WORST_CASE(Omega(n^1),O(n^2))
proof of input_ScmS6NvOEM.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(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), 155 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 67 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 295 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 106 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 43 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 112 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 15 ms]
    (38) CpxRNTS
    (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 4090 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 3703 ms]
    (44) CpxRNTS
    (45) FinalProof [FINISHED, 0 ms]
    (46) BOUNDS(1, n^2)
(47) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(48) CdtProblem
    (49) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CpxRelTRS
    (51) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CpxRelTRS
    (53) TypeInferenceProof [BOTH BOUNDS(ID, ID), 7 ms]
    (54) typed CpxTrs
    (55) OrderProof [LOWER BOUND(ID), 0 ms]
    (56) typed CpxTrs
    (57) RewriteLemmaProof [LOWER BOUND(ID), 236 ms]
    (58) BEST
        (59) proven lower bound
            (60) LowerBoundPropagationProof [FINISHED, 0 ms]
            (61) BOUNDS(n^1, INF)
        (62) typed CpxTrs
            (63) RewriteLemmaProof [LOWER BOUND(ID), 89 ms]
            (64) typed CpxTrs
            (65) RewriteLemmaProof [LOWER BOUND(ID), 7326 ms]
            (66) typed CpxTrs
            (67) RewriteLemmaProof [LOWER BOUND(ID), 168 ms]
            (68) BOUNDS(1, INF)


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   isLeaf(leaf) -> true
   isLeaf(cons(u, v)) -> false
   left(cons(u, v)) -> u
   right(cons(u, v)) -> v
   concat(leaf, y) -> y
   concat(cons(u, v), y) -> cons(u, concat(v, y))
   less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v)
   if1(b, true, u, v) -> false
   if1(b, false, u, v) -> if2(b, u, v)
   if2(true, u, v) -> true
   if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))

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^2).


The TRS R consists of the following rules:

   isLeaf(leaf) -> true [1]
   isLeaf(cons(u, v)) -> false [1]
   left(cons(u, v)) -> u [1]
   right(cons(u, v)) -> v [1]
   concat(leaf, y) -> y [1]
   concat(cons(u, v), y) -> cons(u, concat(v, y)) [1]
   less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v) [1]
   if1(b, true, u, v) -> false [1]
   if1(b, false, u, v) -> if2(b, u, v) [1]
   if2(true, u, v) -> true [1]
   if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v))) [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:

   isLeaf(leaf) -> true [1]
   isLeaf(cons(u, v)) -> false [1]
   left(cons(u, v)) -> u [1]
   right(cons(u, v)) -> v [1]
   concat(leaf, y) -> y [1]
   concat(cons(u, v), y) -> cons(u, concat(v, y)) [1]
   less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v) [1]
   if1(b, true, u, v) -> false [1]
   if1(b, false, u, v) -> if2(b, u, v) [1]
   if2(true, u, v) -> true [1]
   if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v))) [1]

The TRS has the following type information:
isLeaf :: leaf:cons -> true:false
leaf :: leaf:cons
true :: true:false
cons :: leaf:cons -> leaf:cons -> leaf:cons
false :: true:false
left :: leaf:cons -> leaf:cons
right :: leaf:cons -> leaf:cons
concat :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> true:false
if1 :: true:false -> true:false -> leaf:cons -> leaf:cons -> true:false
if2 :: true:false -> leaf:cons -> leaf:cons -> true:false

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:

   less_leaves_2
   if1_4
   if2_3

(c) The following functions are completely defined:

   isLeaf_1
   concat_2
   left_1
   right_1

Due to the following rules being added:

   left(v0) -> leaf [0]
   right(v0) -> leaf [0]

And the following fresh constants: none

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

(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:

   isLeaf(leaf) -> true [1]
   isLeaf(cons(u, v)) -> false [1]
   left(cons(u, v)) -> u [1]
   right(cons(u, v)) -> v [1]
   concat(leaf, y) -> y [1]
   concat(cons(u, v), y) -> cons(u, concat(v, y)) [1]
   less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v) [1]
   if1(b, true, u, v) -> false [1]
   if1(b, false, u, v) -> if2(b, u, v) [1]
   if2(true, u, v) -> true [1]
   if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v))) [1]
   left(v0) -> leaf [0]
   right(v0) -> leaf [0]

The TRS has the following type information:
isLeaf :: leaf:cons -> true:false
leaf :: leaf:cons
true :: true:false
cons :: leaf:cons -> leaf:cons -> leaf:cons
false :: true:false
left :: leaf:cons -> leaf:cons
right :: leaf:cons -> leaf:cons
concat :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> true:false
if1 :: true:false -> true:false -> leaf:cons -> leaf:cons -> true:false
if2 :: true:false -> leaf:cons -> leaf:cons -> true:false

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:

   isLeaf(leaf) -> true [1]
   isLeaf(cons(u, v)) -> false [1]
   left(cons(u, v)) -> u [1]
   right(cons(u, v)) -> v [1]
   concat(leaf, y) -> y [1]
   concat(cons(u, v), y) -> cons(u, concat(v, y)) [1]
   less_leaves(leaf, leaf) -> if1(true, true, leaf, leaf) [3]
   less_leaves(leaf, cons(u'', v'')) -> if1(true, false, leaf, cons(u'', v'')) [3]
   less_leaves(cons(u', v'), leaf) -> if1(false, true, cons(u', v'), leaf) [3]
   less_leaves(cons(u', v'), cons(u1, v1)) -> if1(false, false, cons(u', v'), cons(u1, v1)) [3]
   if1(b, true, u, v) -> false [1]
   if1(b, false, u, v) -> if2(b, u, v) [1]
   if2(true, u, v) -> true [1]
   if2(false, cons(u2, v2), cons(u4, v4)) -> less_leaves(concat(u2, v2), concat(u4, v4)) [5]
   if2(false, cons(u2, v2), cons(u4, v4)) -> less_leaves(concat(u2, v2), concat(u4, leaf)) [4]
   if2(false, cons(u2, v2), cons(u8, v8)) -> less_leaves(concat(u2, v2), concat(leaf, v8)) [4]
   if2(false, cons(u2, v2), v) -> less_leaves(concat(u2, v2), concat(leaf, leaf)) [3]
   if2(false, cons(u2, v2), cons(u5, v5)) -> less_leaves(concat(u2, leaf), concat(u5, v5)) [4]
   if2(false, cons(u2, v2), cons(u5, v5)) -> less_leaves(concat(u2, leaf), concat(u5, leaf)) [3]
   if2(false, cons(u2, v2), cons(u9, v9)) -> less_leaves(concat(u2, leaf), concat(leaf, v9)) [3]
   if2(false, cons(u2, v2), v) -> less_leaves(concat(u2, leaf), concat(leaf, leaf)) [2]
   if2(false, cons(u3, v3), cons(u6, v6)) -> less_leaves(concat(leaf, v3), concat(u6, v6)) [4]
   if2(false, cons(u3, v3), cons(u6, v6)) -> less_leaves(concat(leaf, v3), concat(u6, leaf)) [3]
   if2(false, cons(u3, v3), cons(u10, v10)) -> less_leaves(concat(leaf, v3), concat(leaf, v10)) [3]
   if2(false, cons(u3, v3), v) -> less_leaves(concat(leaf, v3), concat(leaf, leaf)) [2]
   if2(false, u, cons(u7, v7)) -> less_leaves(concat(leaf, leaf), concat(u7, v7)) [3]
   if2(false, u, cons(u7, v7)) -> less_leaves(concat(leaf, leaf), concat(u7, leaf)) [2]
   if2(false, u, cons(u11, v11)) -> less_leaves(concat(leaf, leaf), concat(leaf, v11)) [2]
   if2(false, u, v) -> less_leaves(concat(leaf, leaf), concat(leaf, leaf)) [1]
   left(v0) -> leaf [0]
   right(v0) -> leaf [0]

The TRS has the following type information:
isLeaf :: leaf:cons -> true:false
leaf :: leaf:cons
true :: true:false
cons :: leaf:cons -> leaf:cons -> leaf:cons
false :: true:false
left :: leaf:cons -> leaf:cons
right :: leaf:cons -> leaf:cons
concat :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> true:false
if1 :: true:false -> true:false -> leaf:cons -> leaf:cons -> true:false
if2 :: true:false -> leaf:cons -> leaf:cons -> true:false

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:

   leaf => 0
   true => 1
   false => 0

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

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

   concat(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   concat(z, z') -{ 1 }-> 1 + u + concat(v, y) :|: z = 1 + u + v, v >= 0, y >= 0, z' = y, u >= 0
   if1(z, z', z'', z1) -{ 1 }-> if2(b, u, v) :|: b >= 0, z1 = v, v >= 0, z = b, z'' = u, z' = 0, u >= 0
   if1(z, z', z'', z1) -{ 1 }-> 0 :|: b >= 0, z1 = v, v >= 0, z' = 1, z = b, z'' = u, u >= 0
   if2(z, z', z'') -{ 5 }-> less_leaves(concat(u2, v2), concat(u4, v4)) :|: v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, v2), concat(u4, 0)) :|: v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, v2), concat(0, v8)) :|: v8 >= 0, u8 >= 0, z' = 1 + u2 + v2, z'' = 1 + u8 + v8, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, v2), concat(0, 0)) :|: z'' = v, v >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, 0), concat(u5, v5)) :|: z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, 0), concat(u5, 0)) :|: z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, 0), concat(0, v9)) :|: z' = 1 + u2 + v2, z'' = 1 + u9 + v9, u9 >= 0, v9 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(u2, 0), concat(0, 0)) :|: z'' = v, v >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(0, v3), concat(u6, v6)) :|: u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, v3), concat(u6, 0)) :|: u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, v3), concat(0, v10)) :|: u3 >= 0, z' = 1 + u3 + v3, z'' = 1 + u10 + v10, u10 >= 0, v10 >= 0, z = 0, v3 >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, v3), concat(0, 0)) :|: z'' = v, u3 >= 0, v >= 0, z' = 1 + u3 + v3, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, 0), concat(u7, v7)) :|: u7 >= 0, z' = u, v7 >= 0, z = 0, z'' = 1 + u7 + v7, u >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, 0), concat(u7, 0)) :|: u7 >= 0, z' = u, v7 >= 0, z = 0, z'' = 1 + u7 + v7, u >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, 0), concat(0, v11)) :|: v11 >= 0, z' = u, u11 >= 0, z = 0, z'' = 1 + u11 + v11, u >= 0
   if2(z, z', z'') -{ 1 }-> less_leaves(concat(0, 0), concat(0, 0)) :|: z'' = v, v >= 0, z' = u, z = 0, u >= 0
   if2(z, z', z'') -{ 1 }-> 1 :|: z'' = v, v >= 0, z = 1, z' = u, u >= 0
   isLeaf(z) -{ 1 }-> 1 :|: z = 0
   isLeaf(z) -{ 1 }-> 0 :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 1 }-> u :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   less_leaves(z, z') -{ 3 }-> if1(1, 1, 0, 0) :|: z = 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(1, 0, 0, 1 + u'' + v'') :|: u'' >= 0, v'' >= 0, z' = 1 + u'' + v'', z = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 1, 1 + u' + v', 0) :|: z = 1 + u' + v', u' >= 0, v' >= 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 0, 1 + u' + v', 1 + u1 + v1) :|: z = 1 + u' + v', z' = 1 + u1 + v1, u' >= 0, v' >= 0, u1 >= 0, v1 >= 0
   right(z) -{ 1 }-> v :|: z = 1 + u + v, v >= 0, u >= 0
   right(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

(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:

   concat(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   concat(z, z') -{ 1 }-> 1 + u + concat(v, z') :|: z = 1 + u + v, v >= 0, z' >= 0, u >= 0
   if1(z, z', z'', z1) -{ 1 }-> if2(z, z'', z1) :|: z >= 0, z1 >= 0, z' = 0, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z' = 1, z'' >= 0
   if2(z, z', z'') -{ 5 }-> less_leaves(concat(u2, v2), concat(u4, v4)) :|: v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, v2), concat(u4, 0)) :|: v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, v2), concat(0, v8)) :|: v8 >= 0, u8 >= 0, z' = 1 + u2 + v2, z'' = 1 + u8 + v8, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, v2), concat(0, 0)) :|: z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, 0), concat(u5, v5)) :|: z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, 0), concat(u5, 0)) :|: z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, 0), concat(0, v9)) :|: z' = 1 + u2 + v2, z'' = 1 + u9 + v9, u9 >= 0, v9 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(u2, 0), concat(0, 0)) :|: z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(0, v3), concat(u6, v6)) :|: u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, v3), concat(u6, 0)) :|: u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, v3), concat(0, v10)) :|: u3 >= 0, z' = 1 + u3 + v3, z'' = 1 + u10 + v10, u10 >= 0, v10 >= 0, z = 0, v3 >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, v3), concat(0, 0)) :|: u3 >= 0, z'' >= 0, z' = 1 + u3 + v3, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, 0), concat(u7, v7)) :|: u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, 0), concat(u7, 0)) :|: u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, 0), concat(0, v11)) :|: v11 >= 0, u11 >= 0, z = 0, z'' = 1 + u11 + v11, z' >= 0
   if2(z, z', z'') -{ 1 }-> less_leaves(concat(0, 0), concat(0, 0)) :|: z'' >= 0, z = 0, z' >= 0
   if2(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   isLeaf(z) -{ 1 }-> 1 :|: z = 0
   isLeaf(z) -{ 1 }-> 0 :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 1 }-> u :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 0 }-> 0 :|: z >= 0
   less_leaves(z, z') -{ 3 }-> if1(1, 1, 0, 0) :|: z = 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(1, 0, 0, 1 + u'' + v'') :|: u'' >= 0, v'' >= 0, z' = 1 + u'' + v'', z = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 1, 1 + u' + v', 0) :|: z = 1 + u' + v', u' >= 0, v' >= 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 0, 1 + u' + v', 1 + u1 + v1) :|: z = 1 + u' + v', z' = 1 + u1 + v1, u' >= 0, v' >= 0, u1 >= 0, v1 >= 0
   right(z) -{ 1 }-> v :|: z = 1 + u + v, v >= 0, u >= 0
   right(z) -{ 0 }-> 0 :|: z >= 0


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

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

   { isLeaf }
   { concat }
   { right }
   { left }
   { if2, less_leaves, if1 }

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

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

   concat(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   concat(z, z') -{ 1 }-> 1 + u + concat(v, z') :|: z = 1 + u + v, v >= 0, z' >= 0, u >= 0
   if1(z, z', z'', z1) -{ 1 }-> if2(z, z'', z1) :|: z >= 0, z1 >= 0, z' = 0, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z' = 1, z'' >= 0
   if2(z, z', z'') -{ 5 }-> less_leaves(concat(u2, v2), concat(u4, v4)) :|: v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, v2), concat(u4, 0)) :|: v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, v2), concat(0, v8)) :|: v8 >= 0, u8 >= 0, z' = 1 + u2 + v2, z'' = 1 + u8 + v8, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, v2), concat(0, 0)) :|: z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, 0), concat(u5, v5)) :|: z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, 0), concat(u5, 0)) :|: z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, 0), concat(0, v9)) :|: z' = 1 + u2 + v2, z'' = 1 + u9 + v9, u9 >= 0, v9 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(u2, 0), concat(0, 0)) :|: z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(0, v3), concat(u6, v6)) :|: u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, v3), concat(u6, 0)) :|: u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, v3), concat(0, v10)) :|: u3 >= 0, z' = 1 + u3 + v3, z'' = 1 + u10 + v10, u10 >= 0, v10 >= 0, z = 0, v3 >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, v3), concat(0, 0)) :|: u3 >= 0, z'' >= 0, z' = 1 + u3 + v3, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, 0), concat(u7, v7)) :|: u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, 0), concat(u7, 0)) :|: u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, 0), concat(0, v11)) :|: v11 >= 0, u11 >= 0, z = 0, z'' = 1 + u11 + v11, z' >= 0
   if2(z, z', z'') -{ 1 }-> less_leaves(concat(0, 0), concat(0, 0)) :|: z'' >= 0, z = 0, z' >= 0
   if2(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   isLeaf(z) -{ 1 }-> 1 :|: z = 0
   isLeaf(z) -{ 1 }-> 0 :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 1 }-> u :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 0 }-> 0 :|: z >= 0
   less_leaves(z, z') -{ 3 }-> if1(1, 1, 0, 0) :|: z = 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(1, 0, 0, 1 + u'' + v'') :|: u'' >= 0, v'' >= 0, z' = 1 + u'' + v'', z = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 1, 1 + u' + v', 0) :|: z = 1 + u' + v', u' >= 0, v' >= 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 0, 1 + u' + v', 1 + u1 + v1) :|: z = 1 + u' + v', z' = 1 + u1 + v1, u' >= 0, v' >= 0, u1 >= 0, v1 >= 0
   right(z) -{ 1 }-> v :|: z = 1 + u + v, v >= 0, u >= 0
   right(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {isLeaf}, {concat}, {right}, {left}, {if2,less_leaves,if1}

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

(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:

   concat(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   concat(z, z') -{ 1 }-> 1 + u + concat(v, z') :|: z = 1 + u + v, v >= 0, z' >= 0, u >= 0
   if1(z, z', z'', z1) -{ 1 }-> if2(z, z'', z1) :|: z >= 0, z1 >= 0, z' = 0, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z' = 1, z'' >= 0
   if2(z, z', z'') -{ 5 }-> less_leaves(concat(u2, v2), concat(u4, v4)) :|: v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, v2), concat(u4, 0)) :|: v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, v2), concat(0, v8)) :|: v8 >= 0, u8 >= 0, z' = 1 + u2 + v2, z'' = 1 + u8 + v8, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, v2), concat(0, 0)) :|: z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, 0), concat(u5, v5)) :|: z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, 0), concat(u5, 0)) :|: z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, 0), concat(0, v9)) :|: z' = 1 + u2 + v2, z'' = 1 + u9 + v9, u9 >= 0, v9 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(u2, 0), concat(0, 0)) :|: z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(0, v3), concat(u6, v6)) :|: u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, v3), concat(u6, 0)) :|: u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, v3), concat(0, v10)) :|: u3 >= 0, z' = 1 + u3 + v3, z'' = 1 + u10 + v10, u10 >= 0, v10 >= 0, z = 0, v3 >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, v3), concat(0, 0)) :|: u3 >= 0, z'' >= 0, z' = 1 + u3 + v3, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, 0), concat(u7, v7)) :|: u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, 0), concat(u7, 0)) :|: u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, 0), concat(0, v11)) :|: v11 >= 0, u11 >= 0, z = 0, z'' = 1 + u11 + v11, z' >= 0
   if2(z, z', z'') -{ 1 }-> less_leaves(concat(0, 0), concat(0, 0)) :|: z'' >= 0, z = 0, z' >= 0
   if2(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   isLeaf(z) -{ 1 }-> 1 :|: z = 0
   isLeaf(z) -{ 1 }-> 0 :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 1 }-> u :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 0 }-> 0 :|: z >= 0
   less_leaves(z, z') -{ 3 }-> if1(1, 1, 0, 0) :|: z = 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(1, 0, 0, 1 + u'' + v'') :|: u'' >= 0, v'' >= 0, z' = 1 + u'' + v'', z = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 1, 1 + u' + v', 0) :|: z = 1 + u' + v', u' >= 0, v' >= 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 0, 1 + u' + v', 1 + u1 + v1) :|: z = 1 + u' + v', z' = 1 + u1 + v1, u' >= 0, v' >= 0, u1 >= 0, v1 >= 0
   right(z) -{ 1 }-> v :|: z = 1 + u + v, v >= 0, u >= 0
   right(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {isLeaf}, {concat}, {right}, {left}, {if2,less_leaves,if1}

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

(17) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   concat(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   concat(z, z') -{ 1 }-> 1 + u + concat(v, z') :|: z = 1 + u + v, v >= 0, z' >= 0, u >= 0
   if1(z, z', z'', z1) -{ 1 }-> if2(z, z'', z1) :|: z >= 0, z1 >= 0, z' = 0, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z' = 1, z'' >= 0
   if2(z, z', z'') -{ 5 }-> less_leaves(concat(u2, v2), concat(u4, v4)) :|: v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, v2), concat(u4, 0)) :|: v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, v2), concat(0, v8)) :|: v8 >= 0, u8 >= 0, z' = 1 + u2 + v2, z'' = 1 + u8 + v8, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, v2), concat(0, 0)) :|: z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, 0), concat(u5, v5)) :|: z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, 0), concat(u5, 0)) :|: z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, 0), concat(0, v9)) :|: z' = 1 + u2 + v2, z'' = 1 + u9 + v9, u9 >= 0, v9 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(u2, 0), concat(0, 0)) :|: z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(0, v3), concat(u6, v6)) :|: u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, v3), concat(u6, 0)) :|: u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, v3), concat(0, v10)) :|: u3 >= 0, z' = 1 + u3 + v3, z'' = 1 + u10 + v10, u10 >= 0, v10 >= 0, z = 0, v3 >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, v3), concat(0, 0)) :|: u3 >= 0, z'' >= 0, z' = 1 + u3 + v3, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, 0), concat(u7, v7)) :|: u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, 0), concat(u7, 0)) :|: u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, 0), concat(0, v11)) :|: v11 >= 0, u11 >= 0, z = 0, z'' = 1 + u11 + v11, z' >= 0
   if2(z, z', z'') -{ 1 }-> less_leaves(concat(0, 0), concat(0, 0)) :|: z'' >= 0, z = 0, z' >= 0
   if2(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   isLeaf(z) -{ 1 }-> 1 :|: z = 0
   isLeaf(z) -{ 1 }-> 0 :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 1 }-> u :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 0 }-> 0 :|: z >= 0
   less_leaves(z, z') -{ 3 }-> if1(1, 1, 0, 0) :|: z = 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(1, 0, 0, 1 + u'' + v'') :|: u'' >= 0, v'' >= 0, z' = 1 + u'' + v'', z = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 1, 1 + u' + v', 0) :|: z = 1 + u' + v', u' >= 0, v' >= 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 0, 1 + u' + v', 1 + u1 + v1) :|: z = 1 + u' + v', z' = 1 + u1 + v1, u' >= 0, v' >= 0, u1 >= 0, v1 >= 0
   right(z) -{ 1 }-> v :|: z = 1 + u + v, v >= 0, u >= 0
   right(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {isLeaf}, {concat}, {right}, {left}, {if2,less_leaves,if1}
Previous analysis results are:
  isLeaf: runtime: ?, size: O(1) [1]

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   concat(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   concat(z, z') -{ 1 }-> 1 + u + concat(v, z') :|: z = 1 + u + v, v >= 0, z' >= 0, u >= 0
   if1(z, z', z'', z1) -{ 1 }-> if2(z, z'', z1) :|: z >= 0, z1 >= 0, z' = 0, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z' = 1, z'' >= 0
   if2(z, z', z'') -{ 5 }-> less_leaves(concat(u2, v2), concat(u4, v4)) :|: v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, v2), concat(u4, 0)) :|: v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, v2), concat(0, v8)) :|: v8 >= 0, u8 >= 0, z' = 1 + u2 + v2, z'' = 1 + u8 + v8, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, v2), concat(0, 0)) :|: z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, 0), concat(u5, v5)) :|: z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, 0), concat(u5, 0)) :|: z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, 0), concat(0, v9)) :|: z' = 1 + u2 + v2, z'' = 1 + u9 + v9, u9 >= 0, v9 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(u2, 0), concat(0, 0)) :|: z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(0, v3), concat(u6, v6)) :|: u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, v3), concat(u6, 0)) :|: u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, v3), concat(0, v10)) :|: u3 >= 0, z' = 1 + u3 + v3, z'' = 1 + u10 + v10, u10 >= 0, v10 >= 0, z = 0, v3 >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, v3), concat(0, 0)) :|: u3 >= 0, z'' >= 0, z' = 1 + u3 + v3, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, 0), concat(u7, v7)) :|: u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, 0), concat(u7, 0)) :|: u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, 0), concat(0, v11)) :|: v11 >= 0, u11 >= 0, z = 0, z'' = 1 + u11 + v11, z' >= 0
   if2(z, z', z'') -{ 1 }-> less_leaves(concat(0, 0), concat(0, 0)) :|: z'' >= 0, z = 0, z' >= 0
   if2(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   isLeaf(z) -{ 1 }-> 1 :|: z = 0
   isLeaf(z) -{ 1 }-> 0 :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 1 }-> u :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 0 }-> 0 :|: z >= 0
   less_leaves(z, z') -{ 3 }-> if1(1, 1, 0, 0) :|: z = 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(1, 0, 0, 1 + u'' + v'') :|: u'' >= 0, v'' >= 0, z' = 1 + u'' + v'', z = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 1, 1 + u' + v', 0) :|: z = 1 + u' + v', u' >= 0, v' >= 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 0, 1 + u' + v', 1 + u1 + v1) :|: z = 1 + u' + v', z' = 1 + u1 + v1, u' >= 0, v' >= 0, u1 >= 0, v1 >= 0
   right(z) -{ 1 }-> v :|: z = 1 + u + v, v >= 0, u >= 0
   right(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {concat}, {right}, {left}, {if2,less_leaves,if1}
Previous analysis results are:
  isLeaf: runtime: O(1) [1], size: O(1) [1]

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

(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:

   concat(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   concat(z, z') -{ 1 }-> 1 + u + concat(v, z') :|: z = 1 + u + v, v >= 0, z' >= 0, u >= 0
   if1(z, z', z'', z1) -{ 1 }-> if2(z, z'', z1) :|: z >= 0, z1 >= 0, z' = 0, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z' = 1, z'' >= 0
   if2(z, z', z'') -{ 5 }-> less_leaves(concat(u2, v2), concat(u4, v4)) :|: v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, v2), concat(u4, 0)) :|: v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, v2), concat(0, v8)) :|: v8 >= 0, u8 >= 0, z' = 1 + u2 + v2, z'' = 1 + u8 + v8, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, v2), concat(0, 0)) :|: z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, 0), concat(u5, v5)) :|: z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, 0), concat(u5, 0)) :|: z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, 0), concat(0, v9)) :|: z' = 1 + u2 + v2, z'' = 1 + u9 + v9, u9 >= 0, v9 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(u2, 0), concat(0, 0)) :|: z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(0, v3), concat(u6, v6)) :|: u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, v3), concat(u6, 0)) :|: u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, v3), concat(0, v10)) :|: u3 >= 0, z' = 1 + u3 + v3, z'' = 1 + u10 + v10, u10 >= 0, v10 >= 0, z = 0, v3 >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, v3), concat(0, 0)) :|: u3 >= 0, z'' >= 0, z' = 1 + u3 + v3, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, 0), concat(u7, v7)) :|: u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, 0), concat(u7, 0)) :|: u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, 0), concat(0, v11)) :|: v11 >= 0, u11 >= 0, z = 0, z'' = 1 + u11 + v11, z' >= 0
   if2(z, z', z'') -{ 1 }-> less_leaves(concat(0, 0), concat(0, 0)) :|: z'' >= 0, z = 0, z' >= 0
   if2(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   isLeaf(z) -{ 1 }-> 1 :|: z = 0
   isLeaf(z) -{ 1 }-> 0 :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 1 }-> u :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 0 }-> 0 :|: z >= 0
   less_leaves(z, z') -{ 3 }-> if1(1, 1, 0, 0) :|: z = 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(1, 0, 0, 1 + u'' + v'') :|: u'' >= 0, v'' >= 0, z' = 1 + u'' + v'', z = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 1, 1 + u' + v', 0) :|: z = 1 + u' + v', u' >= 0, v' >= 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 0, 1 + u' + v', 1 + u1 + v1) :|: z = 1 + u' + v', z' = 1 + u1 + v1, u' >= 0, v' >= 0, u1 >= 0, v1 >= 0
   right(z) -{ 1 }-> v :|: z = 1 + u + v, v >= 0, u >= 0
   right(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {concat}, {right}, {left}, {if2,less_leaves,if1}
Previous analysis results are:
  isLeaf: runtime: O(1) [1], size: O(1) [1]

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

(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:

   concat(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   concat(z, z') -{ 1 }-> 1 + u + concat(v, z') :|: z = 1 + u + v, v >= 0, z' >= 0, u >= 0
   if1(z, z', z'', z1) -{ 1 }-> if2(z, z'', z1) :|: z >= 0, z1 >= 0, z' = 0, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z' = 1, z'' >= 0
   if2(z, z', z'') -{ 5 }-> less_leaves(concat(u2, v2), concat(u4, v4)) :|: v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, v2), concat(u4, 0)) :|: v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, v2), concat(0, v8)) :|: v8 >= 0, u8 >= 0, z' = 1 + u2 + v2, z'' = 1 + u8 + v8, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, v2), concat(0, 0)) :|: z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, 0), concat(u5, v5)) :|: z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, 0), concat(u5, 0)) :|: z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, 0), concat(0, v9)) :|: z' = 1 + u2 + v2, z'' = 1 + u9 + v9, u9 >= 0, v9 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(u2, 0), concat(0, 0)) :|: z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(0, v3), concat(u6, v6)) :|: u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, v3), concat(u6, 0)) :|: u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, v3), concat(0, v10)) :|: u3 >= 0, z' = 1 + u3 + v3, z'' = 1 + u10 + v10, u10 >= 0, v10 >= 0, z = 0, v3 >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, v3), concat(0, 0)) :|: u3 >= 0, z'' >= 0, z' = 1 + u3 + v3, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, 0), concat(u7, v7)) :|: u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, 0), concat(u7, 0)) :|: u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, 0), concat(0, v11)) :|: v11 >= 0, u11 >= 0, z = 0, z'' = 1 + u11 + v11, z' >= 0
   if2(z, z', z'') -{ 1 }-> less_leaves(concat(0, 0), concat(0, 0)) :|: z'' >= 0, z = 0, z' >= 0
   if2(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   isLeaf(z) -{ 1 }-> 1 :|: z = 0
   isLeaf(z) -{ 1 }-> 0 :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 1 }-> u :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 0 }-> 0 :|: z >= 0
   less_leaves(z, z') -{ 3 }-> if1(1, 1, 0, 0) :|: z = 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(1, 0, 0, 1 + u'' + v'') :|: u'' >= 0, v'' >= 0, z' = 1 + u'' + v'', z = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 1, 1 + u' + v', 0) :|: z = 1 + u' + v', u' >= 0, v' >= 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 0, 1 + u' + v', 1 + u1 + v1) :|: z = 1 + u' + v', z' = 1 + u1 + v1, u' >= 0, v' >= 0, u1 >= 0, v1 >= 0
   right(z) -{ 1 }-> v :|: z = 1 + u + v, v >= 0, u >= 0
   right(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {concat}, {right}, {left}, {if2,less_leaves,if1}
Previous analysis results are:
  isLeaf: runtime: O(1) [1], size: O(1) [1]
  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:

   concat(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   concat(z, z') -{ 1 }-> 1 + u + concat(v, z') :|: z = 1 + u + v, v >= 0, z' >= 0, u >= 0
   if1(z, z', z'', z1) -{ 1 }-> if2(z, z'', z1) :|: z >= 0, z1 >= 0, z' = 0, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z' = 1, z'' >= 0
   if2(z, z', z'') -{ 5 }-> less_leaves(concat(u2, v2), concat(u4, v4)) :|: v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, v2), concat(u4, 0)) :|: v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, v2), concat(0, v8)) :|: v8 >= 0, u8 >= 0, z' = 1 + u2 + v2, z'' = 1 + u8 + v8, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, v2), concat(0, 0)) :|: z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(u2, 0), concat(u5, v5)) :|: z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, 0), concat(u5, 0)) :|: z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(u2, 0), concat(0, v9)) :|: z' = 1 + u2 + v2, z'' = 1 + u9 + v9, u9 >= 0, v9 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(u2, 0), concat(0, 0)) :|: z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(concat(0, v3), concat(u6, v6)) :|: u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, v3), concat(u6, 0)) :|: u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, v3), concat(0, v10)) :|: u3 >= 0, z' = 1 + u3 + v3, z'' = 1 + u10 + v10, u10 >= 0, v10 >= 0, z = 0, v3 >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, v3), concat(0, 0)) :|: u3 >= 0, z'' >= 0, z' = 1 + u3 + v3, z = 0, v3 >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(concat(0, 0), concat(u7, v7)) :|: u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, 0), concat(u7, 0)) :|: u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 2 }-> less_leaves(concat(0, 0), concat(0, v11)) :|: v11 >= 0, u11 >= 0, z = 0, z'' = 1 + u11 + v11, z' >= 0
   if2(z, z', z'') -{ 1 }-> less_leaves(concat(0, 0), concat(0, 0)) :|: z'' >= 0, z = 0, z' >= 0
   if2(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   isLeaf(z) -{ 1 }-> 1 :|: z = 0
   isLeaf(z) -{ 1 }-> 0 :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 1 }-> u :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 0 }-> 0 :|: z >= 0
   less_leaves(z, z') -{ 3 }-> if1(1, 1, 0, 0) :|: z = 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(1, 0, 0, 1 + u'' + v'') :|: u'' >= 0, v'' >= 0, z' = 1 + u'' + v'', z = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 1, 1 + u' + v', 0) :|: z = 1 + u' + v', u' >= 0, v' >= 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 0, 1 + u' + v', 1 + u1 + v1) :|: z = 1 + u' + v', z' = 1 + u1 + v1, u' >= 0, v' >= 0, u1 >= 0, v1 >= 0
   right(z) -{ 1 }-> v :|: z = 1 + u + v, v >= 0, u >= 0
   right(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {right}, {left}, {if2,less_leaves,if1}
Previous analysis results are:
  isLeaf: runtime: O(1) [1], size: O(1) [1]
  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:

   concat(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   concat(z, z') -{ 2 + v }-> 1 + u + s :|: s >= 0, s <= v + z', z = 1 + u + v, v >= 0, z' >= 0, u >= 0
   if1(z, z', z'', z1) -{ 1 }-> if2(z, z'', z1) :|: z >= 0, z1 >= 0, z' = 0, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z' = 1, z'' >= 0
   if2(z, z', z'') -{ 7 + u2 + u4 }-> less_leaves(s', s'') :|: s' >= 0, s' <= u2 + v2, s'' >= 0, s'' <= u4 + v4, v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u2 + u4 }-> less_leaves(s1, s2) :|: s1 >= 0, s1 <= u2 + v2, s2 >= 0, s2 <= u4 + 0, v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 }-> less_leaves(s11, s12) :|: s11 >= 0, s11 <= u2 + 0, s12 >= 0, s12 <= 0 + v9, z' = 1 + u2 + v2, z'' = 1 + u9 + v9, u9 >= 0, v9 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 + u2 }-> less_leaves(s13, s14) :|: s13 >= 0, s13 <= u2 + 0, s14 >= 0, s14 <= 0 + 0, z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u6 }-> less_leaves(s15, s16) :|: s15 >= 0, s15 <= 0 + v3, s16 >= 0, s16 <= u6 + v6, u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 + u6 }-> less_leaves(s17, s18) :|: s17 >= 0, s17 <= 0 + v3, s18 >= 0, s18 <= u6 + 0, u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 }-> less_leaves(s19, s20) :|: s19 >= 0, s19 <= 0 + v3, s20 >= 0, s20 <= 0 + v10, u3 >= 0, z' = 1 + u3 + v3, z'' = 1 + u10 + v10, u10 >= 0, v10 >= 0, z = 0, v3 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(s21, s22) :|: s21 >= 0, s21 <= 0 + v3, s22 >= 0, s22 <= 0 + 0, u3 >= 0, z'' >= 0, z' = 1 + u3 + v3, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 + u7 }-> less_leaves(s23, s24) :|: s23 >= 0, s23 <= 0 + 0, s24 >= 0, s24 <= u7 + v7, u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 4 + u7 }-> less_leaves(s25, s26) :|: s25 >= 0, s25 <= 0 + 0, s26 >= 0, s26 <= u7 + 0, u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(s27, s28) :|: s27 >= 0, s27 <= 0 + 0, s28 >= 0, s28 <= 0 + v11, v11 >= 0, u11 >= 0, z = 0, z'' = 1 + u11 + v11, z' >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(s29, s30) :|: s29 >= 0, s29 <= 0 + 0, s30 >= 0, s30 <= 0 + 0, z'' >= 0, z = 0, z' >= 0
   if2(z, z', z'') -{ 6 + u2 }-> less_leaves(s3, s4) :|: s3 >= 0, s3 <= u2 + v2, s4 >= 0, s4 <= 0 + v8, v8 >= 0, u8 >= 0, z' = 1 + u2 + v2, z'' = 1 + u8 + v8, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 }-> less_leaves(s5, s6) :|: s5 >= 0, s5 <= u2 + v2, s6 >= 0, s6 <= 0 + 0, z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u2 + u5 }-> less_leaves(s7, s8) :|: s7 >= 0, s7 <= u2 + 0, s8 >= 0, s8 <= u5 + v5, z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 + u5 }-> less_leaves(s9, s10) :|: s9 >= 0, s9 <= u2 + 0, s10 >= 0, s10 <= u5 + 0, z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   isLeaf(z) -{ 1 }-> 1 :|: z = 0
   isLeaf(z) -{ 1 }-> 0 :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 1 }-> u :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 0 }-> 0 :|: z >= 0
   less_leaves(z, z') -{ 3 }-> if1(1, 1, 0, 0) :|: z = 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(1, 0, 0, 1 + u'' + v'') :|: u'' >= 0, v'' >= 0, z' = 1 + u'' + v'', z = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 1, 1 + u' + v', 0) :|: z = 1 + u' + v', u' >= 0, v' >= 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 0, 1 + u' + v', 1 + u1 + v1) :|: z = 1 + u' + v', z' = 1 + u1 + v1, u' >= 0, v' >= 0, u1 >= 0, v1 >= 0
   right(z) -{ 1 }-> v :|: z = 1 + u + v, v >= 0, u >= 0
   right(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {right}, {left}, {if2,less_leaves,if1}
Previous analysis results are:
  isLeaf: runtime: O(1) [1], size: O(1) [1]
  concat: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   concat(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   concat(z, z') -{ 2 + v }-> 1 + u + s :|: s >= 0, s <= v + z', z = 1 + u + v, v >= 0, z' >= 0, u >= 0
   if1(z, z', z'', z1) -{ 1 }-> if2(z, z'', z1) :|: z >= 0, z1 >= 0, z' = 0, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z' = 1, z'' >= 0
   if2(z, z', z'') -{ 7 + u2 + u4 }-> less_leaves(s', s'') :|: s' >= 0, s' <= u2 + v2, s'' >= 0, s'' <= u4 + v4, v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u2 + u4 }-> less_leaves(s1, s2) :|: s1 >= 0, s1 <= u2 + v2, s2 >= 0, s2 <= u4 + 0, v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 }-> less_leaves(s11, s12) :|: s11 >= 0, s11 <= u2 + 0, s12 >= 0, s12 <= 0 + v9, z' = 1 + u2 + v2, z'' = 1 + u9 + v9, u9 >= 0, v9 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 + u2 }-> less_leaves(s13, s14) :|: s13 >= 0, s13 <= u2 + 0, s14 >= 0, s14 <= 0 + 0, z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u6 }-> less_leaves(s15, s16) :|: s15 >= 0, s15 <= 0 + v3, s16 >= 0, s16 <= u6 + v6, u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 + u6 }-> less_leaves(s17, s18) :|: s17 >= 0, s17 <= 0 + v3, s18 >= 0, s18 <= u6 + 0, u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 }-> less_leaves(s19, s20) :|: s19 >= 0, s19 <= 0 + v3, s20 >= 0, s20 <= 0 + v10, u3 >= 0, z' = 1 + u3 + v3, z'' = 1 + u10 + v10, u10 >= 0, v10 >= 0, z = 0, v3 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(s21, s22) :|: s21 >= 0, s21 <= 0 + v3, s22 >= 0, s22 <= 0 + 0, u3 >= 0, z'' >= 0, z' = 1 + u3 + v3, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 + u7 }-> less_leaves(s23, s24) :|: s23 >= 0, s23 <= 0 + 0, s24 >= 0, s24 <= u7 + v7, u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 4 + u7 }-> less_leaves(s25, s26) :|: s25 >= 0, s25 <= 0 + 0, s26 >= 0, s26 <= u7 + 0, u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(s27, s28) :|: s27 >= 0, s27 <= 0 + 0, s28 >= 0, s28 <= 0 + v11, v11 >= 0, u11 >= 0, z = 0, z'' = 1 + u11 + v11, z' >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(s29, s30) :|: s29 >= 0, s29 <= 0 + 0, s30 >= 0, s30 <= 0 + 0, z'' >= 0, z = 0, z' >= 0
   if2(z, z', z'') -{ 6 + u2 }-> less_leaves(s3, s4) :|: s3 >= 0, s3 <= u2 + v2, s4 >= 0, s4 <= 0 + v8, v8 >= 0, u8 >= 0, z' = 1 + u2 + v2, z'' = 1 + u8 + v8, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 }-> less_leaves(s5, s6) :|: s5 >= 0, s5 <= u2 + v2, s6 >= 0, s6 <= 0 + 0, z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u2 + u5 }-> less_leaves(s7, s8) :|: s7 >= 0, s7 <= u2 + 0, s8 >= 0, s8 <= u5 + v5, z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 + u5 }-> less_leaves(s9, s10) :|: s9 >= 0, s9 <= u2 + 0, s10 >= 0, s10 <= u5 + 0, z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   isLeaf(z) -{ 1 }-> 1 :|: z = 0
   isLeaf(z) -{ 1 }-> 0 :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 1 }-> u :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 0 }-> 0 :|: z >= 0
   less_leaves(z, z') -{ 3 }-> if1(1, 1, 0, 0) :|: z = 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(1, 0, 0, 1 + u'' + v'') :|: u'' >= 0, v'' >= 0, z' = 1 + u'' + v'', z = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 1, 1 + u' + v', 0) :|: z = 1 + u' + v', u' >= 0, v' >= 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 0, 1 + u' + v', 1 + u1 + v1) :|: z = 1 + u' + v', z' = 1 + u1 + v1, u' >= 0, v' >= 0, u1 >= 0, v1 >= 0
   right(z) -{ 1 }-> v :|: z = 1 + u + v, v >= 0, u >= 0
   right(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {right}, {left}, {if2,less_leaves,if1}
Previous analysis results are:
  isLeaf: runtime: O(1) [1], size: O(1) [1]
  concat: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  right: runtime: ?, size: O(n^1) [z]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   concat(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   concat(z, z') -{ 2 + v }-> 1 + u + s :|: s >= 0, s <= v + z', z = 1 + u + v, v >= 0, z' >= 0, u >= 0
   if1(z, z', z'', z1) -{ 1 }-> if2(z, z'', z1) :|: z >= 0, z1 >= 0, z' = 0, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z' = 1, z'' >= 0
   if2(z, z', z'') -{ 7 + u2 + u4 }-> less_leaves(s', s'') :|: s' >= 0, s' <= u2 + v2, s'' >= 0, s'' <= u4 + v4, v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u2 + u4 }-> less_leaves(s1, s2) :|: s1 >= 0, s1 <= u2 + v2, s2 >= 0, s2 <= u4 + 0, v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 }-> less_leaves(s11, s12) :|: s11 >= 0, s11 <= u2 + 0, s12 >= 0, s12 <= 0 + v9, z' = 1 + u2 + v2, z'' = 1 + u9 + v9, u9 >= 0, v9 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 + u2 }-> less_leaves(s13, s14) :|: s13 >= 0, s13 <= u2 + 0, s14 >= 0, s14 <= 0 + 0, z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u6 }-> less_leaves(s15, s16) :|: s15 >= 0, s15 <= 0 + v3, s16 >= 0, s16 <= u6 + v6, u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 + u6 }-> less_leaves(s17, s18) :|: s17 >= 0, s17 <= 0 + v3, s18 >= 0, s18 <= u6 + 0, u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 }-> less_leaves(s19, s20) :|: s19 >= 0, s19 <= 0 + v3, s20 >= 0, s20 <= 0 + v10, u3 >= 0, z' = 1 + u3 + v3, z'' = 1 + u10 + v10, u10 >= 0, v10 >= 0, z = 0, v3 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(s21, s22) :|: s21 >= 0, s21 <= 0 + v3, s22 >= 0, s22 <= 0 + 0, u3 >= 0, z'' >= 0, z' = 1 + u3 + v3, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 + u7 }-> less_leaves(s23, s24) :|: s23 >= 0, s23 <= 0 + 0, s24 >= 0, s24 <= u7 + v7, u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 4 + u7 }-> less_leaves(s25, s26) :|: s25 >= 0, s25 <= 0 + 0, s26 >= 0, s26 <= u7 + 0, u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(s27, s28) :|: s27 >= 0, s27 <= 0 + 0, s28 >= 0, s28 <= 0 + v11, v11 >= 0, u11 >= 0, z = 0, z'' = 1 + u11 + v11, z' >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(s29, s30) :|: s29 >= 0, s29 <= 0 + 0, s30 >= 0, s30 <= 0 + 0, z'' >= 0, z = 0, z' >= 0
   if2(z, z', z'') -{ 6 + u2 }-> less_leaves(s3, s4) :|: s3 >= 0, s3 <= u2 + v2, s4 >= 0, s4 <= 0 + v8, v8 >= 0, u8 >= 0, z' = 1 + u2 + v2, z'' = 1 + u8 + v8, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 }-> less_leaves(s5, s6) :|: s5 >= 0, s5 <= u2 + v2, s6 >= 0, s6 <= 0 + 0, z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u2 + u5 }-> less_leaves(s7, s8) :|: s7 >= 0, s7 <= u2 + 0, s8 >= 0, s8 <= u5 + v5, z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 + u5 }-> less_leaves(s9, s10) :|: s9 >= 0, s9 <= u2 + 0, s10 >= 0, s10 <= u5 + 0, z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   isLeaf(z) -{ 1 }-> 1 :|: z = 0
   isLeaf(z) -{ 1 }-> 0 :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 1 }-> u :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 0 }-> 0 :|: z >= 0
   less_leaves(z, z') -{ 3 }-> if1(1, 1, 0, 0) :|: z = 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(1, 0, 0, 1 + u'' + v'') :|: u'' >= 0, v'' >= 0, z' = 1 + u'' + v'', z = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 1, 1 + u' + v', 0) :|: z = 1 + u' + v', u' >= 0, v' >= 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 0, 1 + u' + v', 1 + u1 + v1) :|: z = 1 + u' + v', z' = 1 + u1 + v1, u' >= 0, v' >= 0, u1 >= 0, v1 >= 0
   right(z) -{ 1 }-> v :|: z = 1 + u + v, v >= 0, u >= 0
   right(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {left}, {if2,less_leaves,if1}
Previous analysis results are:
  isLeaf: runtime: O(1) [1], size: O(1) [1]
  concat: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  right: runtime: O(1) [1], size: O(n^1) [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:

   concat(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   concat(z, z') -{ 2 + v }-> 1 + u + s :|: s >= 0, s <= v + z', z = 1 + u + v, v >= 0, z' >= 0, u >= 0
   if1(z, z', z'', z1) -{ 1 }-> if2(z, z'', z1) :|: z >= 0, z1 >= 0, z' = 0, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z' = 1, z'' >= 0
   if2(z, z', z'') -{ 7 + u2 + u4 }-> less_leaves(s', s'') :|: s' >= 0, s' <= u2 + v2, s'' >= 0, s'' <= u4 + v4, v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u2 + u4 }-> less_leaves(s1, s2) :|: s1 >= 0, s1 <= u2 + v2, s2 >= 0, s2 <= u4 + 0, v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 }-> less_leaves(s11, s12) :|: s11 >= 0, s11 <= u2 + 0, s12 >= 0, s12 <= 0 + v9, z' = 1 + u2 + v2, z'' = 1 + u9 + v9, u9 >= 0, v9 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 + u2 }-> less_leaves(s13, s14) :|: s13 >= 0, s13 <= u2 + 0, s14 >= 0, s14 <= 0 + 0, z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u6 }-> less_leaves(s15, s16) :|: s15 >= 0, s15 <= 0 + v3, s16 >= 0, s16 <= u6 + v6, u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 + u6 }-> less_leaves(s17, s18) :|: s17 >= 0, s17 <= 0 + v3, s18 >= 0, s18 <= u6 + 0, u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 }-> less_leaves(s19, s20) :|: s19 >= 0, s19 <= 0 + v3, s20 >= 0, s20 <= 0 + v10, u3 >= 0, z' = 1 + u3 + v3, z'' = 1 + u10 + v10, u10 >= 0, v10 >= 0, z = 0, v3 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(s21, s22) :|: s21 >= 0, s21 <= 0 + v3, s22 >= 0, s22 <= 0 + 0, u3 >= 0, z'' >= 0, z' = 1 + u3 + v3, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 + u7 }-> less_leaves(s23, s24) :|: s23 >= 0, s23 <= 0 + 0, s24 >= 0, s24 <= u7 + v7, u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 4 + u7 }-> less_leaves(s25, s26) :|: s25 >= 0, s25 <= 0 + 0, s26 >= 0, s26 <= u7 + 0, u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(s27, s28) :|: s27 >= 0, s27 <= 0 + 0, s28 >= 0, s28 <= 0 + v11, v11 >= 0, u11 >= 0, z = 0, z'' = 1 + u11 + v11, z' >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(s29, s30) :|: s29 >= 0, s29 <= 0 + 0, s30 >= 0, s30 <= 0 + 0, z'' >= 0, z = 0, z' >= 0
   if2(z, z', z'') -{ 6 + u2 }-> less_leaves(s3, s4) :|: s3 >= 0, s3 <= u2 + v2, s4 >= 0, s4 <= 0 + v8, v8 >= 0, u8 >= 0, z' = 1 + u2 + v2, z'' = 1 + u8 + v8, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 }-> less_leaves(s5, s6) :|: s5 >= 0, s5 <= u2 + v2, s6 >= 0, s6 <= 0 + 0, z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u2 + u5 }-> less_leaves(s7, s8) :|: s7 >= 0, s7 <= u2 + 0, s8 >= 0, s8 <= u5 + v5, z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 + u5 }-> less_leaves(s9, s10) :|: s9 >= 0, s9 <= u2 + 0, s10 >= 0, s10 <= u5 + 0, z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   isLeaf(z) -{ 1 }-> 1 :|: z = 0
   isLeaf(z) -{ 1 }-> 0 :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 1 }-> u :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 0 }-> 0 :|: z >= 0
   less_leaves(z, z') -{ 3 }-> if1(1, 1, 0, 0) :|: z = 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(1, 0, 0, 1 + u'' + v'') :|: u'' >= 0, v'' >= 0, z' = 1 + u'' + v'', z = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 1, 1 + u' + v', 0) :|: z = 1 + u' + v', u' >= 0, v' >= 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 0, 1 + u' + v', 1 + u1 + v1) :|: z = 1 + u' + v', z' = 1 + u1 + v1, u' >= 0, v' >= 0, u1 >= 0, v1 >= 0
   right(z) -{ 1 }-> v :|: z = 1 + u + v, v >= 0, u >= 0
   right(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {left}, {if2,less_leaves,if1}
Previous analysis results are:
  isLeaf: runtime: O(1) [1], size: O(1) [1]
  concat: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  right: runtime: O(1) [1], size: O(n^1) [z]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: left
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:

   concat(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   concat(z, z') -{ 2 + v }-> 1 + u + s :|: s >= 0, s <= v + z', z = 1 + u + v, v >= 0, z' >= 0, u >= 0
   if1(z, z', z'', z1) -{ 1 }-> if2(z, z'', z1) :|: z >= 0, z1 >= 0, z' = 0, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z' = 1, z'' >= 0
   if2(z, z', z'') -{ 7 + u2 + u4 }-> less_leaves(s', s'') :|: s' >= 0, s' <= u2 + v2, s'' >= 0, s'' <= u4 + v4, v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u2 + u4 }-> less_leaves(s1, s2) :|: s1 >= 0, s1 <= u2 + v2, s2 >= 0, s2 <= u4 + 0, v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 }-> less_leaves(s11, s12) :|: s11 >= 0, s11 <= u2 + 0, s12 >= 0, s12 <= 0 + v9, z' = 1 + u2 + v2, z'' = 1 + u9 + v9, u9 >= 0, v9 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 + u2 }-> less_leaves(s13, s14) :|: s13 >= 0, s13 <= u2 + 0, s14 >= 0, s14 <= 0 + 0, z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u6 }-> less_leaves(s15, s16) :|: s15 >= 0, s15 <= 0 + v3, s16 >= 0, s16 <= u6 + v6, u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 + u6 }-> less_leaves(s17, s18) :|: s17 >= 0, s17 <= 0 + v3, s18 >= 0, s18 <= u6 + 0, u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 }-> less_leaves(s19, s20) :|: s19 >= 0, s19 <= 0 + v3, s20 >= 0, s20 <= 0 + v10, u3 >= 0, z' = 1 + u3 + v3, z'' = 1 + u10 + v10, u10 >= 0, v10 >= 0, z = 0, v3 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(s21, s22) :|: s21 >= 0, s21 <= 0 + v3, s22 >= 0, s22 <= 0 + 0, u3 >= 0, z'' >= 0, z' = 1 + u3 + v3, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 + u7 }-> less_leaves(s23, s24) :|: s23 >= 0, s23 <= 0 + 0, s24 >= 0, s24 <= u7 + v7, u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 4 + u7 }-> less_leaves(s25, s26) :|: s25 >= 0, s25 <= 0 + 0, s26 >= 0, s26 <= u7 + 0, u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(s27, s28) :|: s27 >= 0, s27 <= 0 + 0, s28 >= 0, s28 <= 0 + v11, v11 >= 0, u11 >= 0, z = 0, z'' = 1 + u11 + v11, z' >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(s29, s30) :|: s29 >= 0, s29 <= 0 + 0, s30 >= 0, s30 <= 0 + 0, z'' >= 0, z = 0, z' >= 0
   if2(z, z', z'') -{ 6 + u2 }-> less_leaves(s3, s4) :|: s3 >= 0, s3 <= u2 + v2, s4 >= 0, s4 <= 0 + v8, v8 >= 0, u8 >= 0, z' = 1 + u2 + v2, z'' = 1 + u8 + v8, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 }-> less_leaves(s5, s6) :|: s5 >= 0, s5 <= u2 + v2, s6 >= 0, s6 <= 0 + 0, z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u2 + u5 }-> less_leaves(s7, s8) :|: s7 >= 0, s7 <= u2 + 0, s8 >= 0, s8 <= u5 + v5, z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 + u5 }-> less_leaves(s9, s10) :|: s9 >= 0, s9 <= u2 + 0, s10 >= 0, s10 <= u5 + 0, z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   isLeaf(z) -{ 1 }-> 1 :|: z = 0
   isLeaf(z) -{ 1 }-> 0 :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 1 }-> u :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 0 }-> 0 :|: z >= 0
   less_leaves(z, z') -{ 3 }-> if1(1, 1, 0, 0) :|: z = 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(1, 0, 0, 1 + u'' + v'') :|: u'' >= 0, v'' >= 0, z' = 1 + u'' + v'', z = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 1, 1 + u' + v', 0) :|: z = 1 + u' + v', u' >= 0, v' >= 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 0, 1 + u' + v', 1 + u1 + v1) :|: z = 1 + u' + v', z' = 1 + u1 + v1, u' >= 0, v' >= 0, u1 >= 0, v1 >= 0
   right(z) -{ 1 }-> v :|: z = 1 + u + v, v >= 0, u >= 0
   right(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {left}, {if2,less_leaves,if1}
Previous analysis results are:
  isLeaf: runtime: O(1) [1], size: O(1) [1]
  concat: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  right: runtime: O(1) [1], size: O(n^1) [z]
  left: runtime: ?, size: O(n^1) [z]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   concat(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   concat(z, z') -{ 2 + v }-> 1 + u + s :|: s >= 0, s <= v + z', z = 1 + u + v, v >= 0, z' >= 0, u >= 0
   if1(z, z', z'', z1) -{ 1 }-> if2(z, z'', z1) :|: z >= 0, z1 >= 0, z' = 0, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z' = 1, z'' >= 0
   if2(z, z', z'') -{ 7 + u2 + u4 }-> less_leaves(s', s'') :|: s' >= 0, s' <= u2 + v2, s'' >= 0, s'' <= u4 + v4, v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u2 + u4 }-> less_leaves(s1, s2) :|: s1 >= 0, s1 <= u2 + v2, s2 >= 0, s2 <= u4 + 0, v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 }-> less_leaves(s11, s12) :|: s11 >= 0, s11 <= u2 + 0, s12 >= 0, s12 <= 0 + v9, z' = 1 + u2 + v2, z'' = 1 + u9 + v9, u9 >= 0, v9 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 + u2 }-> less_leaves(s13, s14) :|: s13 >= 0, s13 <= u2 + 0, s14 >= 0, s14 <= 0 + 0, z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u6 }-> less_leaves(s15, s16) :|: s15 >= 0, s15 <= 0 + v3, s16 >= 0, s16 <= u6 + v6, u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 + u6 }-> less_leaves(s17, s18) :|: s17 >= 0, s17 <= 0 + v3, s18 >= 0, s18 <= u6 + 0, u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 }-> less_leaves(s19, s20) :|: s19 >= 0, s19 <= 0 + v3, s20 >= 0, s20 <= 0 + v10, u3 >= 0, z' = 1 + u3 + v3, z'' = 1 + u10 + v10, u10 >= 0, v10 >= 0, z = 0, v3 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(s21, s22) :|: s21 >= 0, s21 <= 0 + v3, s22 >= 0, s22 <= 0 + 0, u3 >= 0, z'' >= 0, z' = 1 + u3 + v3, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 + u7 }-> less_leaves(s23, s24) :|: s23 >= 0, s23 <= 0 + 0, s24 >= 0, s24 <= u7 + v7, u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 4 + u7 }-> less_leaves(s25, s26) :|: s25 >= 0, s25 <= 0 + 0, s26 >= 0, s26 <= u7 + 0, u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(s27, s28) :|: s27 >= 0, s27 <= 0 + 0, s28 >= 0, s28 <= 0 + v11, v11 >= 0, u11 >= 0, z = 0, z'' = 1 + u11 + v11, z' >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(s29, s30) :|: s29 >= 0, s29 <= 0 + 0, s30 >= 0, s30 <= 0 + 0, z'' >= 0, z = 0, z' >= 0
   if2(z, z', z'') -{ 6 + u2 }-> less_leaves(s3, s4) :|: s3 >= 0, s3 <= u2 + v2, s4 >= 0, s4 <= 0 + v8, v8 >= 0, u8 >= 0, z' = 1 + u2 + v2, z'' = 1 + u8 + v8, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 }-> less_leaves(s5, s6) :|: s5 >= 0, s5 <= u2 + v2, s6 >= 0, s6 <= 0 + 0, z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u2 + u5 }-> less_leaves(s7, s8) :|: s7 >= 0, s7 <= u2 + 0, s8 >= 0, s8 <= u5 + v5, z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 + u5 }-> less_leaves(s9, s10) :|: s9 >= 0, s9 <= u2 + 0, s10 >= 0, s10 <= u5 + 0, z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   isLeaf(z) -{ 1 }-> 1 :|: z = 0
   isLeaf(z) -{ 1 }-> 0 :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 1 }-> u :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 0 }-> 0 :|: z >= 0
   less_leaves(z, z') -{ 3 }-> if1(1, 1, 0, 0) :|: z = 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(1, 0, 0, 1 + u'' + v'') :|: u'' >= 0, v'' >= 0, z' = 1 + u'' + v'', z = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 1, 1 + u' + v', 0) :|: z = 1 + u' + v', u' >= 0, v' >= 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 0, 1 + u' + v', 1 + u1 + v1) :|: z = 1 + u' + v', z' = 1 + u1 + v1, u' >= 0, v' >= 0, u1 >= 0, v1 >= 0
   right(z) -{ 1 }-> v :|: z = 1 + u + v, v >= 0, u >= 0
   right(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {if2,less_leaves,if1}
Previous analysis results are:
  isLeaf: runtime: O(1) [1], size: O(1) [1]
  concat: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  right: runtime: O(1) [1], size: O(n^1) [z]
  left: runtime: O(1) [1], size: O(n^1) [z]

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

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

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

   concat(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   concat(z, z') -{ 2 + v }-> 1 + u + s :|: s >= 0, s <= v + z', z = 1 + u + v, v >= 0, z' >= 0, u >= 0
   if1(z, z', z'', z1) -{ 1 }-> if2(z, z'', z1) :|: z >= 0, z1 >= 0, z' = 0, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z' = 1, z'' >= 0
   if2(z, z', z'') -{ 7 + u2 + u4 }-> less_leaves(s', s'') :|: s' >= 0, s' <= u2 + v2, s'' >= 0, s'' <= u4 + v4, v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u2 + u4 }-> less_leaves(s1, s2) :|: s1 >= 0, s1 <= u2 + v2, s2 >= 0, s2 <= u4 + 0, v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 }-> less_leaves(s11, s12) :|: s11 >= 0, s11 <= u2 + 0, s12 >= 0, s12 <= 0 + v9, z' = 1 + u2 + v2, z'' = 1 + u9 + v9, u9 >= 0, v9 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 + u2 }-> less_leaves(s13, s14) :|: s13 >= 0, s13 <= u2 + 0, s14 >= 0, s14 <= 0 + 0, z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u6 }-> less_leaves(s15, s16) :|: s15 >= 0, s15 <= 0 + v3, s16 >= 0, s16 <= u6 + v6, u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 + u6 }-> less_leaves(s17, s18) :|: s17 >= 0, s17 <= 0 + v3, s18 >= 0, s18 <= u6 + 0, u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 }-> less_leaves(s19, s20) :|: s19 >= 0, s19 <= 0 + v3, s20 >= 0, s20 <= 0 + v10, u3 >= 0, z' = 1 + u3 + v3, z'' = 1 + u10 + v10, u10 >= 0, v10 >= 0, z = 0, v3 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(s21, s22) :|: s21 >= 0, s21 <= 0 + v3, s22 >= 0, s22 <= 0 + 0, u3 >= 0, z'' >= 0, z' = 1 + u3 + v3, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 + u7 }-> less_leaves(s23, s24) :|: s23 >= 0, s23 <= 0 + 0, s24 >= 0, s24 <= u7 + v7, u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 4 + u7 }-> less_leaves(s25, s26) :|: s25 >= 0, s25 <= 0 + 0, s26 >= 0, s26 <= u7 + 0, u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(s27, s28) :|: s27 >= 0, s27 <= 0 + 0, s28 >= 0, s28 <= 0 + v11, v11 >= 0, u11 >= 0, z = 0, z'' = 1 + u11 + v11, z' >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(s29, s30) :|: s29 >= 0, s29 <= 0 + 0, s30 >= 0, s30 <= 0 + 0, z'' >= 0, z = 0, z' >= 0
   if2(z, z', z'') -{ 6 + u2 }-> less_leaves(s3, s4) :|: s3 >= 0, s3 <= u2 + v2, s4 >= 0, s4 <= 0 + v8, v8 >= 0, u8 >= 0, z' = 1 + u2 + v2, z'' = 1 + u8 + v8, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 }-> less_leaves(s5, s6) :|: s5 >= 0, s5 <= u2 + v2, s6 >= 0, s6 <= 0 + 0, z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u2 + u5 }-> less_leaves(s7, s8) :|: s7 >= 0, s7 <= u2 + 0, s8 >= 0, s8 <= u5 + v5, z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 + u5 }-> less_leaves(s9, s10) :|: s9 >= 0, s9 <= u2 + 0, s10 >= 0, s10 <= u5 + 0, z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   isLeaf(z) -{ 1 }-> 1 :|: z = 0
   isLeaf(z) -{ 1 }-> 0 :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 1 }-> u :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 0 }-> 0 :|: z >= 0
   less_leaves(z, z') -{ 3 }-> if1(1, 1, 0, 0) :|: z = 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(1, 0, 0, 1 + u'' + v'') :|: u'' >= 0, v'' >= 0, z' = 1 + u'' + v'', z = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 1, 1 + u' + v', 0) :|: z = 1 + u' + v', u' >= 0, v' >= 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 0, 1 + u' + v', 1 + u1 + v1) :|: z = 1 + u' + v', z' = 1 + u1 + v1, u' >= 0, v' >= 0, u1 >= 0, v1 >= 0
   right(z) -{ 1 }-> v :|: z = 1 + u + v, v >= 0, u >= 0
   right(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {if2,less_leaves,if1}
Previous analysis results are:
  isLeaf: runtime: O(1) [1], size: O(1) [1]
  concat: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  right: runtime: O(1) [1], size: O(n^1) [z]
  left: runtime: O(1) [1], size: O(n^1) [z]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: if2
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

Computed SIZE bound using CoFloCo for: if1
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:

   concat(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   concat(z, z') -{ 2 + v }-> 1 + u + s :|: s >= 0, s <= v + z', z = 1 + u + v, v >= 0, z' >= 0, u >= 0
   if1(z, z', z'', z1) -{ 1 }-> if2(z, z'', z1) :|: z >= 0, z1 >= 0, z' = 0, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z' = 1, z'' >= 0
   if2(z, z', z'') -{ 7 + u2 + u4 }-> less_leaves(s', s'') :|: s' >= 0, s' <= u2 + v2, s'' >= 0, s'' <= u4 + v4, v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u2 + u4 }-> less_leaves(s1, s2) :|: s1 >= 0, s1 <= u2 + v2, s2 >= 0, s2 <= u4 + 0, v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 }-> less_leaves(s11, s12) :|: s11 >= 0, s11 <= u2 + 0, s12 >= 0, s12 <= 0 + v9, z' = 1 + u2 + v2, z'' = 1 + u9 + v9, u9 >= 0, v9 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 + u2 }-> less_leaves(s13, s14) :|: s13 >= 0, s13 <= u2 + 0, s14 >= 0, s14 <= 0 + 0, z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u6 }-> less_leaves(s15, s16) :|: s15 >= 0, s15 <= 0 + v3, s16 >= 0, s16 <= u6 + v6, u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 + u6 }-> less_leaves(s17, s18) :|: s17 >= 0, s17 <= 0 + v3, s18 >= 0, s18 <= u6 + 0, u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 }-> less_leaves(s19, s20) :|: s19 >= 0, s19 <= 0 + v3, s20 >= 0, s20 <= 0 + v10, u3 >= 0, z' = 1 + u3 + v3, z'' = 1 + u10 + v10, u10 >= 0, v10 >= 0, z = 0, v3 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(s21, s22) :|: s21 >= 0, s21 <= 0 + v3, s22 >= 0, s22 <= 0 + 0, u3 >= 0, z'' >= 0, z' = 1 + u3 + v3, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 + u7 }-> less_leaves(s23, s24) :|: s23 >= 0, s23 <= 0 + 0, s24 >= 0, s24 <= u7 + v7, u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 4 + u7 }-> less_leaves(s25, s26) :|: s25 >= 0, s25 <= 0 + 0, s26 >= 0, s26 <= u7 + 0, u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(s27, s28) :|: s27 >= 0, s27 <= 0 + 0, s28 >= 0, s28 <= 0 + v11, v11 >= 0, u11 >= 0, z = 0, z'' = 1 + u11 + v11, z' >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(s29, s30) :|: s29 >= 0, s29 <= 0 + 0, s30 >= 0, s30 <= 0 + 0, z'' >= 0, z = 0, z' >= 0
   if2(z, z', z'') -{ 6 + u2 }-> less_leaves(s3, s4) :|: s3 >= 0, s3 <= u2 + v2, s4 >= 0, s4 <= 0 + v8, v8 >= 0, u8 >= 0, z' = 1 + u2 + v2, z'' = 1 + u8 + v8, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 }-> less_leaves(s5, s6) :|: s5 >= 0, s5 <= u2 + v2, s6 >= 0, s6 <= 0 + 0, z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u2 + u5 }-> less_leaves(s7, s8) :|: s7 >= 0, s7 <= u2 + 0, s8 >= 0, s8 <= u5 + v5, z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 + u5 }-> less_leaves(s9, s10) :|: s9 >= 0, s9 <= u2 + 0, s10 >= 0, s10 <= u5 + 0, z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   isLeaf(z) -{ 1 }-> 1 :|: z = 0
   isLeaf(z) -{ 1 }-> 0 :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 1 }-> u :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 0 }-> 0 :|: z >= 0
   less_leaves(z, z') -{ 3 }-> if1(1, 1, 0, 0) :|: z = 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(1, 0, 0, 1 + u'' + v'') :|: u'' >= 0, v'' >= 0, z' = 1 + u'' + v'', z = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 1, 1 + u' + v', 0) :|: z = 1 + u' + v', u' >= 0, v' >= 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 0, 1 + u' + v', 1 + u1 + v1) :|: z = 1 + u' + v', z' = 1 + u1 + v1, u' >= 0, v' >= 0, u1 >= 0, v1 >= 0
   right(z) -{ 1 }-> v :|: z = 1 + u + v, v >= 0, u >= 0
   right(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {if2,less_leaves,if1}
Previous analysis results are:
  isLeaf: runtime: O(1) [1], size: O(1) [1]
  concat: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  right: runtime: O(1) [1], size: O(n^1) [z]
  left: runtime: O(1) [1], size: O(n^1) [z]
  if2: runtime: ?, size: O(1) [1]
  less_leaves: runtime: ?, size: O(1) [1]
  if1: runtime: ?, size: O(1) [1]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: if2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 235 + 238*z' + 16*z'*z'' + 8*z'^2 + 238*z'' + 8*z''^2

Computed RUNTIME bound using KoAT for: less_leaves
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 486 + 476*z + 32*z*z' + 16*z^2 + 476*z' + 16*z'^2

Computed RUNTIME bound using KoAT for: if1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 237 + 238*z'' + 16*z''*z1 + 8*z''^2 + 238*z1 + 8*z1^2

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

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

   concat(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   concat(z, z') -{ 2 + v }-> 1 + u + s :|: s >= 0, s <= v + z', z = 1 + u + v, v >= 0, z' >= 0, u >= 0
   if1(z, z', z'', z1) -{ 1 }-> if2(z, z'', z1) :|: z >= 0, z1 >= 0, z' = 0, z'' >= 0
   if1(z, z', z'', z1) -{ 1 }-> 0 :|: z >= 0, z1 >= 0, z' = 1, z'' >= 0
   if2(z, z', z'') -{ 7 + u2 + u4 }-> less_leaves(s', s'') :|: s' >= 0, s' <= u2 + v2, s'' >= 0, s'' <= u4 + v4, v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u2 + u4 }-> less_leaves(s1, s2) :|: s1 >= 0, s1 <= u2 + v2, s2 >= 0, s2 <= u4 + 0, v4 >= 0, z' = 1 + u2 + v2, z'' = 1 + u4 + v4, u2 >= 0, u4 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 }-> less_leaves(s11, s12) :|: s11 >= 0, s11 <= u2 + 0, s12 >= 0, s12 <= 0 + v9, z' = 1 + u2 + v2, z'' = 1 + u9 + v9, u9 >= 0, v9 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 4 + u2 }-> less_leaves(s13, s14) :|: s13 >= 0, s13 <= u2 + 0, s14 >= 0, s14 <= 0 + 0, z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u6 }-> less_leaves(s15, s16) :|: s15 >= 0, s15 <= 0 + v3, s16 >= 0, s16 <= u6 + v6, u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 + u6 }-> less_leaves(s17, s18) :|: s17 >= 0, s17 <= 0 + v3, s18 >= 0, s18 <= u6 + 0, u3 >= 0, z' = 1 + u3 + v3, u6 >= 0, v6 >= 0, z'' = 1 + u6 + v6, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 }-> less_leaves(s19, s20) :|: s19 >= 0, s19 <= 0 + v3, s20 >= 0, s20 <= 0 + v10, u3 >= 0, z' = 1 + u3 + v3, z'' = 1 + u10 + v10, u10 >= 0, v10 >= 0, z = 0, v3 >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(s21, s22) :|: s21 >= 0, s21 <= 0 + v3, s22 >= 0, s22 <= 0 + 0, u3 >= 0, z'' >= 0, z' = 1 + u3 + v3, z = 0, v3 >= 0
   if2(z, z', z'') -{ 5 + u7 }-> less_leaves(s23, s24) :|: s23 >= 0, s23 <= 0 + 0, s24 >= 0, s24 <= u7 + v7, u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 4 + u7 }-> less_leaves(s25, s26) :|: s25 >= 0, s25 <= 0 + 0, s26 >= 0, s26 <= u7 + 0, u7 >= 0, v7 >= 0, z = 0, z'' = 1 + u7 + v7, z' >= 0
   if2(z, z', z'') -{ 4 }-> less_leaves(s27, s28) :|: s27 >= 0, s27 <= 0 + 0, s28 >= 0, s28 <= 0 + v11, v11 >= 0, u11 >= 0, z = 0, z'' = 1 + u11 + v11, z' >= 0
   if2(z, z', z'') -{ 3 }-> less_leaves(s29, s30) :|: s29 >= 0, s29 <= 0 + 0, s30 >= 0, s30 <= 0 + 0, z'' >= 0, z = 0, z' >= 0
   if2(z, z', z'') -{ 6 + u2 }-> less_leaves(s3, s4) :|: s3 >= 0, s3 <= u2 + v2, s4 >= 0, s4 <= 0 + v8, v8 >= 0, u8 >= 0, z' = 1 + u2 + v2, z'' = 1 + u8 + v8, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 }-> less_leaves(s5, s6) :|: s5 >= 0, s5 <= u2 + v2, s6 >= 0, s6 <= 0 + 0, z'' >= 0, z' = 1 + u2 + v2, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 6 + u2 + u5 }-> less_leaves(s7, s8) :|: s7 >= 0, s7 <= u2 + 0, s8 >= 0, s8 <= u5 + v5, z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 5 + u2 + u5 }-> less_leaves(s9, s10) :|: s9 >= 0, s9 <= u2 + 0, s10 >= 0, s10 <= u5 + 0, z' = 1 + u2 + v2, z'' = 1 + u5 + v5, u5 >= 0, v5 >= 0, u2 >= 0, z = 0, v2 >= 0
   if2(z, z', z'') -{ 1 }-> 1 :|: z'' >= 0, z = 1, z' >= 0
   isLeaf(z) -{ 1 }-> 1 :|: z = 0
   isLeaf(z) -{ 1 }-> 0 :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 1 }-> u :|: z = 1 + u + v, v >= 0, u >= 0
   left(z) -{ 0 }-> 0 :|: z >= 0
   less_leaves(z, z') -{ 3 }-> if1(1, 1, 0, 0) :|: z = 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(1, 0, 0, 1 + u'' + v'') :|: u'' >= 0, v'' >= 0, z' = 1 + u'' + v'', z = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 1, 1 + u' + v', 0) :|: z = 1 + u' + v', u' >= 0, v' >= 0, z' = 0
   less_leaves(z, z') -{ 3 }-> if1(0, 0, 1 + u' + v', 1 + u1 + v1) :|: z = 1 + u' + v', z' = 1 + u1 + v1, u' >= 0, v' >= 0, u1 >= 0, v1 >= 0
   right(z) -{ 1 }-> v :|: z = 1 + u + v, v >= 0, u >= 0
   right(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  isLeaf: runtime: O(1) [1], size: O(1) [1]
  concat: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  right: runtime: O(1) [1], size: O(n^1) [z]
  left: runtime: O(1) [1], size: O(n^1) [z]
  if2: runtime: O(n^2) [235 + 238*z' + 16*z'*z'' + 8*z'^2 + 238*z'' + 8*z''^2], size: O(1) [1]
  less_leaves: runtime: O(n^2) [486 + 476*z + 32*z*z' + 16*z^2 + 476*z' + 16*z'^2], size: O(1) [1]
  if1: runtime: O(n^2) [237 + 238*z'' + 16*z''*z1 + 8*z''^2 + 238*z1 + 8*z1^2], size: O(1) [1]

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

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

(46)
BOUNDS(1, n^2)

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

(47) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   isLeaf(leaf) -> true
   isLeaf(cons(z0, z1)) -> false
   left(cons(z0, z1)) -> z0
   right(cons(z0, z1)) -> z1
   concat(leaf, z0) -> z0
   concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
   less_leaves(z0, z1) -> if1(isLeaf(z0), isLeaf(z1), z0, z1)
   if1(z0, true, z1, z2) -> false
   if1(z0, false, z1, z2) -> if2(z0, z1, z2)
   if2(true, z0, z1) -> true
   if2(false, z0, z1) -> less_leaves(concat(left(z0), right(z0)), concat(left(z1), right(z1)))
Tuples:
   ISLEAF(leaf) -> c
   ISLEAF(cons(z0, z1)) -> c1
   LEFT(cons(z0, z1)) -> c2
   RIGHT(cons(z0, z1)) -> c3
   CONCAT(leaf, z0) -> c4
   CONCAT(cons(z0, z1), z2) -> c5(CONCAT(z1, z2))
   LESS_LEAVES(z0, z1) -> c6(IF1(isLeaf(z0), isLeaf(z1), z0, z1), ISLEAF(z0))
   LESS_LEAVES(z0, z1) -> c7(IF1(isLeaf(z0), isLeaf(z1), z0, z1), ISLEAF(z1))
   IF1(z0, true, z1, z2) -> c8
   IF1(z0, false, z1, z2) -> c9(IF2(z0, z1, z2))
   IF2(true, z0, z1) -> c10
   IF2(false, z0, z1) -> c11(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z0), right(z0)), LEFT(z0))
   IF2(false, z0, z1) -> c12(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z0), right(z0)), RIGHT(z0))
   IF2(false, z0, z1) -> c13(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z1), right(z1)), LEFT(z1))
   IF2(false, z0, z1) -> c14(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z1), right(z1)), RIGHT(z1))
S tuples:
   ISLEAF(leaf) -> c
   ISLEAF(cons(z0, z1)) -> c1
   LEFT(cons(z0, z1)) -> c2
   RIGHT(cons(z0, z1)) -> c3
   CONCAT(leaf, z0) -> c4
   CONCAT(cons(z0, z1), z2) -> c5(CONCAT(z1, z2))
   LESS_LEAVES(z0, z1) -> c6(IF1(isLeaf(z0), isLeaf(z1), z0, z1), ISLEAF(z0))
   LESS_LEAVES(z0, z1) -> c7(IF1(isLeaf(z0), isLeaf(z1), z0, z1), ISLEAF(z1))
   IF1(z0, true, z1, z2) -> c8
   IF1(z0, false, z1, z2) -> c9(IF2(z0, z1, z2))
   IF2(true, z0, z1) -> c10
   IF2(false, z0, z1) -> c11(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z0), right(z0)), LEFT(z0))
   IF2(false, z0, z1) -> c12(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z0), right(z0)), RIGHT(z0))
   IF2(false, z0, z1) -> c13(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z1), right(z1)), LEFT(z1))
   IF2(false, z0, z1) -> c14(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z1), right(z1)), RIGHT(z1))
K tuples:none
Defined Rule Symbols:   isLeaf_1, left_1, right_1, concat_2, less_leaves_2, if1_4, if2_3

Defined Pair Symbols:   ISLEAF_1, LEFT_1, RIGHT_1, CONCAT_2, LESS_LEAVES_2, IF1_4, IF2_3

Compound Symbols:   c, c1, c2, c3, c4, c5_1, c6_2, c7_2, c8, c9_1, c10, c11_3, c12_3, c13_3, c14_3


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

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

(50)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   ISLEAF(leaf) -> c
   ISLEAF(cons(z0, z1)) -> c1
   LEFT(cons(z0, z1)) -> c2
   RIGHT(cons(z0, z1)) -> c3
   CONCAT(leaf, z0) -> c4
   CONCAT(cons(z0, z1), z2) -> c5(CONCAT(z1, z2))
   LESS_LEAVES(z0, z1) -> c6(IF1(isLeaf(z0), isLeaf(z1), z0, z1), ISLEAF(z0))
   LESS_LEAVES(z0, z1) -> c7(IF1(isLeaf(z0), isLeaf(z1), z0, z1), ISLEAF(z1))
   IF1(z0, true, z1, z2) -> c8
   IF1(z0, false, z1, z2) -> c9(IF2(z0, z1, z2))
   IF2(true, z0, z1) -> c10
   IF2(false, z0, z1) -> c11(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z0), right(z0)), LEFT(z0))
   IF2(false, z0, z1) -> c12(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z0), right(z0)), RIGHT(z0))
   IF2(false, z0, z1) -> c13(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z1), right(z1)), LEFT(z1))
   IF2(false, z0, z1) -> c14(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z1), right(z1)), RIGHT(z1))

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

   isLeaf(leaf) -> true
   isLeaf(cons(z0, z1)) -> false
   left(cons(z0, z1)) -> z0
   right(cons(z0, z1)) -> z1
   concat(leaf, z0) -> z0
   concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
   less_leaves(z0, z1) -> if1(isLeaf(z0), isLeaf(z1), z0, z1)
   if1(z0, true, z1, z2) -> false
   if1(z0, false, z1, z2) -> if2(z0, z1, z2)
   if2(true, z0, z1) -> true
   if2(false, z0, z1) -> less_leaves(concat(left(z0), right(z0)), concat(left(z1), right(z1)))

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

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

(52)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   ISLEAF(leaf) -> c
   ISLEAF(cons(z0, z1)) -> c1
   LEFT(cons(z0, z1)) -> c2
   RIGHT(cons(z0, z1)) -> c3
   CONCAT(leaf, z0) -> c4
   CONCAT(cons(z0, z1), z2) -> c5(CONCAT(z1, z2))
   LESS_LEAVES(z0, z1) -> c6(IF1(isLeaf(z0), isLeaf(z1), z0, z1), ISLEAF(z0))
   LESS_LEAVES(z0, z1) -> c7(IF1(isLeaf(z0), isLeaf(z1), z0, z1), ISLEAF(z1))
   IF1(z0, true, z1, z2) -> c8
   IF1(z0, false, z1, z2) -> c9(IF2(z0, z1, z2))
   IF2(true, z0, z1) -> c10
   IF2(false, z0, z1) -> c11(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z0), right(z0)), LEFT(z0))
   IF2(false, z0, z1) -> c12(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z0), right(z0)), RIGHT(z0))
   IF2(false, z0, z1) -> c13(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z1), right(z1)), LEFT(z1))
   IF2(false, z0, z1) -> c14(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z1), right(z1)), RIGHT(z1))

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

   isLeaf(leaf) -> true
   isLeaf(cons(z0, z1)) -> false
   left(cons(z0, z1)) -> z0
   right(cons(z0, z1)) -> z1
   concat(leaf, z0) -> z0
   concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
   less_leaves(z0, z1) -> if1(isLeaf(z0), isLeaf(z1), z0, z1)
   if1(z0, true, z1, z2) -> false
   if1(z0, false, z1, z2) -> if2(z0, z1, z2)
   if2(true, z0, z1) -> true
   if2(false, z0, z1) -> less_leaves(concat(left(z0), right(z0)), concat(left(z1), right(z1)))

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

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

(54)
Obligation:
Innermost TRS:
Rules:
ISLEAF(leaf) -> c
ISLEAF(cons(z0, z1)) -> c1
LEFT(cons(z0, z1)) -> c2
RIGHT(cons(z0, z1)) -> c3
CONCAT(leaf, z0) -> c4
CONCAT(cons(z0, z1), z2) -> c5(CONCAT(z1, z2))
LESS_LEAVES(z0, z1) -> c6(IF1(isLeaf(z0), isLeaf(z1), z0, z1), ISLEAF(z0))
LESS_LEAVES(z0, z1) -> c7(IF1(isLeaf(z0), isLeaf(z1), z0, z1), ISLEAF(z1))
IF1(z0, true, z1, z2) -> c8
IF1(z0, false, z1, z2) -> c9(IF2(z0, z1, z2))
IF2(true, z0, z1) -> c10
IF2(false, z0, z1) -> c11(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z0), right(z0)), LEFT(z0))
IF2(false, z0, z1) -> c12(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z0), right(z0)), RIGHT(z0))
IF2(false, z0, z1) -> c13(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z1), right(z1)), LEFT(z1))
IF2(false, z0, z1) -> c14(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z1), right(z1)), RIGHT(z1))
isLeaf(leaf) -> true
isLeaf(cons(z0, z1)) -> false
left(cons(z0, z1)) -> z0
right(cons(z0, z1)) -> z1
concat(leaf, z0) -> z0
concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
less_leaves(z0, z1) -> if1(isLeaf(z0), isLeaf(z1), z0, z1)
if1(z0, true, z1, z2) -> false
if1(z0, false, z1, z2) -> if2(z0, z1, z2)
if2(true, z0, z1) -> true
if2(false, z0, z1) -> less_leaves(concat(left(z0), right(z0)), concat(left(z1), right(z1)))

Types:
ISLEAF :: leaf:cons -> c:c1
leaf :: leaf:cons
c :: c:c1
cons :: leaf:cons -> leaf:cons -> leaf:cons
c1 :: c:c1
LEFT :: leaf:cons -> c2
c2 :: c2
RIGHT :: leaf:cons -> c3
c3 :: c3
CONCAT :: leaf:cons -> leaf:cons -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
LESS_LEAVES :: leaf:cons -> leaf:cons -> c6:c7
c6 :: c8:c9 -> c:c1 -> c6:c7
IF1 :: true:false -> true:false -> leaf:cons -> leaf:cons -> c8:c9
isLeaf :: leaf:cons -> true:false
c7 :: c8:c9 -> c:c1 -> c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c10:c11:c12:c13:c14 -> c8:c9
IF2 :: true:false -> leaf:cons -> leaf:cons -> c10:c11:c12:c13:c14
c10 :: c10:c11:c12:c13:c14
c11 :: c6:c7 -> c4:c5 -> c2 -> c10:c11:c12:c13:c14
concat :: leaf:cons -> leaf:cons -> leaf:cons
left :: leaf:cons -> leaf:cons
right :: leaf:cons -> leaf:cons
c12 :: c6:c7 -> c4:c5 -> c3 -> c10:c11:c12:c13:c14
c13 :: c6:c7 -> c4:c5 -> c2 -> c10:c11:c12:c13:c14
c14 :: c6:c7 -> c4:c5 -> c3 -> c10:c11:c12:c13:c14
less_leaves :: leaf:cons -> leaf:cons -> true:false
if1 :: true:false -> true:false -> leaf:cons -> leaf:cons -> true:false
if2 :: true:false -> leaf:cons -> leaf:cons -> true:false
hole_c:c11_15 :: c:c1
hole_leaf:cons2_15 :: leaf:cons
hole_c23_15 :: c2
hole_c34_15 :: c3
hole_c4:c55_15 :: c4:c5
hole_c6:c76_15 :: c6:c7
hole_c8:c97_15 :: c8:c9
hole_true:false8_15 :: true:false
hole_c10:c11:c12:c13:c149_15 :: c10:c11:c12:c13:c14
gen_leaf:cons10_15 :: Nat -> leaf:cons
gen_c4:c511_15 :: Nat -> c4:c5

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

(55) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
CONCAT, LESS_LEAVES, concat, less_leaves

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

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

(56)
Obligation:
Innermost TRS:
Rules:
ISLEAF(leaf) -> c
ISLEAF(cons(z0, z1)) -> c1
LEFT(cons(z0, z1)) -> c2
RIGHT(cons(z0, z1)) -> c3
CONCAT(leaf, z0) -> c4
CONCAT(cons(z0, z1), z2) -> c5(CONCAT(z1, z2))
LESS_LEAVES(z0, z1) -> c6(IF1(isLeaf(z0), isLeaf(z1), z0, z1), ISLEAF(z0))
LESS_LEAVES(z0, z1) -> c7(IF1(isLeaf(z0), isLeaf(z1), z0, z1), ISLEAF(z1))
IF1(z0, true, z1, z2) -> c8
IF1(z0, false, z1, z2) -> c9(IF2(z0, z1, z2))
IF2(true, z0, z1) -> c10
IF2(false, z0, z1) -> c11(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z0), right(z0)), LEFT(z0))
IF2(false, z0, z1) -> c12(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z0), right(z0)), RIGHT(z0))
IF2(false, z0, z1) -> c13(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z1), right(z1)), LEFT(z1))
IF2(false, z0, z1) -> c14(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z1), right(z1)), RIGHT(z1))
isLeaf(leaf) -> true
isLeaf(cons(z0, z1)) -> false
left(cons(z0, z1)) -> z0
right(cons(z0, z1)) -> z1
concat(leaf, z0) -> z0
concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
less_leaves(z0, z1) -> if1(isLeaf(z0), isLeaf(z1), z0, z1)
if1(z0, true, z1, z2) -> false
if1(z0, false, z1, z2) -> if2(z0, z1, z2)
if2(true, z0, z1) -> true
if2(false, z0, z1) -> less_leaves(concat(left(z0), right(z0)), concat(left(z1), right(z1)))

Types:
ISLEAF :: leaf:cons -> c:c1
leaf :: leaf:cons
c :: c:c1
cons :: leaf:cons -> leaf:cons -> leaf:cons
c1 :: c:c1
LEFT :: leaf:cons -> c2
c2 :: c2
RIGHT :: leaf:cons -> c3
c3 :: c3
CONCAT :: leaf:cons -> leaf:cons -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
LESS_LEAVES :: leaf:cons -> leaf:cons -> c6:c7
c6 :: c8:c9 -> c:c1 -> c6:c7
IF1 :: true:false -> true:false -> leaf:cons -> leaf:cons -> c8:c9
isLeaf :: leaf:cons -> true:false
c7 :: c8:c9 -> c:c1 -> c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c10:c11:c12:c13:c14 -> c8:c9
IF2 :: true:false -> leaf:cons -> leaf:cons -> c10:c11:c12:c13:c14
c10 :: c10:c11:c12:c13:c14
c11 :: c6:c7 -> c4:c5 -> c2 -> c10:c11:c12:c13:c14
concat :: leaf:cons -> leaf:cons -> leaf:cons
left :: leaf:cons -> leaf:cons
right :: leaf:cons -> leaf:cons
c12 :: c6:c7 -> c4:c5 -> c3 -> c10:c11:c12:c13:c14
c13 :: c6:c7 -> c4:c5 -> c2 -> c10:c11:c12:c13:c14
c14 :: c6:c7 -> c4:c5 -> c3 -> c10:c11:c12:c13:c14
less_leaves :: leaf:cons -> leaf:cons -> true:false
if1 :: true:false -> true:false -> leaf:cons -> leaf:cons -> true:false
if2 :: true:false -> leaf:cons -> leaf:cons -> true:false
hole_c:c11_15 :: c:c1
hole_leaf:cons2_15 :: leaf:cons
hole_c23_15 :: c2
hole_c34_15 :: c3
hole_c4:c55_15 :: c4:c5
hole_c6:c76_15 :: c6:c7
hole_c8:c97_15 :: c8:c9
hole_true:false8_15 :: true:false
hole_c10:c11:c12:c13:c149_15 :: c10:c11:c12:c13:c14
gen_leaf:cons10_15 :: Nat -> leaf:cons
gen_c4:c511_15 :: Nat -> c4:c5


Generator Equations:
gen_leaf:cons10_15(0) <=> leaf
gen_leaf:cons10_15(+(x, 1)) <=> cons(leaf, gen_leaf:cons10_15(x))
gen_c4:c511_15(0) <=> c4
gen_c4:c511_15(+(x, 1)) <=> c5(gen_c4:c511_15(x))


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

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

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

(57) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
CONCAT(gen_leaf:cons10_15(n13_15), gen_leaf:cons10_15(b)) -> gen_c4:c511_15(n13_15), rt in Omega(1 + n13_15)

Induction Base:
CONCAT(gen_leaf:cons10_15(0), gen_leaf:cons10_15(b)) ->_R^Omega(1)
c4

Induction Step:
CONCAT(gen_leaf:cons10_15(+(n13_15, 1)), gen_leaf:cons10_15(b)) ->_R^Omega(1)
c5(CONCAT(gen_leaf:cons10_15(n13_15), gen_leaf:cons10_15(b))) ->_IH
c5(gen_c4:c511_15(c14_15))

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

(58)
Complex Obligation (BEST)

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

(59)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
ISLEAF(leaf) -> c
ISLEAF(cons(z0, z1)) -> c1
LEFT(cons(z0, z1)) -> c2
RIGHT(cons(z0, z1)) -> c3
CONCAT(leaf, z0) -> c4
CONCAT(cons(z0, z1), z2) -> c5(CONCAT(z1, z2))
LESS_LEAVES(z0, z1) -> c6(IF1(isLeaf(z0), isLeaf(z1), z0, z1), ISLEAF(z0))
LESS_LEAVES(z0, z1) -> c7(IF1(isLeaf(z0), isLeaf(z1), z0, z1), ISLEAF(z1))
IF1(z0, true, z1, z2) -> c8
IF1(z0, false, z1, z2) -> c9(IF2(z0, z1, z2))
IF2(true, z0, z1) -> c10
IF2(false, z0, z1) -> c11(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z0), right(z0)), LEFT(z0))
IF2(false, z0, z1) -> c12(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z0), right(z0)), RIGHT(z0))
IF2(false, z0, z1) -> c13(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z1), right(z1)), LEFT(z1))
IF2(false, z0, z1) -> c14(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z1), right(z1)), RIGHT(z1))
isLeaf(leaf) -> true
isLeaf(cons(z0, z1)) -> false
left(cons(z0, z1)) -> z0
right(cons(z0, z1)) -> z1
concat(leaf, z0) -> z0
concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
less_leaves(z0, z1) -> if1(isLeaf(z0), isLeaf(z1), z0, z1)
if1(z0, true, z1, z2) -> false
if1(z0, false, z1, z2) -> if2(z0, z1, z2)
if2(true, z0, z1) -> true
if2(false, z0, z1) -> less_leaves(concat(left(z0), right(z0)), concat(left(z1), right(z1)))

Types:
ISLEAF :: leaf:cons -> c:c1
leaf :: leaf:cons
c :: c:c1
cons :: leaf:cons -> leaf:cons -> leaf:cons
c1 :: c:c1
LEFT :: leaf:cons -> c2
c2 :: c2
RIGHT :: leaf:cons -> c3
c3 :: c3
CONCAT :: leaf:cons -> leaf:cons -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
LESS_LEAVES :: leaf:cons -> leaf:cons -> c6:c7
c6 :: c8:c9 -> c:c1 -> c6:c7
IF1 :: true:false -> true:false -> leaf:cons -> leaf:cons -> c8:c9
isLeaf :: leaf:cons -> true:false
c7 :: c8:c9 -> c:c1 -> c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c10:c11:c12:c13:c14 -> c8:c9
IF2 :: true:false -> leaf:cons -> leaf:cons -> c10:c11:c12:c13:c14
c10 :: c10:c11:c12:c13:c14
c11 :: c6:c7 -> c4:c5 -> c2 -> c10:c11:c12:c13:c14
concat :: leaf:cons -> leaf:cons -> leaf:cons
left :: leaf:cons -> leaf:cons
right :: leaf:cons -> leaf:cons
c12 :: c6:c7 -> c4:c5 -> c3 -> c10:c11:c12:c13:c14
c13 :: c6:c7 -> c4:c5 -> c2 -> c10:c11:c12:c13:c14
c14 :: c6:c7 -> c4:c5 -> c3 -> c10:c11:c12:c13:c14
less_leaves :: leaf:cons -> leaf:cons -> true:false
if1 :: true:false -> true:false -> leaf:cons -> leaf:cons -> true:false
if2 :: true:false -> leaf:cons -> leaf:cons -> true:false
hole_c:c11_15 :: c:c1
hole_leaf:cons2_15 :: leaf:cons
hole_c23_15 :: c2
hole_c34_15 :: c3
hole_c4:c55_15 :: c4:c5
hole_c6:c76_15 :: c6:c7
hole_c8:c97_15 :: c8:c9
hole_true:false8_15 :: true:false
hole_c10:c11:c12:c13:c149_15 :: c10:c11:c12:c13:c14
gen_leaf:cons10_15 :: Nat -> leaf:cons
gen_c4:c511_15 :: Nat -> c4:c5


Generator Equations:
gen_leaf:cons10_15(0) <=> leaf
gen_leaf:cons10_15(+(x, 1)) <=> cons(leaf, gen_leaf:cons10_15(x))
gen_c4:c511_15(0) <=> c4
gen_c4:c511_15(+(x, 1)) <=> c5(gen_c4:c511_15(x))


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

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

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

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

(61)
BOUNDS(n^1, INF)

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

(62)
Obligation:
Innermost TRS:
Rules:
ISLEAF(leaf) -> c
ISLEAF(cons(z0, z1)) -> c1
LEFT(cons(z0, z1)) -> c2
RIGHT(cons(z0, z1)) -> c3
CONCAT(leaf, z0) -> c4
CONCAT(cons(z0, z1), z2) -> c5(CONCAT(z1, z2))
LESS_LEAVES(z0, z1) -> c6(IF1(isLeaf(z0), isLeaf(z1), z0, z1), ISLEAF(z0))
LESS_LEAVES(z0, z1) -> c7(IF1(isLeaf(z0), isLeaf(z1), z0, z1), ISLEAF(z1))
IF1(z0, true, z1, z2) -> c8
IF1(z0, false, z1, z2) -> c9(IF2(z0, z1, z2))
IF2(true, z0, z1) -> c10
IF2(false, z0, z1) -> c11(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z0), right(z0)), LEFT(z0))
IF2(false, z0, z1) -> c12(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z0), right(z0)), RIGHT(z0))
IF2(false, z0, z1) -> c13(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z1), right(z1)), LEFT(z1))
IF2(false, z0, z1) -> c14(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z1), right(z1)), RIGHT(z1))
isLeaf(leaf) -> true
isLeaf(cons(z0, z1)) -> false
left(cons(z0, z1)) -> z0
right(cons(z0, z1)) -> z1
concat(leaf, z0) -> z0
concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
less_leaves(z0, z1) -> if1(isLeaf(z0), isLeaf(z1), z0, z1)
if1(z0, true, z1, z2) -> false
if1(z0, false, z1, z2) -> if2(z0, z1, z2)
if2(true, z0, z1) -> true
if2(false, z0, z1) -> less_leaves(concat(left(z0), right(z0)), concat(left(z1), right(z1)))

Types:
ISLEAF :: leaf:cons -> c:c1
leaf :: leaf:cons
c :: c:c1
cons :: leaf:cons -> leaf:cons -> leaf:cons
c1 :: c:c1
LEFT :: leaf:cons -> c2
c2 :: c2
RIGHT :: leaf:cons -> c3
c3 :: c3
CONCAT :: leaf:cons -> leaf:cons -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
LESS_LEAVES :: leaf:cons -> leaf:cons -> c6:c7
c6 :: c8:c9 -> c:c1 -> c6:c7
IF1 :: true:false -> true:false -> leaf:cons -> leaf:cons -> c8:c9
isLeaf :: leaf:cons -> true:false
c7 :: c8:c9 -> c:c1 -> c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c10:c11:c12:c13:c14 -> c8:c9
IF2 :: true:false -> leaf:cons -> leaf:cons -> c10:c11:c12:c13:c14
c10 :: c10:c11:c12:c13:c14
c11 :: c6:c7 -> c4:c5 -> c2 -> c10:c11:c12:c13:c14
concat :: leaf:cons -> leaf:cons -> leaf:cons
left :: leaf:cons -> leaf:cons
right :: leaf:cons -> leaf:cons
c12 :: c6:c7 -> c4:c5 -> c3 -> c10:c11:c12:c13:c14
c13 :: c6:c7 -> c4:c5 -> c2 -> c10:c11:c12:c13:c14
c14 :: c6:c7 -> c4:c5 -> c3 -> c10:c11:c12:c13:c14
less_leaves :: leaf:cons -> leaf:cons -> true:false
if1 :: true:false -> true:false -> leaf:cons -> leaf:cons -> true:false
if2 :: true:false -> leaf:cons -> leaf:cons -> true:false
hole_c:c11_15 :: c:c1
hole_leaf:cons2_15 :: leaf:cons
hole_c23_15 :: c2
hole_c34_15 :: c3
hole_c4:c55_15 :: c4:c5
hole_c6:c76_15 :: c6:c7
hole_c8:c97_15 :: c8:c9
hole_true:false8_15 :: true:false
hole_c10:c11:c12:c13:c149_15 :: c10:c11:c12:c13:c14
gen_leaf:cons10_15 :: Nat -> leaf:cons
gen_c4:c511_15 :: Nat -> c4:c5


Lemmas:
CONCAT(gen_leaf:cons10_15(n13_15), gen_leaf:cons10_15(b)) -> gen_c4:c511_15(n13_15), rt in Omega(1 + n13_15)


Generator Equations:
gen_leaf:cons10_15(0) <=> leaf
gen_leaf:cons10_15(+(x, 1)) <=> cons(leaf, gen_leaf:cons10_15(x))
gen_c4:c511_15(0) <=> c4
gen_c4:c511_15(+(x, 1)) <=> c5(gen_c4:c511_15(x))


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

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

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

(63) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
concat(gen_leaf:cons10_15(n632_15), gen_leaf:cons10_15(b)) -> gen_leaf:cons10_15(+(n632_15, b)), rt in Omega(0)

Induction Base:
concat(gen_leaf:cons10_15(0), gen_leaf:cons10_15(b)) ->_R^Omega(0)
gen_leaf:cons10_15(b)

Induction Step:
concat(gen_leaf:cons10_15(+(n632_15, 1)), gen_leaf:cons10_15(b)) ->_R^Omega(0)
cons(leaf, concat(gen_leaf:cons10_15(n632_15), gen_leaf:cons10_15(b))) ->_IH
cons(leaf, gen_leaf:cons10_15(+(b, c633_15)))

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

(64)
Obligation:
Innermost TRS:
Rules:
ISLEAF(leaf) -> c
ISLEAF(cons(z0, z1)) -> c1
LEFT(cons(z0, z1)) -> c2
RIGHT(cons(z0, z1)) -> c3
CONCAT(leaf, z0) -> c4
CONCAT(cons(z0, z1), z2) -> c5(CONCAT(z1, z2))
LESS_LEAVES(z0, z1) -> c6(IF1(isLeaf(z0), isLeaf(z1), z0, z1), ISLEAF(z0))
LESS_LEAVES(z0, z1) -> c7(IF1(isLeaf(z0), isLeaf(z1), z0, z1), ISLEAF(z1))
IF1(z0, true, z1, z2) -> c8
IF1(z0, false, z1, z2) -> c9(IF2(z0, z1, z2))
IF2(true, z0, z1) -> c10
IF2(false, z0, z1) -> c11(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z0), right(z0)), LEFT(z0))
IF2(false, z0, z1) -> c12(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z0), right(z0)), RIGHT(z0))
IF2(false, z0, z1) -> c13(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z1), right(z1)), LEFT(z1))
IF2(false, z0, z1) -> c14(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z1), right(z1)), RIGHT(z1))
isLeaf(leaf) -> true
isLeaf(cons(z0, z1)) -> false
left(cons(z0, z1)) -> z0
right(cons(z0, z1)) -> z1
concat(leaf, z0) -> z0
concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
less_leaves(z0, z1) -> if1(isLeaf(z0), isLeaf(z1), z0, z1)
if1(z0, true, z1, z2) -> false
if1(z0, false, z1, z2) -> if2(z0, z1, z2)
if2(true, z0, z1) -> true
if2(false, z0, z1) -> less_leaves(concat(left(z0), right(z0)), concat(left(z1), right(z1)))

Types:
ISLEAF :: leaf:cons -> c:c1
leaf :: leaf:cons
c :: c:c1
cons :: leaf:cons -> leaf:cons -> leaf:cons
c1 :: c:c1
LEFT :: leaf:cons -> c2
c2 :: c2
RIGHT :: leaf:cons -> c3
c3 :: c3
CONCAT :: leaf:cons -> leaf:cons -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
LESS_LEAVES :: leaf:cons -> leaf:cons -> c6:c7
c6 :: c8:c9 -> c:c1 -> c6:c7
IF1 :: true:false -> true:false -> leaf:cons -> leaf:cons -> c8:c9
isLeaf :: leaf:cons -> true:false
c7 :: c8:c9 -> c:c1 -> c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c10:c11:c12:c13:c14 -> c8:c9
IF2 :: true:false -> leaf:cons -> leaf:cons -> c10:c11:c12:c13:c14
c10 :: c10:c11:c12:c13:c14
c11 :: c6:c7 -> c4:c5 -> c2 -> c10:c11:c12:c13:c14
concat :: leaf:cons -> leaf:cons -> leaf:cons
left :: leaf:cons -> leaf:cons
right :: leaf:cons -> leaf:cons
c12 :: c6:c7 -> c4:c5 -> c3 -> c10:c11:c12:c13:c14
c13 :: c6:c7 -> c4:c5 -> c2 -> c10:c11:c12:c13:c14
c14 :: c6:c7 -> c4:c5 -> c3 -> c10:c11:c12:c13:c14
less_leaves :: leaf:cons -> leaf:cons -> true:false
if1 :: true:false -> true:false -> leaf:cons -> leaf:cons -> true:false
if2 :: true:false -> leaf:cons -> leaf:cons -> true:false
hole_c:c11_15 :: c:c1
hole_leaf:cons2_15 :: leaf:cons
hole_c23_15 :: c2
hole_c34_15 :: c3
hole_c4:c55_15 :: c4:c5
hole_c6:c76_15 :: c6:c7
hole_c8:c97_15 :: c8:c9
hole_true:false8_15 :: true:false
hole_c10:c11:c12:c13:c149_15 :: c10:c11:c12:c13:c14
gen_leaf:cons10_15 :: Nat -> leaf:cons
gen_c4:c511_15 :: Nat -> c4:c5


Lemmas:
CONCAT(gen_leaf:cons10_15(n13_15), gen_leaf:cons10_15(b)) -> gen_c4:c511_15(n13_15), rt in Omega(1 + n13_15)
concat(gen_leaf:cons10_15(n632_15), gen_leaf:cons10_15(b)) -> gen_leaf:cons10_15(+(n632_15, b)), rt in Omega(0)


Generator Equations:
gen_leaf:cons10_15(0) <=> leaf
gen_leaf:cons10_15(+(x, 1)) <=> cons(leaf, gen_leaf:cons10_15(x))
gen_c4:c511_15(0) <=> c4
gen_c4:c511_15(+(x, 1)) <=> c5(gen_c4:c511_15(x))


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

(65) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LESS_LEAVES(gen_leaf:cons10_15(n1597_15), gen_leaf:cons10_15(n1597_15)) -> *12_15, rt in Omega(n1597_15)

Induction Base:
LESS_LEAVES(gen_leaf:cons10_15(0), gen_leaf:cons10_15(0))

Induction Step:
LESS_LEAVES(gen_leaf:cons10_15(+(n1597_15, 1)), gen_leaf:cons10_15(+(n1597_15, 1))) ->_R^Omega(1)
c6(IF1(isLeaf(gen_leaf:cons10_15(+(n1597_15, 1))), isLeaf(gen_leaf:cons10_15(+(n1597_15, 1))), gen_leaf:cons10_15(+(n1597_15, 1)), gen_leaf:cons10_15(+(n1597_15, 1))), ISLEAF(gen_leaf:cons10_15(+(n1597_15, 1)))) ->_R^Omega(0)
c6(IF1(false, isLeaf(gen_leaf:cons10_15(+(1, n1597_15))), gen_leaf:cons10_15(+(1, n1597_15)), gen_leaf:cons10_15(+(1, n1597_15))), ISLEAF(gen_leaf:cons10_15(+(1, n1597_15)))) ->_R^Omega(0)
c6(IF1(false, false, gen_leaf:cons10_15(+(1, n1597_15)), gen_leaf:cons10_15(+(1, n1597_15))), ISLEAF(gen_leaf:cons10_15(+(1, n1597_15)))) ->_R^Omega(1)
c6(c9(IF2(false, gen_leaf:cons10_15(+(1, n1597_15)), gen_leaf:cons10_15(+(1, n1597_15)))), ISLEAF(gen_leaf:cons10_15(+(1, n1597_15)))) ->_R^Omega(1)
c6(c9(c11(LESS_LEAVES(concat(left(gen_leaf:cons10_15(+(1, n1597_15))), right(gen_leaf:cons10_15(+(1, n1597_15)))), concat(left(gen_leaf:cons10_15(+(1, n1597_15))), right(gen_leaf:cons10_15(+(1, n1597_15))))), CONCAT(left(gen_leaf:cons10_15(+(1, n1597_15))), right(gen_leaf:cons10_15(+(1, n1597_15)))), LEFT(gen_leaf:cons10_15(+(1, n1597_15))))), ISLEAF(gen_leaf:cons10_15(+(1, n1597_15)))) ->_R^Omega(0)
c6(c9(c11(LESS_LEAVES(concat(leaf, right(gen_leaf:cons10_15(+(1, n1597_15)))), concat(left(gen_leaf:cons10_15(+(1, n1597_15))), right(gen_leaf:cons10_15(+(1, n1597_15))))), CONCAT(left(gen_leaf:cons10_15(+(1, n1597_15))), right(gen_leaf:cons10_15(+(1, n1597_15)))), LEFT(gen_leaf:cons10_15(+(1, n1597_15))))), ISLEAF(gen_leaf:cons10_15(+(1, n1597_15)))) ->_R^Omega(0)
c6(c9(c11(LESS_LEAVES(concat(leaf, gen_leaf:cons10_15(n1597_15)), concat(left(gen_leaf:cons10_15(+(1, n1597_15))), right(gen_leaf:cons10_15(+(1, n1597_15))))), CONCAT(left(gen_leaf:cons10_15(+(1, n1597_15))), right(gen_leaf:cons10_15(+(1, n1597_15)))), LEFT(gen_leaf:cons10_15(+(1, n1597_15))))), ISLEAF(gen_leaf:cons10_15(+(1, n1597_15)))) ->_L^Omega(0)
c6(c9(c11(LESS_LEAVES(gen_leaf:cons10_15(+(0, n1597_15)), concat(left(gen_leaf:cons10_15(+(1, n1597_15))), right(gen_leaf:cons10_15(+(1, n1597_15))))), CONCAT(left(gen_leaf:cons10_15(+(1, n1597_15))), right(gen_leaf:cons10_15(+(1, n1597_15)))), LEFT(gen_leaf:cons10_15(+(1, n1597_15))))), ISLEAF(gen_leaf:cons10_15(+(1, n1597_15)))) ->_R^Omega(0)
c6(c9(c11(LESS_LEAVES(gen_leaf:cons10_15(n1597_15), concat(leaf, right(gen_leaf:cons10_15(+(1, n1597_15))))), CONCAT(left(gen_leaf:cons10_15(+(1, n1597_15))), right(gen_leaf:cons10_15(+(1, n1597_15)))), LEFT(gen_leaf:cons10_15(+(1, n1597_15))))), ISLEAF(gen_leaf:cons10_15(+(1, n1597_15)))) ->_R^Omega(0)
c6(c9(c11(LESS_LEAVES(gen_leaf:cons10_15(n1597_15), concat(leaf, gen_leaf:cons10_15(n1597_15))), CONCAT(left(gen_leaf:cons10_15(+(1, n1597_15))), right(gen_leaf:cons10_15(+(1, n1597_15)))), LEFT(gen_leaf:cons10_15(+(1, n1597_15))))), ISLEAF(gen_leaf:cons10_15(+(1, n1597_15)))) ->_L^Omega(0)
c6(c9(c11(LESS_LEAVES(gen_leaf:cons10_15(n1597_15), gen_leaf:cons10_15(+(0, n1597_15))), CONCAT(left(gen_leaf:cons10_15(+(1, n1597_15))), right(gen_leaf:cons10_15(+(1, n1597_15)))), LEFT(gen_leaf:cons10_15(+(1, n1597_15))))), ISLEAF(gen_leaf:cons10_15(+(1, n1597_15)))) ->_IH
c6(c9(c11(*12_15, CONCAT(left(gen_leaf:cons10_15(+(1, n1597_15))), right(gen_leaf:cons10_15(+(1, n1597_15)))), LEFT(gen_leaf:cons10_15(+(1, n1597_15))))), ISLEAF(gen_leaf:cons10_15(+(1, n1597_15)))) ->_R^Omega(0)
c6(c9(c11(*12_15, CONCAT(leaf, right(gen_leaf:cons10_15(+(1, n1597_15)))), LEFT(gen_leaf:cons10_15(+(1, n1597_15))))), ISLEAF(gen_leaf:cons10_15(+(1, n1597_15)))) ->_R^Omega(0)
c6(c9(c11(*12_15, CONCAT(leaf, gen_leaf:cons10_15(n1597_15)), LEFT(gen_leaf:cons10_15(+(1, n1597_15))))), ISLEAF(gen_leaf:cons10_15(+(1, n1597_15)))) ->_L^Omega(1)
c6(c9(c11(*12_15, gen_c4:c511_15(0), LEFT(gen_leaf:cons10_15(+(1, n1597_15))))), ISLEAF(gen_leaf:cons10_15(+(1, n1597_15)))) ->_R^Omega(1)
c6(c9(c11(*12_15, gen_c4:c511_15(0), c2)), ISLEAF(gen_leaf:cons10_15(+(1, n1597_15)))) ->_R^Omega(1)
c6(c9(c11(*12_15, gen_c4:c511_15(0), c2)), c1)

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

(66)
Obligation:
Innermost TRS:
Rules:
ISLEAF(leaf) -> c
ISLEAF(cons(z0, z1)) -> c1
LEFT(cons(z0, z1)) -> c2
RIGHT(cons(z0, z1)) -> c3
CONCAT(leaf, z0) -> c4
CONCAT(cons(z0, z1), z2) -> c5(CONCAT(z1, z2))
LESS_LEAVES(z0, z1) -> c6(IF1(isLeaf(z0), isLeaf(z1), z0, z1), ISLEAF(z0))
LESS_LEAVES(z0, z1) -> c7(IF1(isLeaf(z0), isLeaf(z1), z0, z1), ISLEAF(z1))
IF1(z0, true, z1, z2) -> c8
IF1(z0, false, z1, z2) -> c9(IF2(z0, z1, z2))
IF2(true, z0, z1) -> c10
IF2(false, z0, z1) -> c11(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z0), right(z0)), LEFT(z0))
IF2(false, z0, z1) -> c12(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z0), right(z0)), RIGHT(z0))
IF2(false, z0, z1) -> c13(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z1), right(z1)), LEFT(z1))
IF2(false, z0, z1) -> c14(LESS_LEAVES(concat(left(z0), right(z0)), concat(left(z1), right(z1))), CONCAT(left(z1), right(z1)), RIGHT(z1))
isLeaf(leaf) -> true
isLeaf(cons(z0, z1)) -> false
left(cons(z0, z1)) -> z0
right(cons(z0, z1)) -> z1
concat(leaf, z0) -> z0
concat(cons(z0, z1), z2) -> cons(z0, concat(z1, z2))
less_leaves(z0, z1) -> if1(isLeaf(z0), isLeaf(z1), z0, z1)
if1(z0, true, z1, z2) -> false
if1(z0, false, z1, z2) -> if2(z0, z1, z2)
if2(true, z0, z1) -> true
if2(false, z0, z1) -> less_leaves(concat(left(z0), right(z0)), concat(left(z1), right(z1)))

Types:
ISLEAF :: leaf:cons -> c:c1
leaf :: leaf:cons
c :: c:c1
cons :: leaf:cons -> leaf:cons -> leaf:cons
c1 :: c:c1
LEFT :: leaf:cons -> c2
c2 :: c2
RIGHT :: leaf:cons -> c3
c3 :: c3
CONCAT :: leaf:cons -> leaf:cons -> c4:c5
c4 :: c4:c5
c5 :: c4:c5 -> c4:c5
LESS_LEAVES :: leaf:cons -> leaf:cons -> c6:c7
c6 :: c8:c9 -> c:c1 -> c6:c7
IF1 :: true:false -> true:false -> leaf:cons -> leaf:cons -> c8:c9
isLeaf :: leaf:cons -> true:false
c7 :: c8:c9 -> c:c1 -> c6:c7
true :: true:false
c8 :: c8:c9
false :: true:false
c9 :: c10:c11:c12:c13:c14 -> c8:c9
IF2 :: true:false -> leaf:cons -> leaf:cons -> c10:c11:c12:c13:c14
c10 :: c10:c11:c12:c13:c14
c11 :: c6:c7 -> c4:c5 -> c2 -> c10:c11:c12:c13:c14
concat :: leaf:cons -> leaf:cons -> leaf:cons
left :: leaf:cons -> leaf:cons
right :: leaf:cons -> leaf:cons
c12 :: c6:c7 -> c4:c5 -> c3 -> c10:c11:c12:c13:c14
c13 :: c6:c7 -> c4:c5 -> c2 -> c10:c11:c12:c13:c14
c14 :: c6:c7 -> c4:c5 -> c3 -> c10:c11:c12:c13:c14
less_leaves :: leaf:cons -> leaf:cons -> true:false
if1 :: true:false -> true:false -> leaf:cons -> leaf:cons -> true:false
if2 :: true:false -> leaf:cons -> leaf:cons -> true:false
hole_c:c11_15 :: c:c1
hole_leaf:cons2_15 :: leaf:cons
hole_c23_15 :: c2
hole_c34_15 :: c3
hole_c4:c55_15 :: c4:c5
hole_c6:c76_15 :: c6:c7
hole_c8:c97_15 :: c8:c9
hole_true:false8_15 :: true:false
hole_c10:c11:c12:c13:c149_15 :: c10:c11:c12:c13:c14
gen_leaf:cons10_15 :: Nat -> leaf:cons
gen_c4:c511_15 :: Nat -> c4:c5


Lemmas:
CONCAT(gen_leaf:cons10_15(n13_15), gen_leaf:cons10_15(b)) -> gen_c4:c511_15(n13_15), rt in Omega(1 + n13_15)
concat(gen_leaf:cons10_15(n632_15), gen_leaf:cons10_15(b)) -> gen_leaf:cons10_15(+(n632_15, b)), rt in Omega(0)
LESS_LEAVES(gen_leaf:cons10_15(n1597_15), gen_leaf:cons10_15(n1597_15)) -> *12_15, rt in Omega(n1597_15)


Generator Equations:
gen_leaf:cons10_15(0) <=> leaf
gen_leaf:cons10_15(+(x, 1)) <=> cons(leaf, gen_leaf:cons10_15(x))
gen_c4:c511_15(0) <=> c4
gen_c4:c511_15(+(x, 1)) <=> c5(gen_c4:c511_15(x))


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

(67) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
less_leaves(gen_leaf:cons10_15(+(1, n143783_15)), gen_leaf:cons10_15(n143783_15)) -> false, rt in Omega(0)

Induction Base:
less_leaves(gen_leaf:cons10_15(+(1, 0)), gen_leaf:cons10_15(0)) ->_R^Omega(0)
if1(isLeaf(gen_leaf:cons10_15(+(1, 0))), isLeaf(gen_leaf:cons10_15(0)), gen_leaf:cons10_15(+(1, 0)), gen_leaf:cons10_15(0)) ->_R^Omega(0)
if1(false, isLeaf(gen_leaf:cons10_15(0)), gen_leaf:cons10_15(1), gen_leaf:cons10_15(0)) ->_R^Omega(0)
if1(false, true, gen_leaf:cons10_15(1), gen_leaf:cons10_15(0)) ->_R^Omega(0)
false

Induction Step:
less_leaves(gen_leaf:cons10_15(+(1, +(n143783_15, 1))), gen_leaf:cons10_15(+(n143783_15, 1))) ->_R^Omega(0)
if1(isLeaf(gen_leaf:cons10_15(+(1, +(n143783_15, 1)))), isLeaf(gen_leaf:cons10_15(+(n143783_15, 1))), gen_leaf:cons10_15(+(1, +(n143783_15, 1))), gen_leaf:cons10_15(+(n143783_15, 1))) ->_R^Omega(0)
if1(false, isLeaf(gen_leaf:cons10_15(+(1, n143783_15))), gen_leaf:cons10_15(+(2, n143783_15)), gen_leaf:cons10_15(+(1, n143783_15))) ->_R^Omega(0)
if1(false, false, gen_leaf:cons10_15(+(2, n143783_15)), gen_leaf:cons10_15(+(1, n143783_15))) ->_R^Omega(0)
if2(false, gen_leaf:cons10_15(+(2, n143783_15)), gen_leaf:cons10_15(+(1, n143783_15))) ->_R^Omega(0)
less_leaves(concat(left(gen_leaf:cons10_15(+(2, n143783_15))), right(gen_leaf:cons10_15(+(2, n143783_15)))), concat(left(gen_leaf:cons10_15(+(1, n143783_15))), right(gen_leaf:cons10_15(+(1, n143783_15))))) ->_R^Omega(0)
less_leaves(concat(leaf, right(gen_leaf:cons10_15(+(2, n143783_15)))), concat(left(gen_leaf:cons10_15(+(1, n143783_15))), right(gen_leaf:cons10_15(+(1, n143783_15))))) ->_R^Omega(0)
less_leaves(concat(leaf, gen_leaf:cons10_15(+(1, n143783_15))), concat(left(gen_leaf:cons10_15(+(1, n143783_15))), right(gen_leaf:cons10_15(+(1, n143783_15))))) ->_L^Omega(0)
less_leaves(gen_leaf:cons10_15(+(0, +(1, n143783_15))), concat(left(gen_leaf:cons10_15(+(1, n143783_15))), right(gen_leaf:cons10_15(+(1, n143783_15))))) ->_R^Omega(0)
less_leaves(gen_leaf:cons10_15(+(1, n143783_15)), concat(leaf, right(gen_leaf:cons10_15(+(1, n143783_15))))) ->_R^Omega(0)
less_leaves(gen_leaf:cons10_15(+(1, n143783_15)), concat(leaf, gen_leaf:cons10_15(n143783_15))) ->_L^Omega(0)
less_leaves(gen_leaf:cons10_15(+(1, n143783_15)), gen_leaf:cons10_15(+(0, n143783_15))) ->_IH
false

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

(68)
BOUNDS(1, INF)