> 1 + quot(0, 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'') -{ 2 }-> 1 + quot(z' - 1, 1 + 0) :|: z' - 1 >= 0, z'' = 1 + 0
   reverse(z') -{ 2 }-> app(app(reverse(x''), 1 + n' + 0), 1 + n + 0) :|: n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 2 }-> app(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 2 }-> 1 + n + shuffle(app(reverse(x1), 1 + n'' + 0)) :|: n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0

Function symbols to be analyzed: {minus}, {concat}, {app}, {quot}, {less_leaves}, {reverse}, {shuffle}

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 1 }-> 1 + u + concat(v, z'') :|: v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, 1 + u'' + concat(v'', z)) :|: z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(1 + u' + concat(v', v), z'' - 1) :|: v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 3 }-> less_leaves(1 + u' + concat(v', v), 1 + u1 + concat(v2, z)) :|: v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 2 }-> 1 + quot(minus(z' - 2, z'' - 2), 1 + (1 + (z'' - 2))) :|: z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 1 }-> 1 + quot(0, 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'') -{ 2 }-> 1 + quot(z' - 1, 1 + 0) :|: z' - 1 >= 0, z'' = 1 + 0
   reverse(z') -{ 2 }-> app(app(reverse(x''), 1 + n' + 0), 1 + n + 0) :|: n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 2 }-> app(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 2 }-> 1 + n + shuffle(app(reverse(x1), 1 + n'' + 0)) :|: n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0

Function symbols to be analyzed: {minus}, {concat}, {app}, {quot}, {less_leaves}, {reverse}, {shuffle}

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 1 }-> 1 + u + concat(v, z'') :|: v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, 1 + u'' + concat(v'', z)) :|: z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(1 + u' + concat(v', v), z'' - 1) :|: v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 3 }-> less_leaves(1 + u' + concat(v', v), 1 + u1 + concat(v2, z)) :|: v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 2 }-> 1 + quot(minus(z' - 2, z'' - 2), 1 + (1 + (z'' - 2))) :|: z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 1 }-> 1 + quot(0, 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'') -{ 2 }-> 1 + quot(z' - 1, 1 + 0) :|: z' - 1 >= 0, z'' = 1 + 0
   reverse(z') -{ 2 }-> app(app(reverse(x''), 1 + n' + 0), 1 + n + 0) :|: n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 2 }-> app(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 2 }-> 1 + n + shuffle(app(reverse(x1), 1 + n'' + 0)) :|: n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0

Function symbols to be analyzed: {minus}, {concat}, {app}, {quot}, {less_leaves}, {reverse}, {shuffle}
Previous analysis results are:
  minus: runtime: ?, size: O(n^1) [z']

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 1 }-> 1 + u + concat(v, z'') :|: v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, 1 + u'' + concat(v'', z)) :|: z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(1 + u' + concat(v', v), z'' - 1) :|: v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 3 }-> less_leaves(1 + u' + concat(v', v), 1 + u1 + concat(v2, z)) :|: v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 2 }-> 1 + quot(minus(z' - 2, z'' - 2), 1 + (1 + (z'' - 2))) :|: z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 1 }-> 1 + quot(0, 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'') -{ 2 }-> 1 + quot(z' - 1, 1 + 0) :|: z' - 1 >= 0, z'' = 1 + 0
   reverse(z') -{ 2 }-> app(app(reverse(x''), 1 + n' + 0), 1 + n + 0) :|: n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 2 }-> app(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 2 }-> 1 + n + shuffle(app(reverse(x1), 1 + n'' + 0)) :|: n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0

Function symbols to be analyzed: {concat}, {app}, {quot}, {less_leaves}, {reverse}, {shuffle}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 1 }-> 1 + u + concat(v, z'') :|: v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, 1 + u'' + concat(v'', z)) :|: z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(1 + u' + concat(v', v), z'' - 1) :|: v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 3 }-> less_leaves(1 + u' + concat(v', v), 1 + u1 + concat(v2, z)) :|: v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 1 + z'' }-> 1 + quot(s', 1 + (1 + (z'' - 2))) :|: s' >= 0, s' <= z' - 2, z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 1 }-> 1 + quot(0, 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'') -{ 2 }-> 1 + quot(z' - 1, 1 + 0) :|: z' - 1 >= 0, z'' = 1 + 0
   reverse(z') -{ 2 }-> app(app(reverse(x''), 1 + n' + 0), 1 + n + 0) :|: n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 2 }-> app(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 2 }-> 1 + n + shuffle(app(reverse(x1), 1 + n'' + 0)) :|: n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0

Function symbols to be analyzed: {concat}, {app}, {quot}, {less_leaves}, {reverse}, {shuffle}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 1 }-> 1 + u + concat(v, z'') :|: v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, 1 + u'' + concat(v'', z)) :|: z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(1 + u' + concat(v', v), z'' - 1) :|: v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 3 }-> less_leaves(1 + u' + concat(v', v), 1 + u1 + concat(v2, z)) :|: v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 1 + z'' }-> 1 + quot(s', 1 + (1 + (z'' - 2))) :|: s' >= 0, s' <= z' - 2, z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 1 }-> 1 + quot(0, 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'') -{ 2 }-> 1 + quot(z' - 1, 1 + 0) :|: z' - 1 >= 0, z'' = 1 + 0
   reverse(z') -{ 2 }-> app(app(reverse(x''), 1 + n' + 0), 1 + n + 0) :|: n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 2 }-> app(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 2 }-> 1 + n + shuffle(app(reverse(x1), 1 + n'' + 0)) :|: n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0

Function symbols to be analyzed: {concat}, {app}, {quot}, {less_leaves}, {reverse}, {shuffle}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  concat: runtime: ?, size: O(n^1) [z' + z'']

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 1 }-> 1 + u + concat(v, z'') :|: v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, 1 + u'' + concat(v'', z)) :|: z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(1 + u' + concat(v', v), z'' - 1) :|: v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 3 }-> less_leaves(1 + u' + concat(v', v), 1 + u1 + concat(v2, z)) :|: v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 1 + z'' }-> 1 + quot(s', 1 + (1 + (z'' - 2))) :|: s' >= 0, s' <= z' - 2, z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 1 }-> 1 + quot(0, 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'') -{ 2 }-> 1 + quot(z' - 1, 1 + 0) :|: z' - 1 >= 0, z'' = 1 + 0
   reverse(z') -{ 2 }-> app(app(reverse(x''), 1 + n' + 0), 1 + n + 0) :|: n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 2 }-> app(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 2 }-> 1 + n + shuffle(app(reverse(x1), 1 + n'' + 0)) :|: n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0

Function symbols to be analyzed: {app}, {quot}, {less_leaves}, {reverse}, {shuffle}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  concat: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 2 + v }-> 1 + u + s'' :|: s'' >= 0, s'' <= v + z'', v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 4 + v'' }-> less_leaves(z' - 1, 1 + u'' + s1) :|: s1 >= 0, s1 <= v'' + z, z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 4 + v' }-> less_leaves(1 + u' + s2, z'' - 1) :|: s2 >= 0, s2 <= v' + v, v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 5 + v' + v2 }-> less_leaves(1 + u' + s3, 1 + u1 + s4) :|: s3 >= 0, s3 <= v' + v, s4 >= 0, s4 <= v2 + z, v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 1 + z'' }-> 1 + quot(s', 1 + (1 + (z'' - 2))) :|: s' >= 0, s' <= z' - 2, z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 1 }-> 1 + quot(0, 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'') -{ 2 }-> 1 + quot(z' - 1, 1 + 0) :|: z' - 1 >= 0, z'' = 1 + 0
   reverse(z') -{ 2 }-> app(app(reverse(x''), 1 + n' + 0), 1 + n + 0) :|: n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 2 }-> app(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 2 }-> 1 + n + shuffle(app(reverse(x1), 1 + n'' + 0)) :|: n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0

Function symbols to be analyzed: {app}, {quot}, {less_leaves}, {reverse}, {shuffle}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  concat: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']

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

(31) 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''

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 2 + v }-> 1 + u + s'' :|: s'' >= 0, s'' <= v + z'', v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 4 + v'' }-> less_leaves(z' - 1, 1 + u'' + s1) :|: s1 >= 0, s1 <= v'' + z, z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 4 + v' }-> less_leaves(1 + u' + s2, z'' - 1) :|: s2 >= 0, s2 <= v' + v, v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 5 + v' + v2 }-> less_leaves(1 + u' + s3, 1 + u1 + s4) :|: s3 >= 0, s3 <= v' + v, s4 >= 0, s4 <= v2 + z, v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 1 + z'' }-> 1 + quot(s', 1 + (1 + (z'' - 2))) :|: s' >= 0, s' <= z' - 2, z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 1 }-> 1 + quot(0, 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'') -{ 2 }-> 1 + quot(z' - 1, 1 + 0) :|: z' - 1 >= 0, z'' = 1 + 0
   reverse(z') -{ 2 }-> app(app(reverse(x''), 1 + n' + 0), 1 + n + 0) :|: n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 2 }-> app(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 2 }-> 1 + n + shuffle(app(reverse(x1), 1 + n'' + 0)) :|: n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0

Function symbols to be analyzed: {app}, {quot}, {less_leaves}, {reverse}, {shuffle}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  concat: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  app: runtime: ?, size: O(n^1) [z' + z'']

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

(33) 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'

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 2 + v }-> 1 + u + s'' :|: s'' >= 0, s'' <= v + z'', v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 4 + v'' }-> less_leaves(z' - 1, 1 + u'' + s1) :|: s1 >= 0, s1 <= v'' + z, z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 4 + v' }-> less_leaves(1 + u' + s2, z'' - 1) :|: s2 >= 0, s2 <= v' + v, v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 5 + v' + v2 }-> less_leaves(1 + u' + s3, 1 + u1 + s4) :|: s3 >= 0, s3 <= v' + v, s4 >= 0, s4 <= v2 + z, v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 1 + z'' }-> 1 + quot(s', 1 + (1 + (z'' - 2))) :|: s' >= 0, s' <= z' - 2, z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 1 }-> 1 + quot(0, 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'') -{ 2 }-> 1 + quot(z' - 1, 1 + 0) :|: z' - 1 >= 0, z'' = 1 + 0
   reverse(z') -{ 2 }-> app(app(reverse(x''), 1 + n' + 0), 1 + n + 0) :|: n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 2 }-> app(0, 1 + (z' - 1) + 0) :|: z' - 1 >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 2 }-> 1 + n + shuffle(app(reverse(x1), 1 + n'' + 0)) :|: n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0

Function symbols to be analyzed: {quot}, {less_leaves}, {reverse}, {shuffle}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  concat: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 2 + v }-> 1 + u + s'' :|: s'' >= 0, s'' <= v + z'', v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 4 + v'' }-> less_leaves(z' - 1, 1 + u'' + s1) :|: s1 >= 0, s1 <= v'' + z, z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 4 + v' }-> less_leaves(1 + u' + s2, z'' - 1) :|: s2 >= 0, s2 <= v' + v, v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 5 + v' + v2 }-> less_leaves(1 + u' + s3, 1 + u1 + s4) :|: s3 >= 0, s3 <= v' + v, s4 >= 0, s4 <= v2 + z, v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 1 + z'' }-> 1 + quot(s', 1 + (1 + (z'' - 2))) :|: s' >= 0, s' <= z' - 2, z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 1 }-> 1 + quot(0, 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'') -{ 2 }-> 1 + quot(z' - 1, 1 + 0) :|: z' - 1 >= 0, z'' = 1 + 0
   reverse(z') -{ 3 }-> s6 :|: s6 >= 0, s6 <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x''), 1 + n' + 0), 1 + n + 0) :|: n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 2 }-> 1 + n + shuffle(app(reverse(x1), 1 + n'' + 0)) :|: n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0

Function symbols to be analyzed: {quot}, {less_leaves}, {reverse}, {shuffle}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  concat: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 2 + v }-> 1 + u + s'' :|: s'' >= 0, s'' <= v + z'', v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 4 + v'' }-> less_leaves(z' - 1, 1 + u'' + s1) :|: s1 >= 0, s1 <= v'' + z, z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 4 + v' }-> less_leaves(1 + u' + s2, z'' - 1) :|: s2 >= 0, s2 <= v' + v, v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 5 + v' + v2 }-> less_leaves(1 + u' + s3, 1 + u1 + s4) :|: s3 >= 0, s3 <= v' + v, s4 >= 0, s4 <= v2 + z, v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 1 + z'' }-> 1 + quot(s', 1 + (1 + (z'' - 2))) :|: s' >= 0, s' <= z' - 2, z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 1 }-> 1 + quot(0, 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'') -{ 2 }-> 1 + quot(z' - 1, 1 + 0) :|: z' - 1 >= 0, z'' = 1 + 0
   reverse(z') -{ 3 }-> s6 :|: s6 >= 0, s6 <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x''), 1 + n' + 0), 1 + n + 0) :|: n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 2 }-> 1 + n + shuffle(app(reverse(x1), 1 + n'' + 0)) :|: n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0

Function symbols to be analyzed: {quot}, {less_leaves}, {reverse}, {shuffle}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  concat: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  quot: runtime: ?, size: O(n^1) [z']

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 2 + v }-> 1 + u + s'' :|: s'' >= 0, s'' <= v + z'', v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 4 + v'' }-> less_leaves(z' - 1, 1 + u'' + s1) :|: s1 >= 0, s1 <= v'' + z, z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 4 + v' }-> less_leaves(1 + u' + s2, z'' - 1) :|: s2 >= 0, s2 <= v' + v, v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 5 + v' + v2 }-> less_leaves(1 + u' + s3, 1 + u1 + s4) :|: s3 >= 0, s3 <= v' + v, s4 >= 0, s4 <= v2 + z, v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 1 + z'' }-> 1 + quot(s', 1 + (1 + (z'' - 2))) :|: s' >= 0, s' <= z' - 2, z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 1 }-> 1 + quot(0, 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'') -{ 2 }-> 1 + quot(z' - 1, 1 + 0) :|: z' - 1 >= 0, z'' = 1 + 0
   reverse(z') -{ 3 }-> s6 :|: s6 >= 0, s6 <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x''), 1 + n' + 0), 1 + n + 0) :|: n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 2 }-> 1 + n + shuffle(app(reverse(x1), 1 + n'' + 0)) :|: n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0

Function symbols to be analyzed: {less_leaves}, {reverse}, {shuffle}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  concat: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  quot: runtime: O(n^2) [4 + 2*z' + z'*z'' + z''], size: O(n^1) [z']

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 2 + v }-> 1 + u + s'' :|: s'' >= 0, s'' <= v + z'', v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 4 + v'' }-> less_leaves(z' - 1, 1 + u'' + s1) :|: s1 >= 0, s1 <= v'' + z, z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 4 + v' }-> less_leaves(1 + u' + s2, z'' - 1) :|: s2 >= 0, s2 <= v' + v, v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 5 + v' + v2 }-> less_leaves(1 + u' + s3, 1 + u1 + s4) :|: s3 >= 0, s3 <= v' + v, s4 >= 0, s4 <= v2 + z, v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 4 + 3*z' }-> 1 + s7 :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z'' = 1 + 0
   quot(z', z'') -{ 5 + 2*s' + s'*z'' + 2*z'' }-> 1 + s8 :|: s8 >= 0, s8 <= s', s' >= 0, s' <= z' - 2, z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 5 + z'' }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z'' - 1 >= 0
   reverse(z') -{ 3 }-> s6 :|: s6 >= 0, s6 <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x''), 1 + n' + 0), 1 + n + 0) :|: n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 2 }-> 1 + n + shuffle(app(reverse(x1), 1 + n'' + 0)) :|: n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0

Function symbols to be analyzed: {less_leaves}, {reverse}, {shuffle}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  concat: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  quot: runtime: O(n^2) [4 + 2*z' + z'*z'' + z''], size: O(n^1) [z']

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 2 + v }-> 1 + u + s'' :|: s'' >= 0, s'' <= v + z'', v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 4 + v'' }-> less_leaves(z' - 1, 1 + u'' + s1) :|: s1 >= 0, s1 <= v'' + z, z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 4 + v' }-> less_leaves(1 + u' + s2, z'' - 1) :|: s2 >= 0, s2 <= v' + v, v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 5 + v' + v2 }-> less_leaves(1 + u' + s3, 1 + u1 + s4) :|: s3 >= 0, s3 <= v' + v, s4 >= 0, s4 <= v2 + z, v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 4 + 3*z' }-> 1 + s7 :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z'' = 1 + 0
   quot(z', z'') -{ 5 + 2*s' + s'*z'' + 2*z'' }-> 1 + s8 :|: s8 >= 0, s8 <= s', s' >= 0, s' <= z' - 2, z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 5 + z'' }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z'' - 1 >= 0
   reverse(z') -{ 3 }-> s6 :|: s6 >= 0, s6 <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x''), 1 + n' + 0), 1 + n + 0) :|: n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 2 }-> 1 + n + shuffle(app(reverse(x1), 1 + n'' + 0)) :|: n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0

Function symbols to be analyzed: {less_leaves}, {reverse}, {shuffle}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  concat: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  quot: runtime: O(n^2) [4 + 2*z' + z'*z'' + z''], size: O(n^1) [z']
  less_leaves: runtime: ?, size: O(1) [1]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: less_leaves
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 1 + 3*z' + z'*z'' + z'^2

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 2 + v }-> 1 + u + s'' :|: s'' >= 0, s'' <= v + z'', v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 }-> less_leaves(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 4 + v'' }-> less_leaves(z' - 1, 1 + u'' + s1) :|: s1 >= 0, s1 <= v'' + z, z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 4 + v' }-> less_leaves(1 + u' + s2, z'' - 1) :|: s2 >= 0, s2 <= v' + v, v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 5 + v' + v2 }-> less_leaves(1 + u' + s3, 1 + u1 + s4) :|: s3 >= 0, s3 <= v' + v, s4 >= 0, s4 <= v2 + z, v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 4 + 3*z' }-> 1 + s7 :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z'' = 1 + 0
   quot(z', z'') -{ 5 + 2*s' + s'*z'' + 2*z'' }-> 1 + s8 :|: s8 >= 0, s8 <= s', s' >= 0, s' <= z' - 2, z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 5 + z'' }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z'' - 1 >= 0
   reverse(z') -{ 3 }-> s6 :|: s6 >= 0, s6 <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x''), 1 + n' + 0), 1 + n + 0) :|: n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 2 }-> 1 + n + shuffle(app(reverse(x1), 1 + n'' + 0)) :|: n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0

Function symbols to be analyzed: {reverse}, {shuffle}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  concat: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  quot: runtime: O(n^2) [4 + 2*z' + z'*z'' + z''], size: O(n^1) [z']
  less_leaves: runtime: O(n^2) [1 + 3*z' + z'*z'' + z'^2], size: O(1) [1]

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 2 + v }-> 1 + u + s'' :|: s'' >= 0, s'' <= v + z'', v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 + z'*z'' + z'^2 + -1*z'' }-> s10 :|: s10 >= 0, s10 <= 1, z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 2 + -1*s1 + s1*z' + -1*u'' + u''*z' + v'' + 2*z' + z'^2 }-> s11 :|: s11 >= 0, s11 <= 1, s1 >= 0, s1 <= v'' + z, z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 8 + 4*s2 + 2*s2*u' + s2*z'' + s2^2 + 4*u' + u'*z'' + u'^2 + v' + z'' }-> s12 :|: s12 >= 0, s12 <= 1, s2 >= 0, s2 <= v' + v, v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 11 + 6*s3 + s3*s4 + 2*s3*u' + s3*u1 + s3^2 + s4 + s4*u' + 6*u' + u'*u1 + u'^2 + u1 + v' + v2 }-> s13 :|: s13 >= 0, s13 <= 1, s3 >= 0, s3 <= v' + v, s4 >= 0, s4 <= v2 + z, v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 4 + 3*z' }-> 1 + s7 :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z'' = 1 + 0
   quot(z', z'') -{ 5 + 2*s' + s'*z'' + 2*z'' }-> 1 + s8 :|: s8 >= 0, s8 <= s', s' >= 0, s' <= z' - 2, z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 5 + z'' }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z'' - 1 >= 0
   reverse(z') -{ 3 }-> s6 :|: s6 >= 0, s6 <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x''), 1 + n' + 0), 1 + n + 0) :|: n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 2 }-> 1 + n + shuffle(app(reverse(x1), 1 + n'' + 0)) :|: n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0

Function symbols to be analyzed: {reverse}, {shuffle}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  concat: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  quot: runtime: O(n^2) [4 + 2*z' + z'*z'' + z''], size: O(n^1) [z']
  less_leaves: runtime: O(n^2) [1 + 3*z' + z'*z'' + z'^2], size: O(1) [1]

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

(49) 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'

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 2 + v }-> 1 + u + s'' :|: s'' >= 0, s'' <= v + z'', v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 + z'*z'' + z'^2 + -1*z'' }-> s10 :|: s10 >= 0, s10 <= 1, z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 2 + -1*s1 + s1*z' + -1*u'' + u''*z' + v'' + 2*z' + z'^2 }-> s11 :|: s11 >= 0, s11 <= 1, s1 >= 0, s1 <= v'' + z, z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 8 + 4*s2 + 2*s2*u' + s2*z'' + s2^2 + 4*u' + u'*z'' + u'^2 + v' + z'' }-> s12 :|: s12 >= 0, s12 <= 1, s2 >= 0, s2 <= v' + v, v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 11 + 6*s3 + s3*s4 + 2*s3*u' + s3*u1 + s3^2 + s4 + s4*u' + 6*u' + u'*u1 + u'^2 + u1 + v' + v2 }-> s13 :|: s13 >= 0, s13 <= 1, s3 >= 0, s3 <= v' + v, s4 >= 0, s4 <= v2 + z, v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 4 + 3*z' }-> 1 + s7 :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z'' = 1 + 0
   quot(z', z'') -{ 5 + 2*s' + s'*z'' + 2*z'' }-> 1 + s8 :|: s8 >= 0, s8 <= s', s' >= 0, s' <= z' - 2, z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 5 + z'' }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z'' - 1 >= 0
   reverse(z') -{ 3 }-> s6 :|: s6 >= 0, s6 <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x''), 1 + n' + 0), 1 + n + 0) :|: n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 2 }-> 1 + n + shuffle(app(reverse(x1), 1 + n'' + 0)) :|: n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0

Function symbols to be analyzed: {reverse}, {shuffle}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  concat: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  quot: runtime: O(n^2) [4 + 2*z' + z'*z'' + z''], size: O(n^1) [z']
  less_leaves: runtime: O(n^2) [1 + 3*z' + z'*z'' + z'^2], size: O(1) [1]
  reverse: runtime: ?, size: O(n^1) [z']

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 2 + v }-> 1 + u + s'' :|: s'' >= 0, s'' <= v + z'', v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 + z'*z'' + z'^2 + -1*z'' }-> s10 :|: s10 >= 0, s10 <= 1, z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 2 + -1*s1 + s1*z' + -1*u'' + u''*z' + v'' + 2*z' + z'^2 }-> s11 :|: s11 >= 0, s11 <= 1, s1 >= 0, s1 <= v'' + z, z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 8 + 4*s2 + 2*s2*u' + s2*z'' + s2^2 + 4*u' + u'*z'' + u'^2 + v' + z'' }-> s12 :|: s12 >= 0, s12 <= 1, s2 >= 0, s2 <= v' + v, v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 11 + 6*s3 + s3*s4 + 2*s3*u' + s3*u1 + s3^2 + s4 + s4*u' + 6*u' + u'*u1 + u'^2 + u1 + v' + v2 }-> s13 :|: s13 >= 0, s13 <= 1, s3 >= 0, s3 <= v' + v, s4 >= 0, s4 <= v2 + z, v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 4 + 3*z' }-> 1 + s7 :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z'' = 1 + 0
   quot(z', z'') -{ 5 + 2*s' + s'*z'' + 2*z'' }-> 1 + s8 :|: s8 >= 0, s8 <= s', s' >= 0, s' <= z' - 2, z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 5 + z'' }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z'' - 1 >= 0
   reverse(z') -{ 3 }-> s6 :|: s6 >= 0, s6 <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 2 }-> app(app(reverse(x''), 1 + n' + 0), 1 + n + 0) :|: n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 2 }-> 1 + n + shuffle(app(reverse(x1), 1 + n'' + 0)) :|: n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0

Function symbols to be analyzed: {shuffle}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  concat: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  quot: runtime: O(n^2) [4 + 2*z' + z'*z'' + z''], size: O(n^1) [z']
  less_leaves: runtime: O(n^2) [1 + 3*z' + z'*z'' + z'^2], size: O(1) [1]
  reverse: runtime: O(n^2) [4 + 3*z' + 2*z'^2], size: O(n^1) [z']

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 2 + v }-> 1 + u + s'' :|: s'' >= 0, s'' <= v + z'', v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 + z'*z'' + z'^2 + -1*z'' }-> s10 :|: s10 >= 0, s10 <= 1, z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 2 + -1*s1 + s1*z' + -1*u'' + u''*z' + v'' + 2*z' + z'^2 }-> s11 :|: s11 >= 0, s11 <= 1, s1 >= 0, s1 <= v'' + z, z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 8 + 4*s2 + 2*s2*u' + s2*z'' + s2^2 + 4*u' + u'*z'' + u'^2 + v' + z'' }-> s12 :|: s12 >= 0, s12 <= 1, s2 >= 0, s2 <= v' + v, v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 11 + 6*s3 + s3*s4 + 2*s3*u' + s3*u1 + s3^2 + s4 + s4*u' + 6*u' + u'*u1 + u'^2 + u1 + v' + v2 }-> s13 :|: s13 >= 0, s13 <= 1, s3 >= 0, s3 <= v' + v, s4 >= 0, s4 <= v2 + z, v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 4 + 3*z' }-> 1 + s7 :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z'' = 1 + 0
   quot(z', z'') -{ 5 + 2*s' + s'*z'' + 2*z'' }-> 1 + s8 :|: s8 >= 0, s8 <= s', s' >= 0, s' <= z' - 2, z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 5 + z'' }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z'' - 1 >= 0
   reverse(z') -{ 8 + s14 + s15 + 3*x'' + 2*x''^2 }-> s16 :|: s14 >= 0, s14 <= x'', s15 >= 0, s15 <= s14 + (1 + n' + 0), s16 >= 0, s16 <= s15 + (1 + n + 0), n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 3 }-> s6 :|: s6 >= 0, s6 <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 7 + s17 + 3*x1 + 2*x1^2 }-> 1 + n + shuffle(s18) :|: s17 >= 0, s17 <= x1, s18 >= 0, s18 <= s17 + (1 + n'' + 0), n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0

Function symbols to be analyzed: {shuffle}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  concat: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  quot: runtime: O(n^2) [4 + 2*z' + z'*z'' + z''], size: O(n^1) [z']
  less_leaves: runtime: O(n^2) [1 + 3*z' + z'*z'' + z'^2], size: O(1) [1]
  reverse: runtime: O(n^2) [4 + 3*z' + 2*z'^2], size: O(n^1) [z']

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 2 + v }-> 1 + u + s'' :|: s'' >= 0, s'' <= v + z'', v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 + z'*z'' + z'^2 + -1*z'' }-> s10 :|: s10 >= 0, s10 <= 1, z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 2 + -1*s1 + s1*z' + -1*u'' + u''*z' + v'' + 2*z' + z'^2 }-> s11 :|: s11 >= 0, s11 <= 1, s1 >= 0, s1 <= v'' + z, z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 8 + 4*s2 + 2*s2*u' + s2*z'' + s2^2 + 4*u' + u'*z'' + u'^2 + v' + z'' }-> s12 :|: s12 >= 0, s12 <= 1, s2 >= 0, s2 <= v' + v, v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 11 + 6*s3 + s3*s4 + 2*s3*u' + s3*u1 + s3^2 + s4 + s4*u' + 6*u' + u'*u1 + u'^2 + u1 + v' + v2 }-> s13 :|: s13 >= 0, s13 <= 1, s3 >= 0, s3 <= v' + v, s4 >= 0, s4 <= v2 + z, v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 4 + 3*z' }-> 1 + s7 :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z'' = 1 + 0
   quot(z', z'') -{ 5 + 2*s' + s'*z'' + 2*z'' }-> 1 + s8 :|: s8 >= 0, s8 <= s', s' >= 0, s' <= z' - 2, z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 5 + z'' }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z'' - 1 >= 0
   reverse(z') -{ 8 + s14 + s15 + 3*x'' + 2*x''^2 }-> s16 :|: s14 >= 0, s14 <= x'', s15 >= 0, s15 <= s14 + (1 + n' + 0), s16 >= 0, s16 <= s15 + (1 + n + 0), n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 3 }-> s6 :|: s6 >= 0, s6 <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 7 + s17 + 3*x1 + 2*x1^2 }-> 1 + n + shuffle(s18) :|: s17 >= 0, s17 <= x1, s18 >= 0, s18 <= s17 + (1 + n'' + 0), n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0

Function symbols to be analyzed: {shuffle}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  concat: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  quot: runtime: O(n^2) [4 + 2*z' + z'*z'' + z''], size: O(n^1) [z']
  less_leaves: runtime: O(n^2) [1 + 3*z' + z'*z'' + z'^2], size: O(1) [1]
  reverse: runtime: O(n^2) [4 + 3*z' + 2*z'^2], size: O(n^1) [z']
  shuffle: runtime: ?, size: O(n^1) [z']

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

(57) 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: 1 + 9*z' + 4*z'^2 + 2*z'^3

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + s5 :|: s5 >= 0, s5 <= x + z'', n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   concat(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   concat(z', z'') -{ 2 + v }-> 1 + u + s'' :|: s'' >= 0, s'' <= v + z'', v >= 0, z' = 1 + u + v, z'' >= 0, u >= 0
   less_leaves(z', z'') -{ 3 + z'*z'' + z'^2 + -1*z'' }-> s10 :|: s10 >= 0, s10 <= 1, z' - 1 >= 0, z'' - 1 >= 0
   less_leaves(z', z'') -{ 2 + -1*s1 + s1*z' + -1*u'' + u''*z' + v'' + 2*z' + z'^2 }-> s11 :|: s11 >= 0, s11 <= 1, s1 >= 0, s1 <= v'' + z, z' - 1 >= 0, z >= 0, z'' = 1 + (1 + u'' + v'') + z, u'' >= 0, v'' >= 0
   less_leaves(z', z'') -{ 8 + 4*s2 + 2*s2*u' + s2*z'' + s2^2 + 4*u' + u'*z'' + u'^2 + v' + z'' }-> s12 :|: s12 >= 0, s12 <= 1, s2 >= 0, s2 <= v' + v, v >= 0, z'' - 1 >= 0, u' >= 0, v' >= 0, z' = 1 + (1 + u' + v') + v
   less_leaves(z', z'') -{ 11 + 6*s3 + s3*s4 + 2*s3*u' + s3*u1 + s3^2 + s4 + s4*u' + 6*u' + u'*u1 + u'^2 + u1 + v' + v2 }-> s13 :|: s13 >= 0, s13 <= 1, s3 >= 0, s3 <= v' + v, s4 >= 0, s4 <= v2 + z, v >= 0, z >= 0, u' >= 0, v' >= 0, u1 >= 0, z' = 1 + (1 + u' + v') + v, z'' = 1 + (1 + u1 + v2) + z, v2 >= 0
   less_leaves(z', z'') -{ 1 }-> 1 :|: z >= 0, z'' = 1 + w + z, w >= 0, z' = 0
   less_leaves(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s :|: s >= 0, s <= z' - 1, z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   quot(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   quot(z', z'') -{ 4 + 3*z' }-> 1 + s7 :|: s7 >= 0, s7 <= z' - 1, z' - 1 >= 0, z'' = 1 + 0
   quot(z', z'') -{ 5 + 2*s' + s'*z'' + 2*z'' }-> 1 + s8 :|: s8 >= 0, s8 <= s', s' >= 0, s' <= z' - 2, z' - 2 >= 0, z'' - 2 >= 0
   quot(z', z'') -{ 5 + z'' }-> 1 + s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z'' - 1 >= 0
   reverse(z') -{ 8 + s14 + s15 + 3*x'' + 2*x''^2 }-> s16 :|: s14 >= 0, s14 <= x'', s15 >= 0, s15 <= s14 + (1 + n' + 0), s16 >= 0, s16 <= s15 + (1 + n + 0), n >= 0, n' >= 0, z' = 1 + n + (1 + n' + x''), x'' >= 0
   reverse(z') -{ 3 }-> s6 :|: s6 >= 0, s6 <= 0 + (1 + (z' - 1) + 0), z' - 1 >= 0
   reverse(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 1 }-> 0 :|: z' = 0
   shuffle(z') -{ 7 + s17 + 3*x1 + 2*x1^2 }-> 1 + n + shuffle(s18) :|: s17 >= 0, s17 <= x1, s18 >= 0, s18 <= s17 + (1 + n'' + 0), n >= 0, x1 >= 0, n'' >= 0, z' = 1 + n + (1 + n'' + x1)
   shuffle(z') -{ 2 }-> 1 + (z' - 1) + shuffle(0) :|: z' - 1 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  concat: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  app: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  quot: runtime: O(n^2) [4 + 2*z' + z'*z'' + z''], size: O(n^1) [z']
  less_leaves: runtime: O(n^2) [1 + 3*z' + z'*z'' + z'^2], size: O(1) [1]
  reverse: runtime: O(n^2) [4 + 3*z' + 2*z'^2], size: O(n^1) [z']
  shuffle: runtime: O(n^3) [1 + 9*z' + 4*z'^2 + 2*z'^3], size: O(n^1) [z']

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

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

(60)
BOUNDS(1, n^3)

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

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

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


The TRS R consists of the following rules:

   minus(x, 0') -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0', s(y)) -> 0'
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   app(nil, y) -> y
   app(add(n, x), y) -> add(n, app(x, y))
   reverse(nil) -> nil
   reverse(add(n, x)) -> app(reverse(x), add(n, nil))
   shuffle(nil) -> nil
   shuffle(add(n, x)) -> add(n, shuffle(reverse(x)))
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))
   less_leaves(x, leaf) -> false
   less_leaves(leaf, cons(w, z)) -> true
   less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

(63) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
add/0

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

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


The TRS R consists of the following rules:

   minus(x, 0') -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0', s(y)) -> 0'
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   app(nil, y) -> y
   app(add(x), y) -> add(app(x, y))
   reverse(nil) -> nil
   reverse(add(x)) -> app(reverse(x), add(nil))
   shuffle(nil) -> nil
   shuffle(add(x)) -> add(shuffle(reverse(x)))
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))
   less_leaves(x, leaf) -> false
   less_leaves(leaf, cons(w, z)) -> true
   less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

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

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

(66)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:add6_0 :: Nat -> nil:add
gen_leaf:cons7_0 :: Nat -> leaf:cons

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

(67) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
minus, quot, app, reverse, shuffle, concat, less_leaves

They will be analysed ascendingly in the following order:
minus < quot
app < reverse
reverse < shuffle
concat < less_leaves

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

(68)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:add6_0 :: Nat -> nil:add
gen_leaf:cons7_0 :: Nat -> leaf:cons


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:add6_0(0) <=> nil
gen_nil:add6_0(+(x, 1)) <=> add(gen_nil:add6_0(x))
gen_leaf:cons7_0(0) <=> leaf
gen_leaf:cons7_0(+(x, 1)) <=> cons(leaf, gen_leaf:cons7_0(x))


The following defined symbols remain to be analysed:
minus, quot, app, reverse, shuffle, concat, less_leaves

They will be analysed ascendingly in the following order:
minus < quot
app < reverse
reverse < shuffle
concat < less_leaves

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

(69) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) -> gen_0':s5_0(0), rt in Omega(1 + n9_0)

Induction Base:
minus(gen_0':s5_0(0), gen_0':s5_0(0)) ->_R^Omega(1)
gen_0':s5_0(0)

Induction Step:
minus(gen_0':s5_0(+(n9_0, 1)), gen_0':s5_0(+(n9_0, 1))) ->_R^Omega(1)
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) ->_IH
gen_0':s5_0(0)

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

(70)
Complex Obligation (BEST)

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

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

TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:add6_0 :: Nat -> nil:add
gen_leaf:cons7_0 :: Nat -> leaf:cons


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:add6_0(0) <=> nil
gen_nil:add6_0(+(x, 1)) <=> add(gen_nil:add6_0(x))
gen_leaf:cons7_0(0) <=> leaf
gen_leaf:cons7_0(+(x, 1)) <=> cons(leaf, gen_leaf:cons7_0(x))


The following defined symbols remain to be analysed:
minus, quot, app, reverse, shuffle, concat, less_leaves

They will be analysed ascendingly in the following order:
minus < quot
app < reverse
reverse < shuffle
concat < less_leaves

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

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

(73)
BOUNDS(n^1, INF)

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

(74)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:add6_0 :: Nat -> nil:add
gen_leaf:cons7_0 :: Nat -> leaf:cons


Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) -> gen_0':s5_0(0), rt in Omega(1 + n9_0)


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:add6_0(0) <=> nil
gen_nil:add6_0(+(x, 1)) <=> add(gen_nil:add6_0(x))
gen_leaf:cons7_0(0) <=> leaf
gen_leaf:cons7_0(+(x, 1)) <=> cons(leaf, gen_leaf:cons7_0(x))


The following defined symbols remain to be analysed:
quot, app, reverse, shuffle, concat, less_leaves

They will be analysed ascendingly in the following order:
app < reverse
reverse < shuffle
concat < less_leaves

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

(75) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
app(gen_nil:add6_0(n589_0), gen_nil:add6_0(b)) -> gen_nil:add6_0(+(n589_0, b)), rt in Omega(1 + n589_0)

Induction Base:
app(gen_nil:add6_0(0), gen_nil:add6_0(b)) ->_R^Omega(1)
gen_nil:add6_0(b)

Induction Step:
app(gen_nil:add6_0(+(n589_0, 1)), gen_nil:add6_0(b)) ->_R^Omega(1)
add(app(gen_nil:add6_0(n589_0), gen_nil:add6_0(b))) ->_IH
add(gen_nil:add6_0(+(b, c590_0)))

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

(76)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:add6_0 :: Nat -> nil:add
gen_leaf:cons7_0 :: Nat -> leaf:cons


Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) -> gen_0':s5_0(0), rt in Omega(1 + n9_0)
app(gen_nil:add6_0(n589_0), gen_nil:add6_0(b)) -> gen_nil:add6_0(+(n589_0, b)), rt in Omega(1 + n589_0)


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:add6_0(0) <=> nil
gen_nil:add6_0(+(x, 1)) <=> add(gen_nil:add6_0(x))
gen_leaf:cons7_0(0) <=> leaf
gen_leaf:cons7_0(+(x, 1)) <=> cons(leaf, gen_leaf:cons7_0(x))


The following defined symbols remain to be analysed:
reverse, shuffle, concat, less_leaves

They will be analysed ascendingly in the following order:
reverse < shuffle
concat < less_leaves

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

(77) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
reverse(gen_nil:add6_0(n1508_0)) -> gen_nil:add6_0(n1508_0), rt in Omega(1 + n1508_0 + n1508_0^2)

Induction Base:
reverse(gen_nil:add6_0(0)) ->_R^Omega(1)
nil

Induction Step:
reverse(gen_nil:add6_0(+(n1508_0, 1))) ->_R^Omega(1)
app(reverse(gen_nil:add6_0(n1508_0)), add(nil)) ->_IH
app(gen_nil:add6_0(c1509_0), add(nil)) ->_L^Omega(1 + n1508_0)
gen_nil:add6_0(+(n1508_0, +(0, 1)))

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

(78)
Complex Obligation (BEST)

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

(79)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:add6_0 :: Nat -> nil:add
gen_leaf:cons7_0 :: Nat -> leaf:cons


Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) -> gen_0':s5_0(0), rt in Omega(1 + n9_0)
app(gen_nil:add6_0(n589_0), gen_nil:add6_0(b)) -> gen_nil:add6_0(+(n589_0, b)), rt in Omega(1 + n589_0)


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:add6_0(0) <=> nil
gen_nil:add6_0(+(x, 1)) <=> add(gen_nil:add6_0(x))
gen_leaf:cons7_0(0) <=> leaf
gen_leaf:cons7_0(+(x, 1)) <=> cons(leaf, gen_leaf:cons7_0(x))


The following defined symbols remain to be analysed:
reverse, shuffle, concat, less_leaves

They will be analysed ascendingly in the following order:
reverse < shuffle
concat < less_leaves

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

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

(81)
BOUNDS(n^2, INF)

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

(82)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:add6_0 :: Nat -> nil:add
gen_leaf:cons7_0 :: Nat -> leaf:cons


Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) -> gen_0':s5_0(0), rt in Omega(1 + n9_0)
app(gen_nil:add6_0(n589_0), gen_nil:add6_0(b)) -> gen_nil:add6_0(+(n589_0, b)), rt in Omega(1 + n589_0)
reverse(gen_nil:add6_0(n1508_0)) -> gen_nil:add6_0(n1508_0), rt in Omega(1 + n1508_0 + n1508_0^2)


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:add6_0(0) <=> nil
gen_nil:add6_0(+(x, 1)) <=> add(gen_nil:add6_0(x))
gen_leaf:cons7_0(0) <=> leaf
gen_leaf:cons7_0(+(x, 1)) <=> cons(leaf, gen_leaf:cons7_0(x))


The following defined symbols remain to be analysed:
shuffle, concat, less_leaves

They will be analysed ascendingly in the following order:
concat < less_leaves

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

(83) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
shuffle(gen_nil:add6_0(n1776_0)) -> gen_nil:add6_0(n1776_0), rt in Omega(1 + n1776_0 + n1776_0^2 + n1776_0^3)

Induction Base:
shuffle(gen_nil:add6_0(0)) ->_R^Omega(1)
nil

Induction Step:
shuffle(gen_nil:add6_0(+(n1776_0, 1))) ->_R^Omega(1)
add(shuffle(reverse(gen_nil:add6_0(n1776_0)))) ->_L^Omega(1 + n1776_0 + n1776_0^2)
add(shuffle(gen_nil:add6_0(n1776_0))) ->_IH
add(gen_nil:add6_0(c1777_0))

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

(84)
Complex Obligation (BEST)

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

(85)
Obligation:
Proved the lower bound n^3 for the following obligation:

TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:add6_0 :: Nat -> nil:add
gen_leaf:cons7_0 :: Nat -> leaf:cons


Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) -> gen_0':s5_0(0), rt in Omega(1 + n9_0)
app(gen_nil:add6_0(n589_0), gen_nil:add6_0(b)) -> gen_nil:add6_0(+(n589_0, b)), rt in Omega(1 + n589_0)
reverse(gen_nil:add6_0(n1508_0)) -> gen_nil:add6_0(n1508_0), rt in Omega(1 + n1508_0 + n1508_0^2)


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:add6_0(0) <=> nil
gen_nil:add6_0(+(x, 1)) <=> add(gen_nil:add6_0(x))
gen_leaf:cons7_0(0) <=> leaf
gen_leaf:cons7_0(+(x, 1)) <=> cons(leaf, gen_leaf:cons7_0(x))


The following defined symbols remain to be analysed:
shuffle, concat, less_leaves

They will be analysed ascendingly in the following order:
concat < less_leaves

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

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

(87)
BOUNDS(n^3, INF)

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

(88)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:add6_0 :: Nat -> nil:add
gen_leaf:cons7_0 :: Nat -> leaf:cons


Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) -> gen_0':s5_0(0), rt in Omega(1 + n9_0)
app(gen_nil:add6_0(n589_0), gen_nil:add6_0(b)) -> gen_nil:add6_0(+(n589_0, b)), rt in Omega(1 + n589_0)
reverse(gen_nil:add6_0(n1508_0)) -> gen_nil:add6_0(n1508_0), rt in Omega(1 + n1508_0 + n1508_0^2)
shuffle(gen_nil:add6_0(n1776_0)) -> gen_nil:add6_0(n1776_0), rt in Omega(1 + n1776_0 + n1776_0^2 + n1776_0^3)


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:add6_0(0) <=> nil
gen_nil:add6_0(+(x, 1)) <=> add(gen_nil:add6_0(x))
gen_leaf:cons7_0(0) <=> leaf
gen_leaf:cons7_0(+(x, 1)) <=> cons(leaf, gen_leaf:cons7_0(x))


The following defined symbols remain to be analysed:
concat, less_leaves

They will be analysed ascendingly in the following order:
concat < less_leaves

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

(89) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
concat(gen_leaf:cons7_0(n1977_0), gen_leaf:cons7_0(b)) -> gen_leaf:cons7_0(+(n1977_0, b)), rt in Omega(1 + n1977_0)

Induction Base:
concat(gen_leaf:cons7_0(0), gen_leaf:cons7_0(b)) ->_R^Omega(1)
gen_leaf:cons7_0(b)

Induction Step:
concat(gen_leaf:cons7_0(+(n1977_0, 1)), gen_leaf:cons7_0(b)) ->_R^Omega(1)
cons(leaf, concat(gen_leaf:cons7_0(n1977_0), gen_leaf:cons7_0(b))) ->_IH
cons(leaf, gen_leaf:cons7_0(+(b, c1978_0)))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(90)
Obligation:
TRS:
Rules:
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
app(nil, y) -> y
app(add(x), y) -> add(app(x, y))
reverse(nil) -> nil
reverse(add(x)) -> app(reverse(x), add(nil))
shuffle(nil) -> nil
shuffle(add(x)) -> add(shuffle(reverse(x)))
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(x, leaf) -> false
less_leaves(leaf, cons(w, z)) -> true
less_leaves(cons(u, v), cons(w, z)) -> less_leaves(concat(u, v), concat(w, z))

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
app :: nil:add -> nil:add -> nil:add
nil :: nil:add
add :: nil:add -> nil:add
reverse :: nil:add -> nil:add
shuffle :: nil:add -> nil:add
concat :: leaf:cons -> leaf:cons -> leaf:cons
leaf :: leaf:cons
cons :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> false:true
false :: false:true
true :: false:true
hole_0':s1_0 :: 0':s
hole_nil:add2_0 :: nil:add
hole_leaf:cons3_0 :: leaf:cons
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:add6_0 :: Nat -> nil:add
gen_leaf:cons7_0 :: Nat -> leaf:cons


Lemmas:
minus(gen_0':s5_0(n9_0), gen_0':s5_0(n9_0)) -> gen_0':s5_0(0), rt in Omega(1 + n9_0)
app(gen_nil:add6_0(n589_0), gen_nil:add6_0(b)) -> gen_nil:add6_0(+(n589_0, b)), rt in Omega(1 + n589_0)
reverse(gen_nil:add6_0(n1508_0)) -> gen_nil:add6_0(n1508_0), rt in Omega(1 + n1508_0 + n1508_0^2)
shuffle(gen_nil:add6_0(n1776_0)) -> gen_nil:add6_0(n1776_0), rt in Omega(1 + n1776_0 + n1776_0^2 + n1776_0^3)
concat(gen_leaf:cons7_0(n1977_0), gen_leaf:cons7_0(b)) -> gen_leaf:cons7_0(+(n1977_0, b)), rt in Omega(1 + n1977_0)


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:add6_0(0) <=> nil
gen_nil:add6_0(+(x, 1)) <=> add(gen_nil:add6_0(x))
gen_leaf:cons7_0(0) <=> leaf
gen_leaf:cons7_0(+(x, 1)) <=> cons(leaf, gen_leaf:cons7_0(x))


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

(91) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
less_leaves(gen_leaf:cons7_0(n3068_0), gen_leaf:cons7_0(n3068_0)) -> false, rt in Omega(1 + n3068_0)

Induction Base:
less_leaves(gen_leaf:cons7_0(0), gen_leaf:cons7_0(0)) ->_R^Omega(1)
false

Induction Step:
less_leaves(gen_leaf:cons7_0(+(n3068_0, 1)), gen_leaf:cons7_0(+(n3068_0, 1))) ->_R^Omega(1)
less_leaves(concat(leaf, gen_leaf:cons7_0(n3068_0)), concat(leaf, gen_leaf:cons7_0(n3068_0))) ->_L^Omega(1)
less_leaves(gen_leaf:cons7_0(+(0, n3068_0)), concat(leaf, gen_leaf:cons7_0(n3068_0))) ->_L^Omega(1)
less_leaves(gen_leaf:cons7_0(n3068_0), gen_leaf:cons7_0(+(0, n3068_0))) ->_IH
false

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(92)
BOUNDS(1, INF)