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


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WORST_CASE(Omega(n^2), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 0 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 251 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs
        (13) RewriteLemmaProof [LOWER BOUND(ID), 193 ms]
        (14) proven lower bound
        (15) LowerBoundPropagationProof [FINISHED, 0 ms]
        (16) BOUNDS(n^2, INF)


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

(0)
Obligation:
The Runtime Complexity (full) 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: FULL
----------------------------------------

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

(2)
Obligation:
The Runtime Complexity (full) 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: FULL
----------------------------------------

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

(4)
Obligation:
TRS:
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')

Types:
isEmpty :: empty:node -> true:false
empty :: empty:node
true :: true:false
node :: empty:node -> empty:node -> empty:node
false :: true:false
left :: empty:node -> empty:node
right :: empty:node -> empty:node
inc :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
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_true:false1_0 :: true:false
hole_empty:node2_0 :: empty:node
hole_0':s3_0 :: 0':s
gen_empty:node4_0 :: Nat -> empty:node
gen_0':s5_0 :: Nat -> 0':s

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

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

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

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

(6)
Obligation:
TRS:
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')

Types:
isEmpty :: empty:node -> true:false
empty :: empty:node
true :: true:false
node :: empty:node -> empty:node -> empty:node
false :: true:false
left :: empty:node -> empty:node
right :: empty:node -> empty:node
inc :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
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_true:false1_0 :: true:false
hole_empty:node2_0 :: empty:node
hole_0':s3_0 :: 0':s
gen_empty:node4_0 :: Nat -> empty:node
gen_0':s5_0 :: Nat -> 0':s


Generator Equations:
gen_empty:node4_0(0) <=> empty
gen_empty:node4_0(+(x, 1)) <=> node(empty, gen_empty:node4_0(x))
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


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

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

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
inc(gen_0':s5_0(n7_0)) -> gen_0':s5_0(+(1, n7_0)), rt in Omega(1 + n7_0)

Induction Base:
inc(gen_0':s5_0(0)) ->_R^Omega(1)
s(0')

Induction Step:
inc(gen_0':s5_0(+(n7_0, 1))) ->_R^Omega(1)
s(inc(gen_0':s5_0(n7_0))) ->_IH
s(gen_0':s5_0(+(1, c8_0)))

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

(8)
Complex Obligation (BEST)

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

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
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')

Types:
isEmpty :: empty:node -> true:false
empty :: empty:node
true :: true:false
node :: empty:node -> empty:node -> empty:node
false :: true:false
left :: empty:node -> empty:node
right :: empty:node -> empty:node
inc :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
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_true:false1_0 :: true:false
hole_empty:node2_0 :: empty:node
hole_0':s3_0 :: 0':s
gen_empty:node4_0 :: Nat -> empty:node
gen_0':s5_0 :: Nat -> 0':s


Generator Equations:
gen_empty:node4_0(0) <=> empty
gen_empty:node4_0(+(x, 1)) <=> node(empty, gen_empty:node4_0(x))
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


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

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

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
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')

Types:
isEmpty :: empty:node -> true:false
empty :: empty:node
true :: true:false
node :: empty:node -> empty:node -> empty:node
false :: true:false
left :: empty:node -> empty:node
right :: empty:node -> empty:node
inc :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
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_true:false1_0 :: true:false
hole_empty:node2_0 :: empty:node
hole_0':s3_0 :: 0':s
gen_empty:node4_0 :: Nat -> empty:node
gen_0':s5_0 :: Nat -> 0':s


Lemmas:
inc(gen_0':s5_0(n7_0)) -> gen_0':s5_0(+(1, n7_0)), rt in Omega(1 + n7_0)


Generator Equations:
gen_empty:node4_0(0) <=> empty
gen_empty:node4_0(+(x, 1)) <=> node(empty, gen_empty:node4_0(x))
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
count(gen_empty:node4_0(n263_0), gen_0':s5_0(b)) -> gen_0':s5_0(+(n263_0, b)), rt in Omega(1 + b + b*n263_0 + n263_0)

Induction Base:
count(gen_empty:node4_0(0), gen_0':s5_0(b)) ->_R^Omega(1)
if(isEmpty(gen_empty:node4_0(0)), isEmpty(left(gen_empty:node4_0(0))), right(gen_empty:node4_0(0)), node(left(left(gen_empty:node4_0(0))), node(right(left(gen_empty:node4_0(0))), right(gen_empty:node4_0(0)))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(true, isEmpty(left(gen_empty:node4_0(0))), right(gen_empty:node4_0(0)), node(left(left(gen_empty:node4_0(0))), node(right(left(gen_empty:node4_0(0))), right(gen_empty:node4_0(0)))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(true, isEmpty(empty), right(gen_empty:node4_0(0)), node(left(left(gen_empty:node4_0(0))), node(right(left(gen_empty:node4_0(0))), right(gen_empty:node4_0(0)))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(true, true, right(gen_empty:node4_0(0)), node(left(left(gen_empty:node4_0(0))), node(right(left(gen_empty:node4_0(0))), right(gen_empty:node4_0(0)))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(true, true, empty, node(left(left(gen_empty:node4_0(0))), node(right(left(gen_empty:node4_0(0))), right(gen_empty:node4_0(0)))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(true, true, empty, node(left(empty), node(right(left(gen_empty:node4_0(0))), right(gen_empty:node4_0(0)))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(true, true, empty, node(empty, node(right(left(gen_empty:node4_0(0))), right(gen_empty:node4_0(0)))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(true, true, empty, node(empty, node(right(empty), right(gen_empty:node4_0(0)))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(true, true, empty, node(empty, node(empty, right(gen_empty:node4_0(0)))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(true, true, empty, node(empty, node(empty, empty)), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_L^Omega(1 + b)
if(true, true, empty, node(empty, node(empty, empty)), gen_0':s5_0(b), gen_0':s5_0(+(1, b))) ->_R^Omega(1)
gen_0':s5_0(b)

Induction Step:
count(gen_empty:node4_0(+(n263_0, 1)), gen_0':s5_0(b)) ->_R^Omega(1)
if(isEmpty(gen_empty:node4_0(+(n263_0, 1))), isEmpty(left(gen_empty:node4_0(+(n263_0, 1)))), right(gen_empty:node4_0(+(n263_0, 1))), node(left(left(gen_empty:node4_0(+(n263_0, 1)))), node(right(left(gen_empty:node4_0(+(n263_0, 1)))), right(gen_empty:node4_0(+(n263_0, 1))))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(false, isEmpty(left(gen_empty:node4_0(+(1, n263_0)))), right(gen_empty:node4_0(+(1, n263_0))), node(left(left(gen_empty:node4_0(+(1, n263_0)))), node(right(left(gen_empty:node4_0(+(1, n263_0)))), right(gen_empty:node4_0(+(1, n263_0))))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(false, isEmpty(empty), right(gen_empty:node4_0(+(1, n263_0))), node(left(left(gen_empty:node4_0(+(1, n263_0)))), node(right(left(gen_empty:node4_0(+(1, n263_0)))), right(gen_empty:node4_0(+(1, n263_0))))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(false, true, right(gen_empty:node4_0(+(1, n263_0))), node(left(left(gen_empty:node4_0(+(1, n263_0)))), node(right(left(gen_empty:node4_0(+(1, n263_0)))), right(gen_empty:node4_0(+(1, n263_0))))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(false, true, gen_empty:node4_0(n263_0), node(left(left(gen_empty:node4_0(+(1, n263_0)))), node(right(left(gen_empty:node4_0(+(1, n263_0)))), right(gen_empty:node4_0(+(1, n263_0))))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(false, true, gen_empty:node4_0(n263_0), node(left(empty), node(right(left(gen_empty:node4_0(+(1, n263_0)))), right(gen_empty:node4_0(+(1, n263_0))))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(false, true, gen_empty:node4_0(n263_0), node(empty, node(right(left(gen_empty:node4_0(+(1, n263_0)))), right(gen_empty:node4_0(+(1, n263_0))))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(false, true, gen_empty:node4_0(n263_0), node(empty, node(right(empty), right(gen_empty:node4_0(+(1, n263_0))))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(false, true, gen_empty:node4_0(n263_0), node(empty, node(empty, right(gen_empty:node4_0(+(1, n263_0))))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_R^Omega(1)
if(false, true, gen_empty:node4_0(n263_0), node(empty, node(empty, gen_empty:node4_0(n263_0))), gen_0':s5_0(b), inc(gen_0':s5_0(b))) ->_L^Omega(1 + b)
if(false, true, gen_empty:node4_0(n263_0), node(empty, node(empty, gen_empty:node4_0(n263_0))), gen_0':s5_0(b), gen_0':s5_0(+(1, b))) ->_R^Omega(1)
count(gen_empty:node4_0(n263_0), gen_0':s5_0(+(1, b))) ->_IH
gen_0':s5_0(+(+(1, b), c264_0))

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

(14)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
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')

Types:
isEmpty :: empty:node -> true:false
empty :: empty:node
true :: true:false
node :: empty:node -> empty:node -> empty:node
false :: true:false
left :: empty:node -> empty:node
right :: empty:node -> empty:node
inc :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
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_true:false1_0 :: true:false
hole_empty:node2_0 :: empty:node
hole_0':s3_0 :: 0':s
gen_empty:node4_0 :: Nat -> empty:node
gen_0':s5_0 :: Nat -> 0':s


Lemmas:
inc(gen_0':s5_0(n7_0)) -> gen_0':s5_0(+(1, n7_0)), rt in Omega(1 + n7_0)


Generator Equations:
gen_empty:node4_0(0) <=> empty
gen_empty:node4_0(+(x, 1)) <=> node(empty, gen_empty:node4_0(x))
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


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

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

(16)
BOUNDS(n^2, INF)