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


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, 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), 276 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), 133 ms]
        (14) BOUNDS(1, INF)


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(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

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

S is empty.
Rewrite Strategy: FULL
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(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

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

S is empty.
Rewrite Strategy: FULL
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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(4)
Obligation:
TRS:
Rules:
isLeaf(leaf) -> true
isLeaf(cons(u, v)) -> false
left(cons(u, v)) -> u
right(cons(u, v)) -> v
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v)
if1(b, true, u, v) -> false
if1(b, false, u, v) -> if2(b, u, v)
if2(true, u, v) -> true
if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))

Types:
isLeaf :: leaf:cons -> true:false
leaf :: leaf:cons
true :: true:false
cons :: leaf:cons -> leaf:cons -> leaf:cons
false :: true:false
left :: leaf:cons -> leaf:cons
right :: leaf:cons -> leaf:cons
concat :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> true:false
if1 :: true:false -> true:false -> leaf:cons -> leaf:cons -> true:false
if2 :: true:false -> leaf:cons -> leaf:cons -> true:false
hole_true:false1_0 :: true:false
hole_leaf:cons2_0 :: leaf:cons
gen_leaf:cons3_0 :: Nat -> leaf:cons

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

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

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(6)
Obligation:
TRS:
Rules:
isLeaf(leaf) -> true
isLeaf(cons(u, v)) -> false
left(cons(u, v)) -> u
right(cons(u, v)) -> v
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v)
if1(b, true, u, v) -> false
if1(b, false, u, v) -> if2(b, u, v)
if2(true, u, v) -> true
if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))

Types:
isLeaf :: leaf:cons -> true:false
leaf :: leaf:cons
true :: true:false
cons :: leaf:cons -> leaf:cons -> leaf:cons
false :: true:false
left :: leaf:cons -> leaf:cons
right :: leaf:cons -> leaf:cons
concat :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> true:false
if1 :: true:false -> true:false -> leaf:cons -> leaf:cons -> true:false
if2 :: true:false -> leaf:cons -> leaf:cons -> true:false
hole_true:false1_0 :: true:false
hole_leaf:cons2_0 :: leaf:cons
gen_leaf:cons3_0 :: Nat -> leaf:cons


Generator Equations:
gen_leaf:cons3_0(0) <=> leaf
gen_leaf:cons3_0(+(x, 1)) <=> cons(leaf, gen_leaf:cons3_0(x))


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

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

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
concat(gen_leaf:cons3_0(n5_0), gen_leaf:cons3_0(b)) -> gen_leaf:cons3_0(+(n5_0, b)), rt in Omega(1 + n5_0)

Induction Base:
concat(gen_leaf:cons3_0(0), gen_leaf:cons3_0(b)) ->_R^Omega(1)
gen_leaf:cons3_0(b)

Induction Step:
concat(gen_leaf:cons3_0(+(n5_0, 1)), gen_leaf:cons3_0(b)) ->_R^Omega(1)
cons(leaf, concat(gen_leaf:cons3_0(n5_0), gen_leaf:cons3_0(b))) ->_IH
cons(leaf, gen_leaf:cons3_0(+(b, c6_0)))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(8)
Complex Obligation (BEST)

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

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

Types:
isLeaf :: leaf:cons -> true:false
leaf :: leaf:cons
true :: true:false
cons :: leaf:cons -> leaf:cons -> leaf:cons
false :: true:false
left :: leaf:cons -> leaf:cons
right :: leaf:cons -> leaf:cons
concat :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> true:false
if1 :: true:false -> true:false -> leaf:cons -> leaf:cons -> true:false
if2 :: true:false -> leaf:cons -> leaf:cons -> true:false
hole_true:false1_0 :: true:false
hole_leaf:cons2_0 :: leaf:cons
gen_leaf:cons3_0 :: Nat -> leaf:cons


Generator Equations:
gen_leaf:cons3_0(0) <=> leaf
gen_leaf:cons3_0(+(x, 1)) <=> cons(leaf, gen_leaf:cons3_0(x))


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

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

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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(11)
BOUNDS(n^1, INF)

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(12)
Obligation:
TRS:
Rules:
isLeaf(leaf) -> true
isLeaf(cons(u, v)) -> false
left(cons(u, v)) -> u
right(cons(u, v)) -> v
concat(leaf, y) -> y
concat(cons(u, v), y) -> cons(u, concat(v, y))
less_leaves(u, v) -> if1(isLeaf(u), isLeaf(v), u, v)
if1(b, true, u, v) -> false
if1(b, false, u, v) -> if2(b, u, v)
if2(true, u, v) -> true
if2(false, u, v) -> less_leaves(concat(left(u), right(u)), concat(left(v), right(v)))

Types:
isLeaf :: leaf:cons -> true:false
leaf :: leaf:cons
true :: true:false
cons :: leaf:cons -> leaf:cons -> leaf:cons
false :: true:false
left :: leaf:cons -> leaf:cons
right :: leaf:cons -> leaf:cons
concat :: leaf:cons -> leaf:cons -> leaf:cons
less_leaves :: leaf:cons -> leaf:cons -> true:false
if1 :: true:false -> true:false -> leaf:cons -> leaf:cons -> true:false
if2 :: true:false -> leaf:cons -> leaf:cons -> true:false
hole_true:false1_0 :: true:false
hole_leaf:cons2_0 :: leaf:cons
gen_leaf:cons3_0 :: Nat -> leaf:cons


Lemmas:
concat(gen_leaf:cons3_0(n5_0), gen_leaf:cons3_0(b)) -> gen_leaf:cons3_0(+(n5_0, b)), rt in Omega(1 + n5_0)


Generator Equations:
gen_leaf:cons3_0(0) <=> leaf
gen_leaf:cons3_0(+(x, 1)) <=> cons(leaf, gen_leaf:cons3_0(x))


The following defined symbols remain to be analysed:
less_leaves
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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
less_leaves(gen_leaf:cons3_0(+(1, n548_0)), gen_leaf:cons3_0(n548_0)) -> false, rt in Omega(1 + n548_0)

Induction Base:
less_leaves(gen_leaf:cons3_0(+(1, 0)), gen_leaf:cons3_0(0)) ->_R^Omega(1)
if1(isLeaf(gen_leaf:cons3_0(+(1, 0))), isLeaf(gen_leaf:cons3_0(0)), gen_leaf:cons3_0(+(1, 0)), gen_leaf:cons3_0(0)) ->_R^Omega(1)
if1(false, isLeaf(gen_leaf:cons3_0(0)), gen_leaf:cons3_0(1), gen_leaf:cons3_0(0)) ->_R^Omega(1)
if1(false, true, gen_leaf:cons3_0(1), gen_leaf:cons3_0(0)) ->_R^Omega(1)
false

Induction Step:
less_leaves(gen_leaf:cons3_0(+(1, +(n548_0, 1))), gen_leaf:cons3_0(+(n548_0, 1))) ->_R^Omega(1)
if1(isLeaf(gen_leaf:cons3_0(+(1, +(n548_0, 1)))), isLeaf(gen_leaf:cons3_0(+(n548_0, 1))), gen_leaf:cons3_0(+(1, +(n548_0, 1))), gen_leaf:cons3_0(+(n548_0, 1))) ->_R^Omega(1)
if1(false, isLeaf(gen_leaf:cons3_0(+(1, n548_0))), gen_leaf:cons3_0(+(2, n548_0)), gen_leaf:cons3_0(+(1, n548_0))) ->_R^Omega(1)
if1(false, false, gen_leaf:cons3_0(+(2, n548_0)), gen_leaf:cons3_0(+(1, n548_0))) ->_R^Omega(1)
if2(false, gen_leaf:cons3_0(+(2, n548_0)), gen_leaf:cons3_0(+(1, n548_0))) ->_R^Omega(1)
less_leaves(concat(left(gen_leaf:cons3_0(+(2, n548_0))), right(gen_leaf:cons3_0(+(2, n548_0)))), concat(left(gen_leaf:cons3_0(+(1, n548_0))), right(gen_leaf:cons3_0(+(1, n548_0))))) ->_R^Omega(1)
less_leaves(concat(leaf, right(gen_leaf:cons3_0(+(2, n548_0)))), concat(left(gen_leaf:cons3_0(+(1, n548_0))), right(gen_leaf:cons3_0(+(1, n548_0))))) ->_R^Omega(1)
less_leaves(concat(leaf, gen_leaf:cons3_0(+(1, n548_0))), concat(left(gen_leaf:cons3_0(+(1, n548_0))), right(gen_leaf:cons3_0(+(1, n548_0))))) ->_L^Omega(1)
less_leaves(gen_leaf:cons3_0(+(0, +(1, n548_0))), concat(left(gen_leaf:cons3_0(+(1, n548_0))), right(gen_leaf:cons3_0(+(1, n548_0))))) ->_R^Omega(1)
less_leaves(gen_leaf:cons3_0(+(1, n548_0)), concat(leaf, right(gen_leaf:cons3_0(+(1, n548_0))))) ->_R^Omega(1)
less_leaves(gen_leaf:cons3_0(+(1, n548_0)), concat(leaf, gen_leaf:cons3_0(n548_0))) ->_L^Omega(1)
less_leaves(gen_leaf:cons3_0(+(1, n548_0)), gen_leaf:cons3_0(+(0, n548_0))) ->_IH
false

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(14)
BOUNDS(1, INF)