/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 250 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), 41 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 7 ms]
        (16) proven lower bound
        (17) LowerBoundPropagationProof [FINISHED, 0 ms]
        (18) 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:

   last(nil) -> 0
   last(cons(x, nil)) -> x
   last(cons(x, cons(y, xs))) -> last(cons(y, xs))
   del(x, nil) -> nil
   del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs)
   if(true, x, y, xs) -> xs
   if(false, x, y, xs) -> cons(y, del(x, xs))
   eq(0, 0) -> true
   eq(0, s(y)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   reverse(nil) -> nil
   reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs))))

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:

   last(nil) -> 0'
   last(cons(x, nil)) -> x
   last(cons(x, cons(y, xs))) -> last(cons(y, xs))
   del(x, nil) -> nil
   del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs)
   if(true, x, y, xs) -> xs
   if(false, x, y, xs) -> cons(y, del(x, xs))
   eq(0', 0') -> true
   eq(0', s(y)) -> false
   eq(s(x), 0') -> false
   eq(s(x), s(y)) -> eq(x, y)
   reverse(nil) -> nil
   reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs))))

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

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

(4)
Obligation:
TRS:
Rules:
last(nil) -> 0'
last(cons(x, nil)) -> x
last(cons(x, cons(y, xs))) -> last(cons(y, xs))
del(x, nil) -> nil
del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs)
if(true, x, y, xs) -> xs
if(false, x, y, xs) -> cons(y, del(x, xs))
eq(0', 0') -> true
eq(0', s(y)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
reverse(nil) -> nil
reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs))))

Types:
last :: nil:cons -> 0':s
nil :: nil:cons
0' :: 0':s
cons :: 0':s -> nil:cons -> nil:cons
del :: 0':s -> nil:cons -> nil:cons
if :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
eq :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
s :: 0':s -> 0':s
reverse :: nil:cons -> nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
last, del, eq, reverse

They will be analysed ascendingly in the following order:
last < reverse
eq < del
del < reverse

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

(6)
Obligation:
TRS:
Rules:
last(nil) -> 0'
last(cons(x, nil)) -> x
last(cons(x, cons(y, xs))) -> last(cons(y, xs))
del(x, nil) -> nil
del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs)
if(true, x, y, xs) -> xs
if(false, x, y, xs) -> cons(y, del(x, xs))
eq(0', 0') -> true
eq(0', s(y)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
reverse(nil) -> nil
reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs))))

Types:
last :: nil:cons -> 0':s
nil :: nil:cons
0' :: 0':s
cons :: 0':s -> nil:cons -> nil:cons
del :: 0':s -> nil:cons -> nil:cons
if :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
eq :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
s :: 0':s -> 0':s
reverse :: nil:cons -> nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:cons5_0(0) <=> nil
gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
last, del, eq, reverse

They will be analysed ascendingly in the following order:
last < reverse
eq < del
del < reverse

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
last(gen_nil:cons5_0(+(1, n7_0))) -> gen_0':s4_0(0), rt in Omega(1 + n7_0)

Induction Base:
last(gen_nil:cons5_0(+(1, 0))) ->_R^Omega(1)
0'

Induction Step:
last(gen_nil:cons5_0(+(1, +(n7_0, 1)))) ->_R^Omega(1)
last(cons(0', gen_nil:cons5_0(n7_0))) ->_IH
gen_0':s4_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:
last(nil) -> 0'
last(cons(x, nil)) -> x
last(cons(x, cons(y, xs))) -> last(cons(y, xs))
del(x, nil) -> nil
del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs)
if(true, x, y, xs) -> xs
if(false, x, y, xs) -> cons(y, del(x, xs))
eq(0', 0') -> true
eq(0', s(y)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
reverse(nil) -> nil
reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs))))

Types:
last :: nil:cons -> 0':s
nil :: nil:cons
0' :: 0':s
cons :: 0':s -> nil:cons -> nil:cons
del :: 0':s -> nil:cons -> nil:cons
if :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
eq :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
s :: 0':s -> 0':s
reverse :: nil:cons -> nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:cons5_0(0) <=> nil
gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
last, del, eq, reverse

They will be analysed ascendingly in the following order:
last < reverse
eq < del
del < reverse

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
last(nil) -> 0'
last(cons(x, nil)) -> x
last(cons(x, cons(y, xs))) -> last(cons(y, xs))
del(x, nil) -> nil
del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs)
if(true, x, y, xs) -> xs
if(false, x, y, xs) -> cons(y, del(x, xs))
eq(0', 0') -> true
eq(0', s(y)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
reverse(nil) -> nil
reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs))))

Types:
last :: nil:cons -> 0':s
nil :: nil:cons
0' :: 0':s
cons :: 0':s -> nil:cons -> nil:cons
del :: 0':s -> nil:cons -> nil:cons
if :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
eq :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
s :: 0':s -> 0':s
reverse :: nil:cons -> nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons


Lemmas:
last(gen_nil:cons5_0(+(1, n7_0))) -> gen_0':s4_0(0), rt in Omega(1 + n7_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:cons5_0(0) <=> nil
gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
eq, del, reverse

They will be analysed ascendingly in the following order:
eq < del
del < reverse

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
eq(gen_0':s4_0(n335_0), gen_0':s4_0(n335_0)) -> true, rt in Omega(1 + n335_0)

Induction Base:
eq(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1)
true

Induction Step:
eq(gen_0':s4_0(+(n335_0, 1)), gen_0':s4_0(+(n335_0, 1))) ->_R^Omega(1)
eq(gen_0':s4_0(n335_0), gen_0':s4_0(n335_0)) ->_IH
true

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

(14)
Obligation:
TRS:
Rules:
last(nil) -> 0'
last(cons(x, nil)) -> x
last(cons(x, cons(y, xs))) -> last(cons(y, xs))
del(x, nil) -> nil
del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs)
if(true, x, y, xs) -> xs
if(false, x, y, xs) -> cons(y, del(x, xs))
eq(0', 0') -> true
eq(0', s(y)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
reverse(nil) -> nil
reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs))))

Types:
last :: nil:cons -> 0':s
nil :: nil:cons
0' :: 0':s
cons :: 0':s -> nil:cons -> nil:cons
del :: 0':s -> nil:cons -> nil:cons
if :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
eq :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
s :: 0':s -> 0':s
reverse :: nil:cons -> nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons


Lemmas:
last(gen_nil:cons5_0(+(1, n7_0))) -> gen_0':s4_0(0), rt in Omega(1 + n7_0)
eq(gen_0':s4_0(n335_0), gen_0':s4_0(n335_0)) -> true, rt in Omega(1 + n335_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:cons5_0(0) <=> nil
gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
del, reverse

They will be analysed ascendingly in the following order:
del < reverse

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
reverse(gen_nil:cons5_0(n984_0)) -> gen_nil:cons5_0(n984_0), rt in Omega(1 + n984_0 + n984_0^2)

Induction Base:
reverse(gen_nil:cons5_0(0)) ->_R^Omega(1)
nil

Induction Step:
reverse(gen_nil:cons5_0(+(n984_0, 1))) ->_R^Omega(1)
cons(last(cons(0', gen_nil:cons5_0(n984_0))), reverse(del(last(cons(0', gen_nil:cons5_0(n984_0))), cons(0', gen_nil:cons5_0(n984_0))))) ->_L^Omega(1 + n984_0)
cons(gen_0':s4_0(0), reverse(del(last(cons(0', gen_nil:cons5_0(n984_0))), cons(0', gen_nil:cons5_0(n984_0))))) ->_L^Omega(1 + n984_0)
cons(gen_0':s4_0(0), reverse(del(gen_0':s4_0(0), cons(0', gen_nil:cons5_0(n984_0))))) ->_R^Omega(1)
cons(gen_0':s4_0(0), reverse(if(eq(gen_0':s4_0(0), 0'), gen_0':s4_0(0), 0', gen_nil:cons5_0(n984_0)))) ->_L^Omega(1)
cons(gen_0':s4_0(0), reverse(if(true, gen_0':s4_0(0), 0', gen_nil:cons5_0(n984_0)))) ->_R^Omega(1)
cons(gen_0':s4_0(0), reverse(gen_nil:cons5_0(n984_0))) ->_IH
cons(gen_0':s4_0(0), gen_nil:cons5_0(c985_0))

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

(16)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
last(nil) -> 0'
last(cons(x, nil)) -> x
last(cons(x, cons(y, xs))) -> last(cons(y, xs))
del(x, nil) -> nil
del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs)
if(true, x, y, xs) -> xs
if(false, x, y, xs) -> cons(y, del(x, xs))
eq(0', 0') -> true
eq(0', s(y)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
reverse(nil) -> nil
reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs))))

Types:
last :: nil:cons -> 0':s
nil :: nil:cons
0' :: 0':s
cons :: 0':s -> nil:cons -> nil:cons
del :: 0':s -> nil:cons -> nil:cons
if :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
eq :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
s :: 0':s -> 0':s
reverse :: nil:cons -> nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons


Lemmas:
last(gen_nil:cons5_0(+(1, n7_0))) -> gen_0':s4_0(0), rt in Omega(1 + n7_0)
eq(gen_0':s4_0(n335_0), gen_0':s4_0(n335_0)) -> true, rt in Omega(1 + n335_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:cons5_0(0) <=> nil
gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))


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

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

(18)
BOUNDS(n^2, INF)