WORST_CASE(Omega(n^1),?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 1396 ms]
(2) CpxRelTRS
(3) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) typed CpxTrs
(7) OrderProof [LOWER BOUND(ID), 5 ms]
(8) typed CpxTrs
(9) RewriteLemmaProof [LOWER BOUND(ID), 289 ms]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 97 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 7104 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 195 ms]
        (20) BOUNDS(1, INF)


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

(0)
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
----------------------------------------

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

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

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

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

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

(6)
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

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

(7) 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

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

(8)
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

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

(9) 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).
----------------------------------------

(10)
Complex Obligation (BEST)

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

(11)
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

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

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

(13)
BOUNDS(n^1, INF)

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

(14)
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

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

(15) 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).
----------------------------------------

(16)
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
----------------------------------------

(17) 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).
----------------------------------------

(18)
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
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(19) 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).
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(20)
BOUNDS(1, INF)