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


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


WORST_CASE(Omega(n^1), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^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), 301 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), 70 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 51 ms]
        (16) BOUNDS(1, INF)


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

(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:

   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   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)
   first(nil) -> 0
   first(cons(x, xs)) -> x
   doublelist(nil) -> nil
   doublelist(cons(x, xs)) -> cons(double(x), doublelist(del(first(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^1, INF).


The TRS R consists of the following rules:

   double(0') -> 0'
   double(s(x)) -> s(s(double(x)))
   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)
   first(nil) -> 0'
   first(cons(x, xs)) -> x
   doublelist(nil) -> nil
   doublelist(cons(x, xs)) -> cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs))))

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

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

(4)
Obligation:
TRS:
Rules:
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
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)
first(nil) -> 0'
first(cons(x, xs)) -> x
doublelist(nil) -> nil
doublelist(cons(x, xs)) -> cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs))))

Types:
double :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
del :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 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
first :: nil:cons -> 0':s
doublelist :: 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:
double, del, eq, doublelist

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

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

(6)
Obligation:
TRS:
Rules:
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
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)
first(nil) -> 0'
first(cons(x, xs)) -> x
doublelist(nil) -> nil
doublelist(cons(x, xs)) -> cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs))))

Types:
double :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
del :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 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
first :: nil:cons -> 0':s
doublelist :: 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:
double, del, eq, doublelist

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

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
double(gen_0':s4_0(n7_0)) -> gen_0':s4_0(*(2, n7_0)), rt in Omega(1 + n7_0)

Induction Base:
double(gen_0':s4_0(0)) ->_R^Omega(1)
0'

Induction Step:
double(gen_0':s4_0(+(n7_0, 1))) ->_R^Omega(1)
s(s(double(gen_0':s4_0(n7_0)))) ->_IH
s(s(gen_0':s4_0(*(2, 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:
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
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)
first(nil) -> 0'
first(cons(x, xs)) -> x
doublelist(nil) -> nil
doublelist(cons(x, xs)) -> cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs))))

Types:
double :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
del :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 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
first :: nil:cons -> 0':s
doublelist :: 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:
double, del, eq, doublelist

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

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
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)
first(nil) -> 0'
first(cons(x, xs)) -> x
doublelist(nil) -> nil
doublelist(cons(x, xs)) -> cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs))))

Types:
double :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
del :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 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
first :: nil:cons -> 0':s
doublelist :: 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:
double(gen_0':s4_0(n7_0)) -> gen_0':s4_0(*(2, n7_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, doublelist

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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
eq(gen_0':s4_0(n275_0), gen_0':s4_0(n275_0)) -> true, rt in Omega(1 + n275_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(+(n275_0, 1)), gen_0':s4_0(+(n275_0, 1))) ->_R^Omega(1)
eq(gen_0':s4_0(n275_0), gen_0':s4_0(n275_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:
double(0') -> 0'
double(s(x)) -> s(s(double(x)))
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)
first(nil) -> 0'
first(cons(x, xs)) -> x
doublelist(nil) -> nil
doublelist(cons(x, xs)) -> cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs))))

Types:
double :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
del :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
cons :: 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
first :: nil:cons -> 0':s
doublelist :: 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:
double(gen_0':s4_0(n7_0)) -> gen_0':s4_0(*(2, n7_0)), rt in Omega(1 + n7_0)
eq(gen_0':s4_0(n275_0), gen_0':s4_0(n275_0)) -> true, rt in Omega(1 + n275_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, doublelist

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

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
doublelist(gen_nil:cons5_0(n930_0)) -> gen_nil:cons5_0(n930_0), rt in Omega(1 + n930_0)

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

Induction Step:
doublelist(gen_nil:cons5_0(+(n930_0, 1))) ->_R^Omega(1)
cons(double(0'), doublelist(del(first(cons(0', gen_nil:cons5_0(n930_0))), cons(0', gen_nil:cons5_0(n930_0))))) ->_L^Omega(1)
cons(gen_0':s4_0(*(2, 0)), doublelist(del(first(cons(0', gen_nil:cons5_0(n930_0))), cons(0', gen_nil:cons5_0(n930_0))))) ->_R^Omega(1)
cons(gen_0':s4_0(0), doublelist(del(0', cons(0', gen_nil:cons5_0(n930_0))))) ->_R^Omega(1)
cons(gen_0':s4_0(0), doublelist(if(eq(0', 0'), 0', 0', gen_nil:cons5_0(n930_0)))) ->_L^Omega(1)
cons(gen_0':s4_0(0), doublelist(if(true, 0', 0', gen_nil:cons5_0(n930_0)))) ->_R^Omega(1)
cons(gen_0':s4_0(0), doublelist(gen_nil:cons5_0(n930_0))) ->_IH
cons(gen_0':s4_0(0), gen_nil:cons5_0(c931_0))

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

(16)
BOUNDS(1, INF)