WORST_CASE(Omega(n^2),?)
proof of input_pLB18o5RGX.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CdtProblem
(3) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) typed CpxTrs
(9) OrderProof [LOWER BOUND(ID), 0 ms]
(10) typed CpxTrs
(11) RewriteLemmaProof [LOWER BOUND(ID), 281 ms]
(12) BEST
    (13) proven lower bound
        (14) LowerBoundPropagationProof [FINISHED, 0 ms]
        (15) BOUNDS(n^1, INF)
    (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 107 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 8884 ms]
        (20) BEST
            (21) proven lower bound
                (22) LowerBoundPropagationProof [FINISHED, 0 ms]
                (23) BOUNDS(n^2, INF)
            (24) typed CpxTrs
                (25) RewriteLemmaProof [LOWER BOUND(ID), 254 ms]
                (26) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   isEmpty(empty) -> true
   isEmpty(node(l, r)) -> false
   left(empty) -> empty
   left(node(l, r)) -> l
   right(empty) -> empty
   right(node(l, r)) -> r
   inc(0) -> s(0)
   inc(s(x)) -> s(inc(x))
   count(n, x) -> if(isEmpty(n), isEmpty(left(n)), right(n), node(left(left(n)), node(right(left(n)), right(n))), x, inc(x))
   if(true, b, n, m, x, y) -> x
   if(false, false, n, m, x, y) -> count(m, x)
   if(false, true, n, m, x, y) -> count(n, y)
   nrOfNodes(n) -> count(n, 0)

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

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

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   isEmpty(empty) -> true
   isEmpty(node(z0, z1)) -> false
   left(empty) -> empty
   left(node(z0, z1)) -> z0
   right(empty) -> empty
   right(node(z0, z1)) -> z1
   inc(0) -> s(0)
   inc(s(z0)) -> s(inc(z0))
   count(z0, z1) -> if(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1))
   if(true, z0, z1, z2, z3, z4) -> z3
   if(false, false, z0, z1, z2, z3) -> count(z1, z2)
   if(false, true, z0, z1, z2, z3) -> count(z0, z3)
   nrOfNodes(z0) -> count(z0, 0)
Tuples:
   ISEMPTY(empty) -> c
   ISEMPTY(node(z0, z1)) -> c1
   LEFT(empty) -> c2
   LEFT(node(z0, z1)) -> c3
   RIGHT(empty) -> c4
   RIGHT(node(z0, z1)) -> c5
   INC(0) -> c6
   INC(s(z0)) -> c7(INC(z0))
   COUNT(z0, z1) -> c8(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), ISEMPTY(z0))
   COUNT(z0, z1) -> c9(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), ISEMPTY(left(z0)), LEFT(z0))
   COUNT(z0, z1) -> c10(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(z0))
   COUNT(z0, z1) -> c11(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), LEFT(left(z0)), LEFT(z0))
   COUNT(z0, z1) -> c12(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(left(z0)), LEFT(z0))
   COUNT(z0, z1) -> c13(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(z0))
   COUNT(z0, z1) -> c14(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), INC(z1))
   IF(true, z0, z1, z2, z3, z4) -> c15
   IF(false, false, z0, z1, z2, z3) -> c16(COUNT(z1, z2))
   IF(false, true, z0, z1, z2, z3) -> c17(COUNT(z0, z3))
   NROFNODES(z0) -> c18(COUNT(z0, 0))
S tuples:
   ISEMPTY(empty) -> c
   ISEMPTY(node(z0, z1)) -> c1
   LEFT(empty) -> c2
   LEFT(node(z0, z1)) -> c3
   RIGHT(empty) -> c4
   RIGHT(node(z0, z1)) -> c5
   INC(0) -> c6
   INC(s(z0)) -> c7(INC(z0))
   COUNT(z0, z1) -> c8(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), ISEMPTY(z0))
   COUNT(z0, z1) -> c9(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), ISEMPTY(left(z0)), LEFT(z0))
   COUNT(z0, z1) -> c10(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(z0))
   COUNT(z0, z1) -> c11(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), LEFT(left(z0)), LEFT(z0))
   COUNT(z0, z1) -> c12(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(left(z0)), LEFT(z0))
   COUNT(z0, z1) -> c13(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(z0))
   COUNT(z0, z1) -> c14(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), INC(z1))
   IF(true, z0, z1, z2, z3, z4) -> c15
   IF(false, false, z0, z1, z2, z3) -> c16(COUNT(z1, z2))
   IF(false, true, z0, z1, z2, z3) -> c17(COUNT(z0, z3))
   NROFNODES(z0) -> c18(COUNT(z0, 0))
K tuples:none
Defined Rule Symbols:   isEmpty_1, left_1, right_1, inc_1, count_2, if_6, nrOfNodes_1

Defined Pair Symbols:   ISEMPTY_1, LEFT_1, RIGHT_1, INC_1, COUNT_2, IF_6, NROFNODES_1

Compound Symbols:   c, c1, c2, c3, c4, c5, c6, c7_1, c8_2, c9_3, c10_2, c11_3, c12_3, c13_2, c14_2, c15, c16_1, c17_1, c18_1


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

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   ISEMPTY(empty) -> c
   ISEMPTY(node(z0, z1)) -> c1
   LEFT(empty) -> c2
   LEFT(node(z0, z1)) -> c3
   RIGHT(empty) -> c4
   RIGHT(node(z0, z1)) -> c5
   INC(0) -> c6
   INC(s(z0)) -> c7(INC(z0))
   COUNT(z0, z1) -> c8(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), ISEMPTY(z0))
   COUNT(z0, z1) -> c9(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), ISEMPTY(left(z0)), LEFT(z0))
   COUNT(z0, z1) -> c10(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(z0))
   COUNT(z0, z1) -> c11(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), LEFT(left(z0)), LEFT(z0))
   COUNT(z0, z1) -> c12(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(left(z0)), LEFT(z0))
   COUNT(z0, z1) -> c13(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(z0))
   COUNT(z0, z1) -> c14(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), INC(z1))
   IF(true, z0, z1, z2, z3, z4) -> c15
   IF(false, false, z0, z1, z2, z3) -> c16(COUNT(z1, z2))
   IF(false, true, z0, z1, z2, z3) -> c17(COUNT(z0, z3))
   NROFNODES(z0) -> c18(COUNT(z0, 0))

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

   isEmpty(empty) -> true
   isEmpty(node(z0, z1)) -> false
   left(empty) -> empty
   left(node(z0, z1)) -> z0
   right(empty) -> empty
   right(node(z0, z1)) -> z1
   inc(0) -> s(0)
   inc(s(z0)) -> s(inc(z0))
   count(z0, z1) -> if(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1))
   if(true, z0, z1, z2, z3, z4) -> z3
   if(false, false, z0, z1, z2, z3) -> count(z1, z2)
   if(false, true, z0, z1, z2, z3) -> count(z0, z3)
   nrOfNodes(z0) -> count(z0, 0)

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

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

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   ISEMPTY(empty) -> c
   ISEMPTY(node(z0, z1)) -> c1
   LEFT(empty) -> c2
   LEFT(node(z0, z1)) -> c3
   RIGHT(empty) -> c4
   RIGHT(node(z0, z1)) -> c5
   INC(0') -> c6
   INC(s(z0)) -> c7(INC(z0))
   COUNT(z0, z1) -> c8(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), ISEMPTY(z0))
   COUNT(z0, z1) -> c9(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), ISEMPTY(left(z0)), LEFT(z0))
   COUNT(z0, z1) -> c10(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(z0))
   COUNT(z0, z1) -> c11(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), LEFT(left(z0)), LEFT(z0))
   COUNT(z0, z1) -> c12(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(left(z0)), LEFT(z0))
   COUNT(z0, z1) -> c13(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(z0))
   COUNT(z0, z1) -> c14(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), INC(z1))
   IF(true, z0, z1, z2, z3, z4) -> c15
   IF(false, false, z0, z1, z2, z3) -> c16(COUNT(z1, z2))
   IF(false, true, z0, z1, z2, z3) -> c17(COUNT(z0, z3))
   NROFNODES(z0) -> c18(COUNT(z0, 0'))

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

   isEmpty(empty) -> true
   isEmpty(node(z0, z1)) -> false
   left(empty) -> empty
   left(node(z0, z1)) -> z0
   right(empty) -> empty
   right(node(z0, z1)) -> z1
   inc(0') -> s(0')
   inc(s(z0)) -> s(inc(z0))
   count(z0, z1) -> if(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1))
   if(true, z0, z1, z2, z3, z4) -> z3
   if(false, false, z0, z1, z2, z3) -> count(z1, z2)
   if(false, true, z0, z1, z2, z3) -> count(z0, z3)
   nrOfNodes(z0) -> count(z0, 0')

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

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

(8)
Obligation:
Innermost TRS:
Rules:
ISEMPTY(empty) -> c
ISEMPTY(node(z0, z1)) -> c1
LEFT(empty) -> c2
LEFT(node(z0, z1)) -> c3
RIGHT(empty) -> c4
RIGHT(node(z0, z1)) -> c5
INC(0') -> c6
INC(s(z0)) -> c7(INC(z0))
COUNT(z0, z1) -> c8(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), ISEMPTY(z0))
COUNT(z0, z1) -> c9(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), ISEMPTY(left(z0)), LEFT(z0))
COUNT(z0, z1) -> c10(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(z0))
COUNT(z0, z1) -> c11(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), LEFT(left(z0)), LEFT(z0))
COUNT(z0, z1) -> c12(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(left(z0)), LEFT(z0))
COUNT(z0, z1) -> c13(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(z0))
COUNT(z0, z1) -> c14(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), INC(z1))
IF(true, z0, z1, z2, z3, z4) -> c15
IF(false, false, z0, z1, z2, z3) -> c16(COUNT(z1, z2))
IF(false, true, z0, z1, z2, z3) -> c17(COUNT(z0, z3))
NROFNODES(z0) -> c18(COUNT(z0, 0'))
isEmpty(empty) -> true
isEmpty(node(z0, z1)) -> false
left(empty) -> empty
left(node(z0, z1)) -> z0
right(empty) -> empty
right(node(z0, z1)) -> z1
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))
count(z0, z1) -> if(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1))
if(true, z0, z1, z2, z3, z4) -> z3
if(false, false, z0, z1, z2, z3) -> count(z1, z2)
if(false, true, z0, z1, z2, z3) -> count(z0, z3)
nrOfNodes(z0) -> count(z0, 0')

Types:
ISEMPTY :: empty:node -> c:c1
empty :: empty:node
c :: c:c1
node :: empty:node -> empty:node -> empty:node
c1 :: c:c1
LEFT :: empty:node -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
RIGHT :: empty:node -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
INC :: 0':s -> c6:c7
0' :: 0':s
c6 :: c6:c7
s :: 0':s -> 0':s
c7 :: c6:c7 -> c6:c7
COUNT :: empty:node -> 0':s -> c8:c9:c10:c11:c12:c13:c14
c8 :: c15:c16:c17 -> c:c1 -> c8:c9:c10:c11:c12:c13:c14
IF :: true:false -> true:false -> empty:node -> empty:node -> 0':s -> 0':s -> c15:c16:c17
isEmpty :: empty:node -> true:false
left :: empty:node -> empty:node
right :: empty:node -> empty:node
inc :: 0':s -> 0':s
c9 :: c15:c16:c17 -> c:c1 -> c2:c3 -> c8:c9:c10:c11:c12:c13:c14
c10 :: c15:c16:c17 -> c4:c5 -> c8:c9:c10:c11:c12:c13:c14
c11 :: c15:c16:c17 -> c2:c3 -> c2:c3 -> c8:c9:c10:c11:c12:c13:c14
c12 :: c15:c16:c17 -> c4:c5 -> c2:c3 -> c8:c9:c10:c11:c12:c13:c14
c13 :: c15:c16:c17 -> c4:c5 -> c8:c9:c10:c11:c12:c13:c14
c14 :: c15:c16:c17 -> c6:c7 -> c8:c9:c10:c11:c12:c13:c14
true :: true:false
c15 :: c15:c16:c17
false :: true:false
c16 :: c8:c9:c10:c11:c12:c13:c14 -> c15:c16:c17
c17 :: c8:c9:c10:c11:c12:c13:c14 -> c15:c16:c17
NROFNODES :: empty:node -> c18
c18 :: c8:c9:c10:c11:c12:c13:c14 -> c18
count :: empty:node -> 0':s -> 0':s
if :: true:false -> true:false -> empty:node -> empty:node -> 0':s -> 0':s -> 0':s
nrOfNodes :: empty:node -> 0':s
hole_c:c11_19 :: c:c1
hole_empty:node2_19 :: empty:node
hole_c2:c33_19 :: c2:c3
hole_c4:c54_19 :: c4:c5
hole_c6:c75_19 :: c6:c7
hole_0':s6_19 :: 0':s
hole_c8:c9:c10:c11:c12:c13:c147_19 :: c8:c9:c10:c11:c12:c13:c14
hole_c15:c16:c178_19 :: c15:c16:c17
hole_true:false9_19 :: true:false
hole_c1810_19 :: c18
gen_empty:node11_19 :: Nat -> empty:node
gen_c6:c712_19 :: Nat -> c6:c7
gen_0':s13_19 :: Nat -> 0':s

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

(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
INC, COUNT, inc, count

They will be analysed ascendingly in the following order:
INC < COUNT
inc < COUNT
inc < count

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

(10)
Obligation:
Innermost TRS:
Rules:
ISEMPTY(empty) -> c
ISEMPTY(node(z0, z1)) -> c1
LEFT(empty) -> c2
LEFT(node(z0, z1)) -> c3
RIGHT(empty) -> c4
RIGHT(node(z0, z1)) -> c5
INC(0') -> c6
INC(s(z0)) -> c7(INC(z0))
COUNT(z0, z1) -> c8(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), ISEMPTY(z0))
COUNT(z0, z1) -> c9(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), ISEMPTY(left(z0)), LEFT(z0))
COUNT(z0, z1) -> c10(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(z0))
COUNT(z0, z1) -> c11(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), LEFT(left(z0)), LEFT(z0))
COUNT(z0, z1) -> c12(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(left(z0)), LEFT(z0))
COUNT(z0, z1) -> c13(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(z0))
COUNT(z0, z1) -> c14(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), INC(z1))
IF(true, z0, z1, z2, z3, z4) -> c15
IF(false, false, z0, z1, z2, z3) -> c16(COUNT(z1, z2))
IF(false, true, z0, z1, z2, z3) -> c17(COUNT(z0, z3))
NROFNODES(z0) -> c18(COUNT(z0, 0'))
isEmpty(empty) -> true
isEmpty(node(z0, z1)) -> false
left(empty) -> empty
left(node(z0, z1)) -> z0
right(empty) -> empty
right(node(z0, z1)) -> z1
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))
count(z0, z1) -> if(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1))
if(true, z0, z1, z2, z3, z4) -> z3
if(false, false, z0, z1, z2, z3) -> count(z1, z2)
if(false, true, z0, z1, z2, z3) -> count(z0, z3)
nrOfNodes(z0) -> count(z0, 0')

Types:
ISEMPTY :: empty:node -> c:c1
empty :: empty:node
c :: c:c1
node :: empty:node -> empty:node -> empty:node
c1 :: c:c1
LEFT :: empty:node -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
RIGHT :: empty:node -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
INC :: 0':s -> c6:c7
0' :: 0':s
c6 :: c6:c7
s :: 0':s -> 0':s
c7 :: c6:c7 -> c6:c7
COUNT :: empty:node -> 0':s -> c8:c9:c10:c11:c12:c13:c14
c8 :: c15:c16:c17 -> c:c1 -> c8:c9:c10:c11:c12:c13:c14
IF :: true:false -> true:false -> empty:node -> empty:node -> 0':s -> 0':s -> c15:c16:c17
isEmpty :: empty:node -> true:false
left :: empty:node -> empty:node
right :: empty:node -> empty:node
inc :: 0':s -> 0':s
c9 :: c15:c16:c17 -> c:c1 -> c2:c3 -> c8:c9:c10:c11:c12:c13:c14
c10 :: c15:c16:c17 -> c4:c5 -> c8:c9:c10:c11:c12:c13:c14
c11 :: c15:c16:c17 -> c2:c3 -> c2:c3 -> c8:c9:c10:c11:c12:c13:c14
c12 :: c15:c16:c17 -> c4:c5 -> c2:c3 -> c8:c9:c10:c11:c12:c13:c14
c13 :: c15:c16:c17 -> c4:c5 -> c8:c9:c10:c11:c12:c13:c14
c14 :: c15:c16:c17 -> c6:c7 -> c8:c9:c10:c11:c12:c13:c14
true :: true:false
c15 :: c15:c16:c17
false :: true:false
c16 :: c8:c9:c10:c11:c12:c13:c14 -> c15:c16:c17
c17 :: c8:c9:c10:c11:c12:c13:c14 -> c15:c16:c17
NROFNODES :: empty:node -> c18
c18 :: c8:c9:c10:c11:c12:c13:c14 -> c18
count :: empty:node -> 0':s -> 0':s
if :: true:false -> true:false -> empty:node -> empty:node -> 0':s -> 0':s -> 0':s
nrOfNodes :: empty:node -> 0':s
hole_c:c11_19 :: c:c1
hole_empty:node2_19 :: empty:node
hole_c2:c33_19 :: c2:c3
hole_c4:c54_19 :: c4:c5
hole_c6:c75_19 :: c6:c7
hole_0':s6_19 :: 0':s
hole_c8:c9:c10:c11:c12:c13:c147_19 :: c8:c9:c10:c11:c12:c13:c14
hole_c15:c16:c178_19 :: c15:c16:c17
hole_true:false9_19 :: true:false
hole_c1810_19 :: c18
gen_empty:node11_19 :: Nat -> empty:node
gen_c6:c712_19 :: Nat -> c6:c7
gen_0':s13_19 :: Nat -> 0':s


Generator Equations:
gen_empty:node11_19(0) <=> empty
gen_empty:node11_19(+(x, 1)) <=> node(empty, gen_empty:node11_19(x))
gen_c6:c712_19(0) <=> c6
gen_c6:c712_19(+(x, 1)) <=> c7(gen_c6:c712_19(x))
gen_0':s13_19(0) <=> 0'
gen_0':s13_19(+(x, 1)) <=> s(gen_0':s13_19(x))


The following defined symbols remain to be analysed:
INC, COUNT, inc, count

They will be analysed ascendingly in the following order:
INC < COUNT
inc < COUNT
inc < count

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

(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
INC(gen_0':s13_19(n15_19)) -> gen_c6:c712_19(n15_19), rt in Omega(1 + n15_19)

Induction Base:
INC(gen_0':s13_19(0)) ->_R^Omega(1)
c6

Induction Step:
INC(gen_0':s13_19(+(n15_19, 1))) ->_R^Omega(1)
c7(INC(gen_0':s13_19(n15_19))) ->_IH
c7(gen_c6:c712_19(c16_19))

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

(12)
Complex Obligation (BEST)

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

(13)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
ISEMPTY(empty) -> c
ISEMPTY(node(z0, z1)) -> c1
LEFT(empty) -> c2
LEFT(node(z0, z1)) -> c3
RIGHT(empty) -> c4
RIGHT(node(z0, z1)) -> c5
INC(0') -> c6
INC(s(z0)) -> c7(INC(z0))
COUNT(z0, z1) -> c8(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), ISEMPTY(z0))
COUNT(z0, z1) -> c9(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), ISEMPTY(left(z0)), LEFT(z0))
COUNT(z0, z1) -> c10(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(z0))
COUNT(z0, z1) -> c11(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), LEFT(left(z0)), LEFT(z0))
COUNT(z0, z1) -> c12(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(left(z0)), LEFT(z0))
COUNT(z0, z1) -> c13(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(z0))
COUNT(z0, z1) -> c14(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), INC(z1))
IF(true, z0, z1, z2, z3, z4) -> c15
IF(false, false, z0, z1, z2, z3) -> c16(COUNT(z1, z2))
IF(false, true, z0, z1, z2, z3) -> c17(COUNT(z0, z3))
NROFNODES(z0) -> c18(COUNT(z0, 0'))
isEmpty(empty) -> true
isEmpty(node(z0, z1)) -> false
left(empty) -> empty
left(node(z0, z1)) -> z0
right(empty) -> empty
right(node(z0, z1)) -> z1
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))
count(z0, z1) -> if(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1))
if(true, z0, z1, z2, z3, z4) -> z3
if(false, false, z0, z1, z2, z3) -> count(z1, z2)
if(false, true, z0, z1, z2, z3) -> count(z0, z3)
nrOfNodes(z0) -> count(z0, 0')

Types:
ISEMPTY :: empty:node -> c:c1
empty :: empty:node
c :: c:c1
node :: empty:node -> empty:node -> empty:node
c1 :: c:c1
LEFT :: empty:node -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
RIGHT :: empty:node -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
INC :: 0':s -> c6:c7
0' :: 0':s
c6 :: c6:c7
s :: 0':s -> 0':s
c7 :: c6:c7 -> c6:c7
COUNT :: empty:node -> 0':s -> c8:c9:c10:c11:c12:c13:c14
c8 :: c15:c16:c17 -> c:c1 -> c8:c9:c10:c11:c12:c13:c14
IF :: true:false -> true:false -> empty:node -> empty:node -> 0':s -> 0':s -> c15:c16:c17
isEmpty :: empty:node -> true:false
left :: empty:node -> empty:node
right :: empty:node -> empty:node
inc :: 0':s -> 0':s
c9 :: c15:c16:c17 -> c:c1 -> c2:c3 -> c8:c9:c10:c11:c12:c13:c14
c10 :: c15:c16:c17 -> c4:c5 -> c8:c9:c10:c11:c12:c13:c14
c11 :: c15:c16:c17 -> c2:c3 -> c2:c3 -> c8:c9:c10:c11:c12:c13:c14
c12 :: c15:c16:c17 -> c4:c5 -> c2:c3 -> c8:c9:c10:c11:c12:c13:c14
c13 :: c15:c16:c17 -> c4:c5 -> c8:c9:c10:c11:c12:c13:c14
c14 :: c15:c16:c17 -> c6:c7 -> c8:c9:c10:c11:c12:c13:c14
true :: true:false
c15 :: c15:c16:c17
false :: true:false
c16 :: c8:c9:c10:c11:c12:c13:c14 -> c15:c16:c17
c17 :: c8:c9:c10:c11:c12:c13:c14 -> c15:c16:c17
NROFNODES :: empty:node -> c18
c18 :: c8:c9:c10:c11:c12:c13:c14 -> c18
count :: empty:node -> 0':s -> 0':s
if :: true:false -> true:false -> empty:node -> empty:node -> 0':s -> 0':s -> 0':s
nrOfNodes :: empty:node -> 0':s
hole_c:c11_19 :: c:c1
hole_empty:node2_19 :: empty:node
hole_c2:c33_19 :: c2:c3
hole_c4:c54_19 :: c4:c5
hole_c6:c75_19 :: c6:c7
hole_0':s6_19 :: 0':s
hole_c8:c9:c10:c11:c12:c13:c147_19 :: c8:c9:c10:c11:c12:c13:c14
hole_c15:c16:c178_19 :: c15:c16:c17
hole_true:false9_19 :: true:false
hole_c1810_19 :: c18
gen_empty:node11_19 :: Nat -> empty:node
gen_c6:c712_19 :: Nat -> c6:c7
gen_0':s13_19 :: Nat -> 0':s


Generator Equations:
gen_empty:node11_19(0) <=> empty
gen_empty:node11_19(+(x, 1)) <=> node(empty, gen_empty:node11_19(x))
gen_c6:c712_19(0) <=> c6
gen_c6:c712_19(+(x, 1)) <=> c7(gen_c6:c712_19(x))
gen_0':s13_19(0) <=> 0'
gen_0':s13_19(+(x, 1)) <=> s(gen_0':s13_19(x))


The following defined symbols remain to be analysed:
INC, COUNT, inc, count

They will be analysed ascendingly in the following order:
INC < COUNT
inc < COUNT
inc < count

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

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

(15)
BOUNDS(n^1, INF)

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

(16)
Obligation:
Innermost TRS:
Rules:
ISEMPTY(empty) -> c
ISEMPTY(node(z0, z1)) -> c1
LEFT(empty) -> c2
LEFT(node(z0, z1)) -> c3
RIGHT(empty) -> c4
RIGHT(node(z0, z1)) -> c5
INC(0') -> c6
INC(s(z0)) -> c7(INC(z0))
COUNT(z0, z1) -> c8(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), ISEMPTY(z0))
COUNT(z0, z1) -> c9(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), ISEMPTY(left(z0)), LEFT(z0))
COUNT(z0, z1) -> c10(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(z0))
COUNT(z0, z1) -> c11(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), LEFT(left(z0)), LEFT(z0))
COUNT(z0, z1) -> c12(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(left(z0)), LEFT(z0))
COUNT(z0, z1) -> c13(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(z0))
COUNT(z0, z1) -> c14(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), INC(z1))
IF(true, z0, z1, z2, z3, z4) -> c15
IF(false, false, z0, z1, z2, z3) -> c16(COUNT(z1, z2))
IF(false, true, z0, z1, z2, z3) -> c17(COUNT(z0, z3))
NROFNODES(z0) -> c18(COUNT(z0, 0'))
isEmpty(empty) -> true
isEmpty(node(z0, z1)) -> false
left(empty) -> empty
left(node(z0, z1)) -> z0
right(empty) -> empty
right(node(z0, z1)) -> z1
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))
count(z0, z1) -> if(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1))
if(true, z0, z1, z2, z3, z4) -> z3
if(false, false, z0, z1, z2, z3) -> count(z1, z2)
if(false, true, z0, z1, z2, z3) -> count(z0, z3)
nrOfNodes(z0) -> count(z0, 0')

Types:
ISEMPTY :: empty:node -> c:c1
empty :: empty:node
c :: c:c1
node :: empty:node -> empty:node -> empty:node
c1 :: c:c1
LEFT :: empty:node -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
RIGHT :: empty:node -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
INC :: 0':s -> c6:c7
0' :: 0':s
c6 :: c6:c7
s :: 0':s -> 0':s
c7 :: c6:c7 -> c6:c7
COUNT :: empty:node -> 0':s -> c8:c9:c10:c11:c12:c13:c14
c8 :: c15:c16:c17 -> c:c1 -> c8:c9:c10:c11:c12:c13:c14
IF :: true:false -> true:false -> empty:node -> empty:node -> 0':s -> 0':s -> c15:c16:c17
isEmpty :: empty:node -> true:false
left :: empty:node -> empty:node
right :: empty:node -> empty:node
inc :: 0':s -> 0':s
c9 :: c15:c16:c17 -> c:c1 -> c2:c3 -> c8:c9:c10:c11:c12:c13:c14
c10 :: c15:c16:c17 -> c4:c5 -> c8:c9:c10:c11:c12:c13:c14
c11 :: c15:c16:c17 -> c2:c3 -> c2:c3 -> c8:c9:c10:c11:c12:c13:c14
c12 :: c15:c16:c17 -> c4:c5 -> c2:c3 -> c8:c9:c10:c11:c12:c13:c14
c13 :: c15:c16:c17 -> c4:c5 -> c8:c9:c10:c11:c12:c13:c14
c14 :: c15:c16:c17 -> c6:c7 -> c8:c9:c10:c11:c12:c13:c14
true :: true:false
c15 :: c15:c16:c17
false :: true:false
c16 :: c8:c9:c10:c11:c12:c13:c14 -> c15:c16:c17
c17 :: c8:c9:c10:c11:c12:c13:c14 -> c15:c16:c17
NROFNODES :: empty:node -> c18
c18 :: c8:c9:c10:c11:c12:c13:c14 -> c18
count :: empty:node -> 0':s -> 0':s
if :: true:false -> true:false -> empty:node -> empty:node -> 0':s -> 0':s -> 0':s
nrOfNodes :: empty:node -> 0':s
hole_c:c11_19 :: c:c1
hole_empty:node2_19 :: empty:node
hole_c2:c33_19 :: c2:c3
hole_c4:c54_19 :: c4:c5
hole_c6:c75_19 :: c6:c7
hole_0':s6_19 :: 0':s
hole_c8:c9:c10:c11:c12:c13:c147_19 :: c8:c9:c10:c11:c12:c13:c14
hole_c15:c16:c178_19 :: c15:c16:c17
hole_true:false9_19 :: true:false
hole_c1810_19 :: c18
gen_empty:node11_19 :: Nat -> empty:node
gen_c6:c712_19 :: Nat -> c6:c7
gen_0':s13_19 :: Nat -> 0':s


Lemmas:
INC(gen_0':s13_19(n15_19)) -> gen_c6:c712_19(n15_19), rt in Omega(1 + n15_19)


Generator Equations:
gen_empty:node11_19(0) <=> empty
gen_empty:node11_19(+(x, 1)) <=> node(empty, gen_empty:node11_19(x))
gen_c6:c712_19(0) <=> c6
gen_c6:c712_19(+(x, 1)) <=> c7(gen_c6:c712_19(x))
gen_0':s13_19(0) <=> 0'
gen_0':s13_19(+(x, 1)) <=> s(gen_0':s13_19(x))


The following defined symbols remain to be analysed:
inc, COUNT, count

They will be analysed ascendingly in the following order:
inc < COUNT
inc < count

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
inc(gen_0':s13_19(n362_19)) -> gen_0':s13_19(+(1, n362_19)), rt in Omega(0)

Induction Base:
inc(gen_0':s13_19(0)) ->_R^Omega(0)
s(0')

Induction Step:
inc(gen_0':s13_19(+(n362_19, 1))) ->_R^Omega(0)
s(inc(gen_0':s13_19(n362_19))) ->_IH
s(gen_0':s13_19(+(1, c363_19)))

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

(18)
Obligation:
Innermost TRS:
Rules:
ISEMPTY(empty) -> c
ISEMPTY(node(z0, z1)) -> c1
LEFT(empty) -> c2
LEFT(node(z0, z1)) -> c3
RIGHT(empty) -> c4
RIGHT(node(z0, z1)) -> c5
INC(0') -> c6
INC(s(z0)) -> c7(INC(z0))
COUNT(z0, z1) -> c8(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), ISEMPTY(z0))
COUNT(z0, z1) -> c9(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), ISEMPTY(left(z0)), LEFT(z0))
COUNT(z0, z1) -> c10(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(z0))
COUNT(z0, z1) -> c11(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), LEFT(left(z0)), LEFT(z0))
COUNT(z0, z1) -> c12(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(left(z0)), LEFT(z0))
COUNT(z0, z1) -> c13(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(z0))
COUNT(z0, z1) -> c14(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), INC(z1))
IF(true, z0, z1, z2, z3, z4) -> c15
IF(false, false, z0, z1, z2, z3) -> c16(COUNT(z1, z2))
IF(false, true, z0, z1, z2, z3) -> c17(COUNT(z0, z3))
NROFNODES(z0) -> c18(COUNT(z0, 0'))
isEmpty(empty) -> true
isEmpty(node(z0, z1)) -> false
left(empty) -> empty
left(node(z0, z1)) -> z0
right(empty) -> empty
right(node(z0, z1)) -> z1
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))
count(z0, z1) -> if(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1))
if(true, z0, z1, z2, z3, z4) -> z3
if(false, false, z0, z1, z2, z3) -> count(z1, z2)
if(false, true, z0, z1, z2, z3) -> count(z0, z3)
nrOfNodes(z0) -> count(z0, 0')

Types:
ISEMPTY :: empty:node -> c:c1
empty :: empty:node
c :: c:c1
node :: empty:node -> empty:node -> empty:node
c1 :: c:c1
LEFT :: empty:node -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
RIGHT :: empty:node -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
INC :: 0':s -> c6:c7
0' :: 0':s
c6 :: c6:c7
s :: 0':s -> 0':s
c7 :: c6:c7 -> c6:c7
COUNT :: empty:node -> 0':s -> c8:c9:c10:c11:c12:c13:c14
c8 :: c15:c16:c17 -> c:c1 -> c8:c9:c10:c11:c12:c13:c14
IF :: true:false -> true:false -> empty:node -> empty:node -> 0':s -> 0':s -> c15:c16:c17
isEmpty :: empty:node -> true:false
left :: empty:node -> empty:node
right :: empty:node -> empty:node
inc :: 0':s -> 0':s
c9 :: c15:c16:c17 -> c:c1 -> c2:c3 -> c8:c9:c10:c11:c12:c13:c14
c10 :: c15:c16:c17 -> c4:c5 -> c8:c9:c10:c11:c12:c13:c14
c11 :: c15:c16:c17 -> c2:c3 -> c2:c3 -> c8:c9:c10:c11:c12:c13:c14
c12 :: c15:c16:c17 -> c4:c5 -> c2:c3 -> c8:c9:c10:c11:c12:c13:c14
c13 :: c15:c16:c17 -> c4:c5 -> c8:c9:c10:c11:c12:c13:c14
c14 :: c15:c16:c17 -> c6:c7 -> c8:c9:c10:c11:c12:c13:c14
true :: true:false
c15 :: c15:c16:c17
false :: true:false
c16 :: c8:c9:c10:c11:c12:c13:c14 -> c15:c16:c17
c17 :: c8:c9:c10:c11:c12:c13:c14 -> c15:c16:c17
NROFNODES :: empty:node -> c18
c18 :: c8:c9:c10:c11:c12:c13:c14 -> c18
count :: empty:node -> 0':s -> 0':s
if :: true:false -> true:false -> empty:node -> empty:node -> 0':s -> 0':s -> 0':s
nrOfNodes :: empty:node -> 0':s
hole_c:c11_19 :: c:c1
hole_empty:node2_19 :: empty:node
hole_c2:c33_19 :: c2:c3
hole_c4:c54_19 :: c4:c5
hole_c6:c75_19 :: c6:c7
hole_0':s6_19 :: 0':s
hole_c8:c9:c10:c11:c12:c13:c147_19 :: c8:c9:c10:c11:c12:c13:c14
hole_c15:c16:c178_19 :: c15:c16:c17
hole_true:false9_19 :: true:false
hole_c1810_19 :: c18
gen_empty:node11_19 :: Nat -> empty:node
gen_c6:c712_19 :: Nat -> c6:c7
gen_0':s13_19 :: Nat -> 0':s


Lemmas:
INC(gen_0':s13_19(n15_19)) -> gen_c6:c712_19(n15_19), rt in Omega(1 + n15_19)
inc(gen_0':s13_19(n362_19)) -> gen_0':s13_19(+(1, n362_19)), rt in Omega(0)


Generator Equations:
gen_empty:node11_19(0) <=> empty
gen_empty:node11_19(+(x, 1)) <=> node(empty, gen_empty:node11_19(x))
gen_c6:c712_19(0) <=> c6
gen_c6:c712_19(+(x, 1)) <=> c7(gen_c6:c712_19(x))
gen_0':s13_19(0) <=> 0'
gen_0':s13_19(+(x, 1)) <=> s(gen_0':s13_19(x))


The following defined symbols remain to be analysed:
COUNT, count
----------------------------------------

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
COUNT(gen_empty:node11_19(n768_19), gen_0':s13_19(b)) -> *14_19, rt in Omega(b*n768_19 + n768_19)

Induction Base:
COUNT(gen_empty:node11_19(0), gen_0':s13_19(b))

Induction Step:
COUNT(gen_empty:node11_19(+(n768_19, 1)), gen_0':s13_19(b)) ->_R^Omega(1)
c14(IF(isEmpty(gen_empty:node11_19(+(n768_19, 1))), isEmpty(left(gen_empty:node11_19(+(n768_19, 1)))), right(gen_empty:node11_19(+(n768_19, 1))), node(left(left(gen_empty:node11_19(+(n768_19, 1)))), node(right(left(gen_empty:node11_19(+(n768_19, 1)))), right(gen_empty:node11_19(+(n768_19, 1))))), gen_0':s13_19(b), inc(gen_0':s13_19(b))), INC(gen_0':s13_19(b))) ->_R^Omega(0)
c14(IF(false, isEmpty(left(gen_empty:node11_19(+(1, n768_19)))), right(gen_empty:node11_19(+(1, n768_19))), node(left(left(gen_empty:node11_19(+(1, n768_19)))), node(right(left(gen_empty:node11_19(+(1, n768_19)))), right(gen_empty:node11_19(+(1, n768_19))))), gen_0':s13_19(b), inc(gen_0':s13_19(b))), INC(gen_0':s13_19(b))) ->_R^Omega(0)
c14(IF(false, isEmpty(empty), right(gen_empty:node11_19(+(1, n768_19))), node(left(left(gen_empty:node11_19(+(1, n768_19)))), node(right(left(gen_empty:node11_19(+(1, n768_19)))), right(gen_empty:node11_19(+(1, n768_19))))), gen_0':s13_19(b), inc(gen_0':s13_19(b))), INC(gen_0':s13_19(b))) ->_R^Omega(0)
c14(IF(false, true, right(gen_empty:node11_19(+(1, n768_19))), node(left(left(gen_empty:node11_19(+(1, n768_19)))), node(right(left(gen_empty:node11_19(+(1, n768_19)))), right(gen_empty:node11_19(+(1, n768_19))))), gen_0':s13_19(b), inc(gen_0':s13_19(b))), INC(gen_0':s13_19(b))) ->_R^Omega(0)
c14(IF(false, true, gen_empty:node11_19(n768_19), node(left(left(gen_empty:node11_19(+(1, n768_19)))), node(right(left(gen_empty:node11_19(+(1, n768_19)))), right(gen_empty:node11_19(+(1, n768_19))))), gen_0':s13_19(b), inc(gen_0':s13_19(b))), INC(gen_0':s13_19(b))) ->_R^Omega(0)
c14(IF(false, true, gen_empty:node11_19(n768_19), node(left(empty), node(right(left(gen_empty:node11_19(+(1, n768_19)))), right(gen_empty:node11_19(+(1, n768_19))))), gen_0':s13_19(b), inc(gen_0':s13_19(b))), INC(gen_0':s13_19(b))) ->_R^Omega(0)
c14(IF(false, true, gen_empty:node11_19(n768_19), node(empty, node(right(left(gen_empty:node11_19(+(1, n768_19)))), right(gen_empty:node11_19(+(1, n768_19))))), gen_0':s13_19(b), inc(gen_0':s13_19(b))), INC(gen_0':s13_19(b))) ->_R^Omega(0)
c14(IF(false, true, gen_empty:node11_19(n768_19), node(empty, node(right(empty), right(gen_empty:node11_19(+(1, n768_19))))), gen_0':s13_19(b), inc(gen_0':s13_19(b))), INC(gen_0':s13_19(b))) ->_R^Omega(0)
c14(IF(false, true, gen_empty:node11_19(n768_19), node(empty, node(empty, right(gen_empty:node11_19(+(1, n768_19))))), gen_0':s13_19(b), inc(gen_0':s13_19(b))), INC(gen_0':s13_19(b))) ->_R^Omega(0)
c14(IF(false, true, gen_empty:node11_19(n768_19), node(empty, node(empty, gen_empty:node11_19(n768_19))), gen_0':s13_19(b), inc(gen_0':s13_19(b))), INC(gen_0':s13_19(b))) ->_L^Omega(0)
c14(IF(false, true, gen_empty:node11_19(n768_19), node(empty, node(empty, gen_empty:node11_19(n768_19))), gen_0':s13_19(b), gen_0':s13_19(+(1, b))), INC(gen_0':s13_19(b))) ->_R^Omega(1)
c14(c17(COUNT(gen_empty:node11_19(n768_19), gen_0':s13_19(+(1, b)))), INC(gen_0':s13_19(b))) ->_IH
c14(c17(*14_19), INC(gen_0':s13_19(+(1, b)))) ->_L^Omega(2 + b)
c14(c17(*14_19), gen_c6:c712_19(+(1, b)))

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

(20)
Complex Obligation (BEST)

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

(21)
Obligation:
Proved the lower bound n^2 for the following obligation:

Innermost TRS:
Rules:
ISEMPTY(empty) -> c
ISEMPTY(node(z0, z1)) -> c1
LEFT(empty) -> c2
LEFT(node(z0, z1)) -> c3
RIGHT(empty) -> c4
RIGHT(node(z0, z1)) -> c5
INC(0') -> c6
INC(s(z0)) -> c7(INC(z0))
COUNT(z0, z1) -> c8(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), ISEMPTY(z0))
COUNT(z0, z1) -> c9(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), ISEMPTY(left(z0)), LEFT(z0))
COUNT(z0, z1) -> c10(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(z0))
COUNT(z0, z1) -> c11(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), LEFT(left(z0)), LEFT(z0))
COUNT(z0, z1) -> c12(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(left(z0)), LEFT(z0))
COUNT(z0, z1) -> c13(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(z0))
COUNT(z0, z1) -> c14(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), INC(z1))
IF(true, z0, z1, z2, z3, z4) -> c15
IF(false, false, z0, z1, z2, z3) -> c16(COUNT(z1, z2))
IF(false, true, z0, z1, z2, z3) -> c17(COUNT(z0, z3))
NROFNODES(z0) -> c18(COUNT(z0, 0'))
isEmpty(empty) -> true
isEmpty(node(z0, z1)) -> false
left(empty) -> empty
left(node(z0, z1)) -> z0
right(empty) -> empty
right(node(z0, z1)) -> z1
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))
count(z0, z1) -> if(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1))
if(true, z0, z1, z2, z3, z4) -> z3
if(false, false, z0, z1, z2, z3) -> count(z1, z2)
if(false, true, z0, z1, z2, z3) -> count(z0, z3)
nrOfNodes(z0) -> count(z0, 0')

Types:
ISEMPTY :: empty:node -> c:c1
empty :: empty:node
c :: c:c1
node :: empty:node -> empty:node -> empty:node
c1 :: c:c1
LEFT :: empty:node -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
RIGHT :: empty:node -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
INC :: 0':s -> c6:c7
0' :: 0':s
c6 :: c6:c7
s :: 0':s -> 0':s
c7 :: c6:c7 -> c6:c7
COUNT :: empty:node -> 0':s -> c8:c9:c10:c11:c12:c13:c14
c8 :: c15:c16:c17 -> c:c1 -> c8:c9:c10:c11:c12:c13:c14
IF :: true:false -> true:false -> empty:node -> empty:node -> 0':s -> 0':s -> c15:c16:c17
isEmpty :: empty:node -> true:false
left :: empty:node -> empty:node
right :: empty:node -> empty:node
inc :: 0':s -> 0':s
c9 :: c15:c16:c17 -> c:c1 -> c2:c3 -> c8:c9:c10:c11:c12:c13:c14
c10 :: c15:c16:c17 -> c4:c5 -> c8:c9:c10:c11:c12:c13:c14
c11 :: c15:c16:c17 -> c2:c3 -> c2:c3 -> c8:c9:c10:c11:c12:c13:c14
c12 :: c15:c16:c17 -> c4:c5 -> c2:c3 -> c8:c9:c10:c11:c12:c13:c14
c13 :: c15:c16:c17 -> c4:c5 -> c8:c9:c10:c11:c12:c13:c14
c14 :: c15:c16:c17 -> c6:c7 -> c8:c9:c10:c11:c12:c13:c14
true :: true:false
c15 :: c15:c16:c17
false :: true:false
c16 :: c8:c9:c10:c11:c12:c13:c14 -> c15:c16:c17
c17 :: c8:c9:c10:c11:c12:c13:c14 -> c15:c16:c17
NROFNODES :: empty:node -> c18
c18 :: c8:c9:c10:c11:c12:c13:c14 -> c18
count :: empty:node -> 0':s -> 0':s
if :: true:false -> true:false -> empty:node -> empty:node -> 0':s -> 0':s -> 0':s
nrOfNodes :: empty:node -> 0':s
hole_c:c11_19 :: c:c1
hole_empty:node2_19 :: empty:node
hole_c2:c33_19 :: c2:c3
hole_c4:c54_19 :: c4:c5
hole_c6:c75_19 :: c6:c7
hole_0':s6_19 :: 0':s
hole_c8:c9:c10:c11:c12:c13:c147_19 :: c8:c9:c10:c11:c12:c13:c14
hole_c15:c16:c178_19 :: c15:c16:c17
hole_true:false9_19 :: true:false
hole_c1810_19 :: c18
gen_empty:node11_19 :: Nat -> empty:node
gen_c6:c712_19 :: Nat -> c6:c7
gen_0':s13_19 :: Nat -> 0':s


Lemmas:
INC(gen_0':s13_19(n15_19)) -> gen_c6:c712_19(n15_19), rt in Omega(1 + n15_19)
inc(gen_0':s13_19(n362_19)) -> gen_0':s13_19(+(1, n362_19)), rt in Omega(0)


Generator Equations:
gen_empty:node11_19(0) <=> empty
gen_empty:node11_19(+(x, 1)) <=> node(empty, gen_empty:node11_19(x))
gen_c6:c712_19(0) <=> c6
gen_c6:c712_19(+(x, 1)) <=> c7(gen_c6:c712_19(x))
gen_0':s13_19(0) <=> 0'
gen_0':s13_19(+(x, 1)) <=> s(gen_0':s13_19(x))


The following defined symbols remain to be analysed:
COUNT, count
----------------------------------------

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

(23)
BOUNDS(n^2, INF)

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

(24)
Obligation:
Innermost TRS:
Rules:
ISEMPTY(empty) -> c
ISEMPTY(node(z0, z1)) -> c1
LEFT(empty) -> c2
LEFT(node(z0, z1)) -> c3
RIGHT(empty) -> c4
RIGHT(node(z0, z1)) -> c5
INC(0') -> c6
INC(s(z0)) -> c7(INC(z0))
COUNT(z0, z1) -> c8(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), ISEMPTY(z0))
COUNT(z0, z1) -> c9(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), ISEMPTY(left(z0)), LEFT(z0))
COUNT(z0, z1) -> c10(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(z0))
COUNT(z0, z1) -> c11(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), LEFT(left(z0)), LEFT(z0))
COUNT(z0, z1) -> c12(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(left(z0)), LEFT(z0))
COUNT(z0, z1) -> c13(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), RIGHT(z0))
COUNT(z0, z1) -> c14(IF(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1)), INC(z1))
IF(true, z0, z1, z2, z3, z4) -> c15
IF(false, false, z0, z1, z2, z3) -> c16(COUNT(z1, z2))
IF(false, true, z0, z1, z2, z3) -> c17(COUNT(z0, z3))
NROFNODES(z0) -> c18(COUNT(z0, 0'))
isEmpty(empty) -> true
isEmpty(node(z0, z1)) -> false
left(empty) -> empty
left(node(z0, z1)) -> z0
right(empty) -> empty
right(node(z0, z1)) -> z1
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))
count(z0, z1) -> if(isEmpty(z0), isEmpty(left(z0)), right(z0), node(left(left(z0)), node(right(left(z0)), right(z0))), z1, inc(z1))
if(true, z0, z1, z2, z3, z4) -> z3
if(false, false, z0, z1, z2, z3) -> count(z1, z2)
if(false, true, z0, z1, z2, z3) -> count(z0, z3)
nrOfNodes(z0) -> count(z0, 0')

Types:
ISEMPTY :: empty:node -> c:c1
empty :: empty:node
c :: c:c1
node :: empty:node -> empty:node -> empty:node
c1 :: c:c1
LEFT :: empty:node -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
RIGHT :: empty:node -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
INC :: 0':s -> c6:c7
0' :: 0':s
c6 :: c6:c7
s :: 0':s -> 0':s
c7 :: c6:c7 -> c6:c7
COUNT :: empty:node -> 0':s -> c8:c9:c10:c11:c12:c13:c14
c8 :: c15:c16:c17 -> c:c1 -> c8:c9:c10:c11:c12:c13:c14
IF :: true:false -> true:false -> empty:node -> empty:node -> 0':s -> 0':s -> c15:c16:c17
isEmpty :: empty:node -> true:false
left :: empty:node -> empty:node
right :: empty:node -> empty:node
inc :: 0':s -> 0':s
c9 :: c15:c16:c17 -> c:c1 -> c2:c3 -> c8:c9:c10:c11:c12:c13:c14
c10 :: c15:c16:c17 -> c4:c5 -> c8:c9:c10:c11:c12:c13:c14
c11 :: c15:c16:c17 -> c2:c3 -> c2:c3 -> c8:c9:c10:c11:c12:c13:c14
c12 :: c15:c16:c17 -> c4:c5 -> c2:c3 -> c8:c9:c10:c11:c12:c13:c14
c13 :: c15:c16:c17 -> c4:c5 -> c8:c9:c10:c11:c12:c13:c14
c14 :: c15:c16:c17 -> c6:c7 -> c8:c9:c10:c11:c12:c13:c14
true :: true:false
c15 :: c15:c16:c17
false :: true:false
c16 :: c8:c9:c10:c11:c12:c13:c14 -> c15:c16:c17
c17 :: c8:c9:c10:c11:c12:c13:c14 -> c15:c16:c17
NROFNODES :: empty:node -> c18
c18 :: c8:c9:c10:c11:c12:c13:c14 -> c18
count :: empty:node -> 0':s -> 0':s
if :: true:false -> true:false -> empty:node -> empty:node -> 0':s -> 0':s -> 0':s
nrOfNodes :: empty:node -> 0':s
hole_c:c11_19 :: c:c1
hole_empty:node2_19 :: empty:node
hole_c2:c33_19 :: c2:c3
hole_c4:c54_19 :: c4:c5
hole_c6:c75_19 :: c6:c7
hole_0':s6_19 :: 0':s
hole_c8:c9:c10:c11:c12:c13:c147_19 :: c8:c9:c10:c11:c12:c13:c14
hole_c15:c16:c178_19 :: c15:c16:c17
hole_true:false9_19 :: true:false
hole_c1810_19 :: c18
gen_empty:node11_19 :: Nat -> empty:node
gen_c6:c712_19 :: Nat -> c6:c7
gen_0':s13_19 :: Nat -> 0':s


Lemmas:
INC(gen_0':s13_19(n15_19)) -> gen_c6:c712_19(n15_19), rt in Omega(1 + n15_19)
inc(gen_0':s13_19(n362_19)) -> gen_0':s13_19(+(1, n362_19)), rt in Omega(0)
COUNT(gen_empty:node11_19(n768_19), gen_0':s13_19(b)) -> *14_19, rt in Omega(b*n768_19 + n768_19)


Generator Equations:
gen_empty:node11_19(0) <=> empty
gen_empty:node11_19(+(x, 1)) <=> node(empty, gen_empty:node11_19(x))
gen_c6:c712_19(0) <=> c6
gen_c6:c712_19(+(x, 1)) <=> c7(gen_c6:c712_19(x))
gen_0':s13_19(0) <=> 0'
gen_0':s13_19(+(x, 1)) <=> s(gen_0':s13_19(x))


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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
count(gen_empty:node11_19(n309708_19), gen_0':s13_19(b)) -> gen_0':s13_19(+(n309708_19, b)), rt in Omega(0)

Induction Base:
count(gen_empty:node11_19(0), gen_0':s13_19(b)) ->_R^Omega(0)
if(isEmpty(gen_empty:node11_19(0)), isEmpty(left(gen_empty:node11_19(0))), right(gen_empty:node11_19(0)), node(left(left(gen_empty:node11_19(0))), node(right(left(gen_empty:node11_19(0))), right(gen_empty:node11_19(0)))), gen_0':s13_19(b), inc(gen_0':s13_19(b))) ->_R^Omega(0)
if(true, isEmpty(left(gen_empty:node11_19(0))), right(gen_empty:node11_19(0)), node(left(left(gen_empty:node11_19(0))), node(right(left(gen_empty:node11_19(0))), right(gen_empty:node11_19(0)))), gen_0':s13_19(b), inc(gen_0':s13_19(b))) ->_R^Omega(0)
if(true, isEmpty(empty), right(gen_empty:node11_19(0)), node(left(left(gen_empty:node11_19(0))), node(right(left(gen_empty:node11_19(0))), right(gen_empty:node11_19(0)))), gen_0':s13_19(b), inc(gen_0':s13_19(b))) ->_R^Omega(0)
if(true, true, right(gen_empty:node11_19(0)), node(left(left(gen_empty:node11_19(0))), node(right(left(gen_empty:node11_19(0))), right(gen_empty:node11_19(0)))), gen_0':s13_19(b), inc(gen_0':s13_19(b))) ->_R^Omega(0)
if(true, true, empty, node(left(left(gen_empty:node11_19(0))), node(right(left(gen_empty:node11_19(0))), right(gen_empty:node11_19(0)))), gen_0':s13_19(b), inc(gen_0':s13_19(b))) ->_R^Omega(0)
if(true, true, empty, node(left(empty), node(right(left(gen_empty:node11_19(0))), right(gen_empty:node11_19(0)))), gen_0':s13_19(b), inc(gen_0':s13_19(b))) ->_R^Omega(0)
if(true, true, empty, node(empty, node(right(left(gen_empty:node11_19(0))), right(gen_empty:node11_19(0)))), gen_0':s13_19(b), inc(gen_0':s13_19(b))) ->_R^Omega(0)
if(true, true, empty, node(empty, node(right(empty), right(gen_empty:node11_19(0)))), gen_0':s13_19(b), inc(gen_0':s13_19(b))) ->_R^Omega(0)
if(true, true, empty, node(empty, node(empty, right(gen_empty:node11_19(0)))), gen_0':s13_19(b), inc(gen_0':s13_19(b))) ->_R^Omega(0)
if(true, true, empty, node(empty, node(empty, empty)), gen_0':s13_19(b), inc(gen_0':s13_19(b))) ->_L^Omega(0)
if(true, true, empty, node(empty, node(empty, empty)), gen_0':s13_19(b), gen_0':s13_19(+(1, b))) ->_R^Omega(0)
gen_0':s13_19(b)

Induction Step:
count(gen_empty:node11_19(+(n309708_19, 1)), gen_0':s13_19(b)) ->_R^Omega(0)
if(isEmpty(gen_empty:node11_19(+(n309708_19, 1))), isEmpty(left(gen_empty:node11_19(+(n309708_19, 1)))), right(gen_empty:node11_19(+(n309708_19, 1))), node(left(left(gen_empty:node11_19(+(n309708_19, 1)))), node(right(left(gen_empty:node11_19(+(n309708_19, 1)))), right(gen_empty:node11_19(+(n309708_19, 1))))), gen_0':s13_19(b), inc(gen_0':s13_19(b))) ->_R^Omega(0)
if(false, isEmpty(left(gen_empty:node11_19(+(1, n309708_19)))), right(gen_empty:node11_19(+(1, n309708_19))), node(left(left(gen_empty:node11_19(+(1, n309708_19)))), node(right(left(gen_empty:node11_19(+(1, n309708_19)))), right(gen_empty:node11_19(+(1, n309708_19))))), gen_0':s13_19(b), inc(gen_0':s13_19(b))) ->_R^Omega(0)
if(false, isEmpty(empty), right(gen_empty:node11_19(+(1, n309708_19))), node(left(left(gen_empty:node11_19(+(1, n309708_19)))), node(right(left(gen_empty:node11_19(+(1, n309708_19)))), right(gen_empty:node11_19(+(1, n309708_19))))), gen_0':s13_19(b), inc(gen_0':s13_19(b))) ->_R^Omega(0)
if(false, true, right(gen_empty:node11_19(+(1, n309708_19))), node(left(left(gen_empty:node11_19(+(1, n309708_19)))), node(right(left(gen_empty:node11_19(+(1, n309708_19)))), right(gen_empty:node11_19(+(1, n309708_19))))), gen_0':s13_19(b), inc(gen_0':s13_19(b))) ->_R^Omega(0)
if(false, true, gen_empty:node11_19(n309708_19), node(left(left(gen_empty:node11_19(+(1, n309708_19)))), node(right(left(gen_empty:node11_19(+(1, n309708_19)))), right(gen_empty:node11_19(+(1, n309708_19))))), gen_0':s13_19(b), inc(gen_0':s13_19(b))) ->_R^Omega(0)
if(false, true, gen_empty:node11_19(n309708_19), node(left(empty), node(right(left(gen_empty:node11_19(+(1, n309708_19)))), right(gen_empty:node11_19(+(1, n309708_19))))), gen_0':s13_19(b), inc(gen_0':s13_19(b))) ->_R^Omega(0)
if(false, true, gen_empty:node11_19(n309708_19), node(empty, node(right(left(gen_empty:node11_19(+(1, n309708_19)))), right(gen_empty:node11_19(+(1, n309708_19))))), gen_0':s13_19(b), inc(gen_0':s13_19(b))) ->_R^Omega(0)
if(false, true, gen_empty:node11_19(n309708_19), node(empty, node(right(empty), right(gen_empty:node11_19(+(1, n309708_19))))), gen_0':s13_19(b), inc(gen_0':s13_19(b))) ->_R^Omega(0)
if(false, true, gen_empty:node11_19(n309708_19), node(empty, node(empty, right(gen_empty:node11_19(+(1, n309708_19))))), gen_0':s13_19(b), inc(gen_0':s13_19(b))) ->_R^Omega(0)
if(false, true, gen_empty:node11_19(n309708_19), node(empty, node(empty, gen_empty:node11_19(n309708_19))), gen_0':s13_19(b), inc(gen_0':s13_19(b))) ->_L^Omega(0)
if(false, true, gen_empty:node11_19(n309708_19), node(empty, node(empty, gen_empty:node11_19(n309708_19))), gen_0':s13_19(b), gen_0':s13_19(+(1, b))) ->_R^Omega(0)
count(gen_empty:node11_19(n309708_19), gen_0':s13_19(+(1, b))) ->_IH
gen_0':s13_19(+(+(1, b), c309709_19))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(26)
BOUNDS(1, INF)