WORST_CASE(?,O(n^3))
proof of input_XxR6p2Geu7.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxWeightedTrs
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxTypedWeightedTrs
(5) CompletionProof [UPPER BOUND(ID), 0 ms]
(6) CpxTypedWeightedCompleteTrs
(7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTypedWeightedCompleteTrs
(9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(10) CpxRNTS
(11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxRNTS
(13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxRNTS
(15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(16) CpxRNTS
(17) IntTrsBoundProof [UPPER BOUND(ID), 357 ms]
(18) CpxRNTS
(19) IntTrsBoundProof [UPPER BOUND(ID), 140 ms]
(20) CpxRNTS
(21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(22) CpxRNTS
(23) IntTrsBoundProof [UPPER BOUND(ID), 187 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 97 ms]
(26) CpxRNTS
(27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(28) CpxRNTS
(29) IntTrsBoundProof [UPPER BOUND(ID), 283 ms]
(30) CpxRNTS
(31) IntTrsBoundProof [UPPER BOUND(ID), 96 ms]
(32) CpxRNTS
(33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(34) CpxRNTS
(35) IntTrsBoundProof [UPPER BOUND(ID), 528 ms]
(36) CpxRNTS
(37) IntTrsBoundProof [UPPER BOUND(ID), 248 ms]
(38) CpxRNTS
(39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(40) CpxRNTS
(41) IntTrsBoundProof [UPPER BOUND(ID), 692 ms]
(42) CpxRNTS
(43) IntTrsBoundProof [UPPER BOUND(ID), 402 ms]
(44) CpxRNTS
(45) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(46) CpxRNTS
(47) IntTrsBoundProof [UPPER BOUND(ID), 321 ms]
(48) CpxRNTS
(49) IntTrsBoundProof [UPPER BOUND(ID), 94 ms]
(50) CpxRNTS
(51) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(52) CpxRNTS
(53) IntTrsBoundProof [UPPER BOUND(ID), 313 ms]
(54) CpxRNTS
(55) IntTrsBoundProof [UPPER BOUND(ID), 44 ms]
(56) CpxRNTS
(57) FinalProof [FINISHED, 0 ms]
(58) BOUNDS(1, n^3)


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

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

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

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


The TRS R consists of the following rules:

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

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

(4)
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
----------------------------------------

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

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

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

   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

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

(6)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

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

(9) 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

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

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   app(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, y) :|: n >= 0, z'' = y, z' = 1 + n + x, x >= 0, y >= 0
   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


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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   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


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

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

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   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}

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   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) 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'

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

(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}
Previous analysis results are:
  minus: runtime: ?, size: O(n^1) [z']

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

(19) 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''

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

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

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 1 }-> 1 + n + app(x, z'') :|: n >= 0, z' = 1 + n + x, x >= 0, z'' >= 0
   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']

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

(23) 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''

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

(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']
  concat: runtime: ?, size: O(n^1) [z' + z'']

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

(25) 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'

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

(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: {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'']

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 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'']

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

(29) 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''

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

(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'']
  app: runtime: ?, size: O(n^1) [z' + z'']

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

(31) 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'

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

(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: {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'']

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + 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'']

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

(35) 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'

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

(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'']
  quot: runtime: ?, size: O(n^1) [z']

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

(37) 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''

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

(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: {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']

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + 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']

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

(41) 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

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

(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']
  less_leaves: runtime: ?, size: O(1) [1]

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

(43) 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

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

(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: {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]

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

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + 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]

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

(47) 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'

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

(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]
  reverse: runtime: ?, size: O(n^1) [z']

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

(49) 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

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

(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: {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']

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

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(53) 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'

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

(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']
  shuffle: runtime: ?, size: O(n^1) [z']

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(55) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: shuffle
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 1 + 9*z' + 4*z'^2 + 2*z'^3

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

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

   app(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   app(z', z'') -{ 2 + x }-> 1 + n + 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']

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(57) FinalProof (FINISHED)
Computed overall runtime complexity
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(58)
BOUNDS(1, n^3)