/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) SlicingProof [LOWER BOUND(ID), 0 ms]
(4) CpxTRS
(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) typed CpxTrs
(7) OrderProof [LOWER BOUND(ID), 0 ms]
(8) typed CpxTrs
(9) RewriteLemmaProof [LOWER BOUND(ID), 306 ms]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 44 ms]
        (16) typed CpxTrs


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

(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:

   half(0) -> 0
   half(s(0)) -> 0
   half(s(s(x))) -> s(half(x))
   lastbit(0) -> 0
   lastbit(s(0)) -> s(0)
   lastbit(s(s(x))) -> lastbit(x)
   zero(0) -> true
   zero(s(x)) -> false
   conv(x) -> conviter(x, cons(0, nil))
   conviter(x, l) -> if(zero(x), x, l)
   if(true, x, l) -> l
   if(false, x, l) -> conviter(half(x), cons(lastbit(x), l))

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:

   half(0') -> 0'
   half(s(0')) -> 0'
   half(s(s(x))) -> s(half(x))
   lastbit(0') -> 0'
   lastbit(s(0')) -> s(0')
   lastbit(s(s(x))) -> lastbit(x)
   zero(0') -> true
   zero(s(x)) -> false
   conv(x) -> conviter(x, cons(0', nil))
   conviter(x, l) -> if(zero(x), x, l)
   if(true, x, l) -> l
   if(false, x, l) -> conviter(half(x), cons(lastbit(x), l))

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

(3) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
cons/1

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

(4)
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:

   half(0') -> 0'
   half(s(0')) -> 0'
   half(s(s(x))) -> s(half(x))
   lastbit(0') -> 0'
   lastbit(s(0')) -> s(0')
   lastbit(s(s(x))) -> lastbit(x)
   zero(0') -> true
   zero(s(x)) -> false
   conv(x) -> conviter(x, cons(0'))
   conviter(x, l) -> if(zero(x), x, l)
   if(true, x, l) -> l
   if(false, x, l) -> conviter(half(x), cons(lastbit(x)))

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

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

(6)
Obligation:
TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
lastbit(0') -> 0'
lastbit(s(0')) -> s(0')
lastbit(s(s(x))) -> lastbit(x)
zero(0') -> true
zero(s(x)) -> false
conv(x) -> conviter(x, cons(0'))
conviter(x, l) -> if(zero(x), x, l)
if(true, x, l) -> l
if(false, x, l) -> conviter(half(x), cons(lastbit(x)))

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
lastbit :: 0':s -> 0':s
zero :: 0':s -> true:false
true :: true:false
false :: true:false
conv :: 0':s -> cons
conviter :: 0':s -> cons -> cons
cons :: 0':s -> cons
if :: true:false -> 0':s -> cons -> cons
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_cons3_0 :: cons
gen_0':s4_0 :: Nat -> 0':s

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

(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
half, lastbit, conviter

They will be analysed ascendingly in the following order:
half < conviter
lastbit < conviter

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

(8)
Obligation:
TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
lastbit(0') -> 0'
lastbit(s(0')) -> s(0')
lastbit(s(s(x))) -> lastbit(x)
zero(0') -> true
zero(s(x)) -> false
conv(x) -> conviter(x, cons(0'))
conviter(x, l) -> if(zero(x), x, l)
if(true, x, l) -> l
if(false, x, l) -> conviter(half(x), cons(lastbit(x)))

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
lastbit :: 0':s -> 0':s
zero :: 0':s -> true:false
true :: true:false
false :: true:false
conv :: 0':s -> cons
conviter :: 0':s -> cons -> cons
cons :: 0':s -> cons
if :: true:false -> 0':s -> cons -> cons
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_cons3_0 :: cons
gen_0':s4_0 :: Nat -> 0':s


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


The following defined symbols remain to be analysed:
half, lastbit, conviter

They will be analysed ascendingly in the following order:
half < conviter
lastbit < conviter

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

(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
half(gen_0':s4_0(*(2, n6_0))) -> gen_0':s4_0(n6_0), rt in Omega(1 + n6_0)

Induction Base:
half(gen_0':s4_0(*(2, 0))) ->_R^Omega(1)
0'

Induction Step:
half(gen_0':s4_0(*(2, +(n6_0, 1)))) ->_R^Omega(1)
s(half(gen_0':s4_0(*(2, n6_0)))) ->_IH
s(gen_0':s4_0(c7_0))

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

(10)
Complex Obligation (BEST)

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

(11)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
lastbit(0') -> 0'
lastbit(s(0')) -> s(0')
lastbit(s(s(x))) -> lastbit(x)
zero(0') -> true
zero(s(x)) -> false
conv(x) -> conviter(x, cons(0'))
conviter(x, l) -> if(zero(x), x, l)
if(true, x, l) -> l
if(false, x, l) -> conviter(half(x), cons(lastbit(x)))

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
lastbit :: 0':s -> 0':s
zero :: 0':s -> true:false
true :: true:false
false :: true:false
conv :: 0':s -> cons
conviter :: 0':s -> cons -> cons
cons :: 0':s -> cons
if :: true:false -> 0':s -> cons -> cons
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_cons3_0 :: cons
gen_0':s4_0 :: Nat -> 0':s


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


The following defined symbols remain to be analysed:
half, lastbit, conviter

They will be analysed ascendingly in the following order:
half < conviter
lastbit < conviter

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

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

(13)
BOUNDS(n^1, INF)

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

(14)
Obligation:
TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
lastbit(0') -> 0'
lastbit(s(0')) -> s(0')
lastbit(s(s(x))) -> lastbit(x)
zero(0') -> true
zero(s(x)) -> false
conv(x) -> conviter(x, cons(0'))
conviter(x, l) -> if(zero(x), x, l)
if(true, x, l) -> l
if(false, x, l) -> conviter(half(x), cons(lastbit(x)))

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
lastbit :: 0':s -> 0':s
zero :: 0':s -> true:false
true :: true:false
false :: true:false
conv :: 0':s -> cons
conviter :: 0':s -> cons -> cons
cons :: 0':s -> cons
if :: true:false -> 0':s -> cons -> cons
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_cons3_0 :: cons
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
half(gen_0':s4_0(*(2, n6_0))) -> gen_0':s4_0(n6_0), rt in Omega(1 + n6_0)


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


The following defined symbols remain to be analysed:
lastbit, conviter

They will be analysed ascendingly in the following order:
lastbit < conviter

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lastbit(gen_0':s4_0(*(2, n308_0))) -> gen_0':s4_0(0), rt in Omega(1 + n308_0)

Induction Base:
lastbit(gen_0':s4_0(*(2, 0))) ->_R^Omega(1)
0'

Induction Step:
lastbit(gen_0':s4_0(*(2, +(n308_0, 1)))) ->_R^Omega(1)
lastbit(gen_0':s4_0(*(2, n308_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).
----------------------------------------

(16)
Obligation:
TRS:
Rules:
half(0') -> 0'
half(s(0')) -> 0'
half(s(s(x))) -> s(half(x))
lastbit(0') -> 0'
lastbit(s(0')) -> s(0')
lastbit(s(s(x))) -> lastbit(x)
zero(0') -> true
zero(s(x)) -> false
conv(x) -> conviter(x, cons(0'))
conviter(x, l) -> if(zero(x), x, l)
if(true, x, l) -> l
if(false, x, l) -> conviter(half(x), cons(lastbit(x)))

Types:
half :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
lastbit :: 0':s -> 0':s
zero :: 0':s -> true:false
true :: true:false
false :: true:false
conv :: 0':s -> cons
conviter :: 0':s -> cons -> cons
cons :: 0':s -> cons
if :: true:false -> 0':s -> cons -> cons
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
hole_cons3_0 :: cons
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
half(gen_0':s4_0(*(2, n6_0))) -> gen_0':s4_0(n6_0), rt in Omega(1 + n6_0)
lastbit(gen_0':s4_0(*(2, n308_0))) -> gen_0':s4_0(0), rt in Omega(1 + n308_0)


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


The following defined symbols remain to be analysed:
conviter