/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^2), ?)
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 249 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), 84 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 331 ms]
        (18) proven lower bound
        (19) LowerBoundPropagationProof [FINISHED, 0 ms]
        (20) 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(nil) -> true
   isEmpty(cons(x, xs)) -> false
   last(cons(x, nil)) -> x
   last(cons(x, cons(y, ys))) -> last(cons(y, ys))
   dropLast(nil) -> nil
   dropLast(cons(x, nil)) -> nil
   dropLast(cons(x, cons(y, ys))) -> cons(x, dropLast(cons(y, ys)))
   append(nil, ys) -> ys
   append(cons(x, xs), ys) -> cons(x, append(xs, ys))
   reverse(xs) -> rev(xs, nil)
   rev(xs, ys) -> if(isEmpty(xs), dropLast(xs), append(ys, last(xs)), ys)
   if(true, xs, ys, zs) -> zs
   if(false, xs, ys, zs) -> rev(xs, ys)

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(nil) -> true
   isEmpty(cons(x, xs)) -> false
   last(cons(x, nil)) -> x
   last(cons(x, cons(y, ys))) -> last(cons(y, ys))
   dropLast(nil) -> nil
   dropLast(cons(x, nil)) -> nil
   dropLast(cons(x, cons(y, ys))) -> cons(x, dropLast(cons(y, ys)))
   append(nil, ys) -> ys
   append(cons(x, xs), ys) -> cons(x, append(xs, ys))
   reverse(xs) -> rev(xs, nil)
   rev(xs, ys) -> if(isEmpty(xs), dropLast(xs), append(ys, last(xs)), ys)
   if(true, xs, ys, zs) -> zs
   if(false, xs, ys, zs) -> rev(xs, ys)

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

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

(4)
Obligation:
TRS:
Rules:
isEmpty(nil) -> true
isEmpty(cons(x, xs)) -> false
last(cons(x, nil)) -> x
last(cons(x, cons(y, ys))) -> last(cons(y, ys))
dropLast(nil) -> nil
dropLast(cons(x, nil)) -> nil
dropLast(cons(x, cons(y, ys))) -> cons(x, dropLast(cons(y, ys)))
append(nil, ys) -> ys
append(cons(x, xs), ys) -> cons(x, append(xs, ys))
reverse(xs) -> rev(xs, nil)
rev(xs, ys) -> if(isEmpty(xs), dropLast(xs), append(ys, last(xs)), ys)
if(true, xs, ys, zs) -> zs
if(false, xs, ys, zs) -> rev(xs, ys)

Types:
isEmpty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: nil:cons -> nil:cons -> nil:cons
false :: true:false
last :: nil:cons -> nil:cons
dropLast :: nil:cons -> nil:cons
append :: nil:cons -> nil:cons -> nil:cons
reverse :: nil:cons -> nil:cons
rev :: nil:cons -> nil:cons -> nil:cons
if :: true:false -> nil:cons -> nil:cons -> nil:cons -> nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
gen_nil:cons3_0 :: Nat -> nil:cons

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
last, dropLast, append, rev

They will be analysed ascendingly in the following order:
last < rev
dropLast < rev
append < rev

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

(6)
Obligation:
TRS:
Rules:
isEmpty(nil) -> true
isEmpty(cons(x, xs)) -> false
last(cons(x, nil)) -> x
last(cons(x, cons(y, ys))) -> last(cons(y, ys))
dropLast(nil) -> nil
dropLast(cons(x, nil)) -> nil
dropLast(cons(x, cons(y, ys))) -> cons(x, dropLast(cons(y, ys)))
append(nil, ys) -> ys
append(cons(x, xs), ys) -> cons(x, append(xs, ys))
reverse(xs) -> rev(xs, nil)
rev(xs, ys) -> if(isEmpty(xs), dropLast(xs), append(ys, last(xs)), ys)
if(true, xs, ys, zs) -> zs
if(false, xs, ys, zs) -> rev(xs, ys)

Types:
isEmpty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: nil:cons -> nil:cons -> nil:cons
false :: true:false
last :: nil:cons -> nil:cons
dropLast :: nil:cons -> nil:cons
append :: nil:cons -> nil:cons -> nil:cons
reverse :: nil:cons -> nil:cons
rev :: nil:cons -> nil:cons -> nil:cons
if :: true:false -> nil:cons -> nil:cons -> nil:cons -> nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
gen_nil:cons3_0 :: Nat -> nil:cons


Generator Equations:
gen_nil:cons3_0(0) <=> nil
gen_nil:cons3_0(+(x, 1)) <=> cons(nil, gen_nil:cons3_0(x))


The following defined symbols remain to be analysed:
last, dropLast, append, rev

They will be analysed ascendingly in the following order:
last < rev
dropLast < rev
append < rev

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
last(gen_nil:cons3_0(+(1, n5_0))) -> gen_nil:cons3_0(0), rt in Omega(1 + n5_0)

Induction Base:
last(gen_nil:cons3_0(+(1, 0))) ->_R^Omega(1)
nil

Induction Step:
last(gen_nil:cons3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1)
last(cons(nil, gen_nil:cons3_0(n5_0))) ->_IH
gen_nil:cons3_0(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(nil) -> true
isEmpty(cons(x, xs)) -> false
last(cons(x, nil)) -> x
last(cons(x, cons(y, ys))) -> last(cons(y, ys))
dropLast(nil) -> nil
dropLast(cons(x, nil)) -> nil
dropLast(cons(x, cons(y, ys))) -> cons(x, dropLast(cons(y, ys)))
append(nil, ys) -> ys
append(cons(x, xs), ys) -> cons(x, append(xs, ys))
reverse(xs) -> rev(xs, nil)
rev(xs, ys) -> if(isEmpty(xs), dropLast(xs), append(ys, last(xs)), ys)
if(true, xs, ys, zs) -> zs
if(false, xs, ys, zs) -> rev(xs, ys)

Types:
isEmpty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: nil:cons -> nil:cons -> nil:cons
false :: true:false
last :: nil:cons -> nil:cons
dropLast :: nil:cons -> nil:cons
append :: nil:cons -> nil:cons -> nil:cons
reverse :: nil:cons -> nil:cons
rev :: nil:cons -> nil:cons -> nil:cons
if :: true:false -> nil:cons -> nil:cons -> nil:cons -> nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
gen_nil:cons3_0 :: Nat -> nil:cons


Generator Equations:
gen_nil:cons3_0(0) <=> nil
gen_nil:cons3_0(+(x, 1)) <=> cons(nil, gen_nil:cons3_0(x))


The following defined symbols remain to be analysed:
last, dropLast, append, rev

They will be analysed ascendingly in the following order:
last < rev
dropLast < rev
append < rev

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
isEmpty(nil) -> true
isEmpty(cons(x, xs)) -> false
last(cons(x, nil)) -> x
last(cons(x, cons(y, ys))) -> last(cons(y, ys))
dropLast(nil) -> nil
dropLast(cons(x, nil)) -> nil
dropLast(cons(x, cons(y, ys))) -> cons(x, dropLast(cons(y, ys)))
append(nil, ys) -> ys
append(cons(x, xs), ys) -> cons(x, append(xs, ys))
reverse(xs) -> rev(xs, nil)
rev(xs, ys) -> if(isEmpty(xs), dropLast(xs), append(ys, last(xs)), ys)
if(true, xs, ys, zs) -> zs
if(false, xs, ys, zs) -> rev(xs, ys)

Types:
isEmpty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: nil:cons -> nil:cons -> nil:cons
false :: true:false
last :: nil:cons -> nil:cons
dropLast :: nil:cons -> nil:cons
append :: nil:cons -> nil:cons -> nil:cons
reverse :: nil:cons -> nil:cons
rev :: nil:cons -> nil:cons -> nil:cons
if :: true:false -> nil:cons -> nil:cons -> nil:cons -> nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
gen_nil:cons3_0 :: Nat -> nil:cons


Lemmas:
last(gen_nil:cons3_0(+(1, n5_0))) -> gen_nil:cons3_0(0), rt in Omega(1 + n5_0)


Generator Equations:
gen_nil:cons3_0(0) <=> nil
gen_nil:cons3_0(+(x, 1)) <=> cons(nil, gen_nil:cons3_0(x))


The following defined symbols remain to be analysed:
dropLast, append, rev

They will be analysed ascendingly in the following order:
dropLast < rev
append < rev

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
dropLast(gen_nil:cons3_0(+(1, n198_0))) -> gen_nil:cons3_0(n198_0), rt in Omega(1 + n198_0)

Induction Base:
dropLast(gen_nil:cons3_0(+(1, 0))) ->_R^Omega(1)
nil

Induction Step:
dropLast(gen_nil:cons3_0(+(1, +(n198_0, 1)))) ->_R^Omega(1)
cons(nil, dropLast(cons(nil, gen_nil:cons3_0(n198_0)))) ->_IH
cons(nil, gen_nil:cons3_0(c199_0))

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

(14)
Obligation:
TRS:
Rules:
isEmpty(nil) -> true
isEmpty(cons(x, xs)) -> false
last(cons(x, nil)) -> x
last(cons(x, cons(y, ys))) -> last(cons(y, ys))
dropLast(nil) -> nil
dropLast(cons(x, nil)) -> nil
dropLast(cons(x, cons(y, ys))) -> cons(x, dropLast(cons(y, ys)))
append(nil, ys) -> ys
append(cons(x, xs), ys) -> cons(x, append(xs, ys))
reverse(xs) -> rev(xs, nil)
rev(xs, ys) -> if(isEmpty(xs), dropLast(xs), append(ys, last(xs)), ys)
if(true, xs, ys, zs) -> zs
if(false, xs, ys, zs) -> rev(xs, ys)

Types:
isEmpty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: nil:cons -> nil:cons -> nil:cons
false :: true:false
last :: nil:cons -> nil:cons
dropLast :: nil:cons -> nil:cons
append :: nil:cons -> nil:cons -> nil:cons
reverse :: nil:cons -> nil:cons
rev :: nil:cons -> nil:cons -> nil:cons
if :: true:false -> nil:cons -> nil:cons -> nil:cons -> nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
gen_nil:cons3_0 :: Nat -> nil:cons


Lemmas:
last(gen_nil:cons3_0(+(1, n5_0))) -> gen_nil:cons3_0(0), rt in Omega(1 + n5_0)
dropLast(gen_nil:cons3_0(+(1, n198_0))) -> gen_nil:cons3_0(n198_0), rt in Omega(1 + n198_0)


Generator Equations:
gen_nil:cons3_0(0) <=> nil
gen_nil:cons3_0(+(x, 1)) <=> cons(nil, gen_nil:cons3_0(x))


The following defined symbols remain to be analysed:
append, rev

They will be analysed ascendingly in the following order:
append < rev

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
append(gen_nil:cons3_0(n545_0), gen_nil:cons3_0(b)) -> gen_nil:cons3_0(+(n545_0, b)), rt in Omega(1 + n545_0)

Induction Base:
append(gen_nil:cons3_0(0), gen_nil:cons3_0(b)) ->_R^Omega(1)
gen_nil:cons3_0(b)

Induction Step:
append(gen_nil:cons3_0(+(n545_0, 1)), gen_nil:cons3_0(b)) ->_R^Omega(1)
cons(nil, append(gen_nil:cons3_0(n545_0), gen_nil:cons3_0(b))) ->_IH
cons(nil, gen_nil:cons3_0(+(b, c546_0)))

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

(16)
Obligation:
TRS:
Rules:
isEmpty(nil) -> true
isEmpty(cons(x, xs)) -> false
last(cons(x, nil)) -> x
last(cons(x, cons(y, ys))) -> last(cons(y, ys))
dropLast(nil) -> nil
dropLast(cons(x, nil)) -> nil
dropLast(cons(x, cons(y, ys))) -> cons(x, dropLast(cons(y, ys)))
append(nil, ys) -> ys
append(cons(x, xs), ys) -> cons(x, append(xs, ys))
reverse(xs) -> rev(xs, nil)
rev(xs, ys) -> if(isEmpty(xs), dropLast(xs), append(ys, last(xs)), ys)
if(true, xs, ys, zs) -> zs
if(false, xs, ys, zs) -> rev(xs, ys)

Types:
isEmpty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: nil:cons -> nil:cons -> nil:cons
false :: true:false
last :: nil:cons -> nil:cons
dropLast :: nil:cons -> nil:cons
append :: nil:cons -> nil:cons -> nil:cons
reverse :: nil:cons -> nil:cons
rev :: nil:cons -> nil:cons -> nil:cons
if :: true:false -> nil:cons -> nil:cons -> nil:cons -> nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
gen_nil:cons3_0 :: Nat -> nil:cons


Lemmas:
last(gen_nil:cons3_0(+(1, n5_0))) -> gen_nil:cons3_0(0), rt in Omega(1 + n5_0)
dropLast(gen_nil:cons3_0(+(1, n198_0))) -> gen_nil:cons3_0(n198_0), rt in Omega(1 + n198_0)
append(gen_nil:cons3_0(n545_0), gen_nil:cons3_0(b)) -> gen_nil:cons3_0(+(n545_0, b)), rt in Omega(1 + n545_0)


Generator Equations:
gen_nil:cons3_0(0) <=> nil
gen_nil:cons3_0(+(x, 1)) <=> cons(nil, gen_nil:cons3_0(x))


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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
rev(gen_nil:cons3_0(n1160_0), gen_nil:cons3_0(0)) -> gen_nil:cons3_0(0), rt in Omega(1 + n1160_0 + n1160_0^2)

Induction Base:
rev(gen_nil:cons3_0(0), gen_nil:cons3_0(0)) ->_R^Omega(1)
if(isEmpty(gen_nil:cons3_0(0)), dropLast(gen_nil:cons3_0(0)), append(gen_nil:cons3_0(0), last(gen_nil:cons3_0(0))), gen_nil:cons3_0(0)) ->_R^Omega(1)
if(true, dropLast(gen_nil:cons3_0(0)), append(gen_nil:cons3_0(0), last(gen_nil:cons3_0(0))), gen_nil:cons3_0(0)) ->_R^Omega(1)
if(true, nil, append(gen_nil:cons3_0(0), last(gen_nil:cons3_0(0))), gen_nil:cons3_0(0)) ->_R^Omega(1)
if(true, nil, last(gen_nil:cons3_0(0)), gen_nil:cons3_0(0)) ->_R^Omega(1)
gen_nil:cons3_0(0)

Induction Step:
rev(gen_nil:cons3_0(+(n1160_0, 1)), gen_nil:cons3_0(0)) ->_R^Omega(1)
if(isEmpty(gen_nil:cons3_0(+(n1160_0, 1))), dropLast(gen_nil:cons3_0(+(n1160_0, 1))), append(gen_nil:cons3_0(0), last(gen_nil:cons3_0(+(n1160_0, 1)))), gen_nil:cons3_0(0)) ->_R^Omega(1)
if(false, dropLast(gen_nil:cons3_0(+(1, n1160_0))), append(gen_nil:cons3_0(0), last(gen_nil:cons3_0(+(1, n1160_0)))), gen_nil:cons3_0(0)) ->_L^Omega(1 + n1160_0)
if(false, gen_nil:cons3_0(n1160_0), append(gen_nil:cons3_0(0), last(gen_nil:cons3_0(+(1, n1160_0)))), gen_nil:cons3_0(0)) ->_L^Omega(1 + n1160_0)
if(false, gen_nil:cons3_0(n1160_0), append(gen_nil:cons3_0(0), gen_nil:cons3_0(0)), gen_nil:cons3_0(0)) ->_L^Omega(1)
if(false, gen_nil:cons3_0(n1160_0), gen_nil:cons3_0(+(0, 0)), gen_nil:cons3_0(0)) ->_R^Omega(1)
rev(gen_nil:cons3_0(n1160_0), gen_nil:cons3_0(0)) ->_IH
gen_nil:cons3_0(0)

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

(18)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
isEmpty(nil) -> true
isEmpty(cons(x, xs)) -> false
last(cons(x, nil)) -> x
last(cons(x, cons(y, ys))) -> last(cons(y, ys))
dropLast(nil) -> nil
dropLast(cons(x, nil)) -> nil
dropLast(cons(x, cons(y, ys))) -> cons(x, dropLast(cons(y, ys)))
append(nil, ys) -> ys
append(cons(x, xs), ys) -> cons(x, append(xs, ys))
reverse(xs) -> rev(xs, nil)
rev(xs, ys) -> if(isEmpty(xs), dropLast(xs), append(ys, last(xs)), ys)
if(true, xs, ys, zs) -> zs
if(false, xs, ys, zs) -> rev(xs, ys)

Types:
isEmpty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: nil:cons -> nil:cons -> nil:cons
false :: true:false
last :: nil:cons -> nil:cons
dropLast :: nil:cons -> nil:cons
append :: nil:cons -> nil:cons -> nil:cons
reverse :: nil:cons -> nil:cons
rev :: nil:cons -> nil:cons -> nil:cons
if :: true:false -> nil:cons -> nil:cons -> nil:cons -> nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
gen_nil:cons3_0 :: Nat -> nil:cons


Lemmas:
last(gen_nil:cons3_0(+(1, n5_0))) -> gen_nil:cons3_0(0), rt in Omega(1 + n5_0)
dropLast(gen_nil:cons3_0(+(1, n198_0))) -> gen_nil:cons3_0(n198_0), rt in Omega(1 + n198_0)
append(gen_nil:cons3_0(n545_0), gen_nil:cons3_0(b)) -> gen_nil:cons3_0(+(n545_0, b)), rt in Omega(1 + n545_0)


Generator Equations:
gen_nil:cons3_0(0) <=> nil
gen_nil:cons3_0(+(x, 1)) <=> cons(nil, gen_nil:cons3_0(x))


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

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

(20)
BOUNDS(n^2, INF)