/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 262 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), 71 ms]
        (14) 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:

   length(nil) -> 0
   length(cons(x, l)) -> s(length(l))
   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   head(cons(x, l)) -> x
   head(nil) -> undefined
   tail(nil) -> nil
   tail(cons(x, l)) -> l
   reverse(l) -> rev(0, l, nil, l)
   rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig)
   if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig)
   if(false, x, l, accu, orig) -> accu

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:

   length(nil) -> 0'
   length(cons(x, l)) -> s(length(l))
   lt(x, 0') -> false
   lt(0', s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   head(cons(x, l)) -> x
   head(nil) -> undefined
   tail(nil) -> nil
   tail(cons(x, l)) -> l
   reverse(l) -> rev(0', l, nil, l)
   rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig)
   if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig)
   if(false, x, l, accu, orig) -> accu

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

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

(4)
Obligation:
TRS:
Rules:
length(nil) -> 0'
length(cons(x, l)) -> s(length(l))
lt(x, 0') -> false
lt(0', s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
head(cons(x, l)) -> x
head(nil) -> undefined
tail(nil) -> nil
tail(cons(x, l)) -> l
reverse(l) -> rev(0', l, nil, l)
rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig)
if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig)
if(false, x, l, accu, orig) -> accu

Types:
length :: nil:cons -> 0':s
nil :: nil:cons
0' :: 0':s
cons :: undefined -> nil:cons -> nil:cons
s :: 0':s -> 0':s
lt :: 0':s -> 0':s -> false:true
false :: false:true
true :: false:true
head :: nil:cons -> undefined
undefined :: undefined
tail :: nil:cons -> nil:cons
reverse :: nil:cons -> nil:cons
rev :: 0':s -> nil:cons -> nil:cons -> nil:cons -> nil:cons
if :: false:true -> 0':s -> nil:cons -> nil:cons -> nil:cons -> nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_undefined3_0 :: undefined
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:cons6_0 :: Nat -> nil:cons

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
length, lt, rev

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

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

(6)
Obligation:
TRS:
Rules:
length(nil) -> 0'
length(cons(x, l)) -> s(length(l))
lt(x, 0') -> false
lt(0', s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
head(cons(x, l)) -> x
head(nil) -> undefined
tail(nil) -> nil
tail(cons(x, l)) -> l
reverse(l) -> rev(0', l, nil, l)
rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig)
if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig)
if(false, x, l, accu, orig) -> accu

Types:
length :: nil:cons -> 0':s
nil :: nil:cons
0' :: 0':s
cons :: undefined -> nil:cons -> nil:cons
s :: 0':s -> 0':s
lt :: 0':s -> 0':s -> false:true
false :: false:true
true :: false:true
head :: nil:cons -> undefined
undefined :: undefined
tail :: nil:cons -> nil:cons
reverse :: nil:cons -> nil:cons
rev :: 0':s -> nil:cons -> nil:cons -> nil:cons -> nil:cons
if :: false:true -> 0':s -> nil:cons -> nil:cons -> nil:cons -> nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_undefined3_0 :: undefined
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:cons6_0 :: Nat -> nil:cons


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:cons6_0(0) <=> nil
gen_nil:cons6_0(+(x, 1)) <=> cons(undefined, gen_nil:cons6_0(x))


The following defined symbols remain to be analysed:
length, lt, rev

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

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length(gen_nil:cons6_0(n8_0)) -> gen_0':s5_0(n8_0), rt in Omega(1 + n8_0)

Induction Base:
length(gen_nil:cons6_0(0)) ->_R^Omega(1)
0'

Induction Step:
length(gen_nil:cons6_0(+(n8_0, 1))) ->_R^Omega(1)
s(length(gen_nil:cons6_0(n8_0))) ->_IH
s(gen_0':s5_0(c9_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:
length(nil) -> 0'
length(cons(x, l)) -> s(length(l))
lt(x, 0') -> false
lt(0', s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
head(cons(x, l)) -> x
head(nil) -> undefined
tail(nil) -> nil
tail(cons(x, l)) -> l
reverse(l) -> rev(0', l, nil, l)
rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig)
if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig)
if(false, x, l, accu, orig) -> accu

Types:
length :: nil:cons -> 0':s
nil :: nil:cons
0' :: 0':s
cons :: undefined -> nil:cons -> nil:cons
s :: 0':s -> 0':s
lt :: 0':s -> 0':s -> false:true
false :: false:true
true :: false:true
head :: nil:cons -> undefined
undefined :: undefined
tail :: nil:cons -> nil:cons
reverse :: nil:cons -> nil:cons
rev :: 0':s -> nil:cons -> nil:cons -> nil:cons -> nil:cons
if :: false:true -> 0':s -> nil:cons -> nil:cons -> nil:cons -> nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_undefined3_0 :: undefined
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:cons6_0 :: Nat -> nil:cons


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:cons6_0(0) <=> nil
gen_nil:cons6_0(+(x, 1)) <=> cons(undefined, gen_nil:cons6_0(x))


The following defined symbols remain to be analysed:
length, lt, rev

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

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
length(nil) -> 0'
length(cons(x, l)) -> s(length(l))
lt(x, 0') -> false
lt(0', s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
head(cons(x, l)) -> x
head(nil) -> undefined
tail(nil) -> nil
tail(cons(x, l)) -> l
reverse(l) -> rev(0', l, nil, l)
rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig)
if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig)
if(false, x, l, accu, orig) -> accu

Types:
length :: nil:cons -> 0':s
nil :: nil:cons
0' :: 0':s
cons :: undefined -> nil:cons -> nil:cons
s :: 0':s -> 0':s
lt :: 0':s -> 0':s -> false:true
false :: false:true
true :: false:true
head :: nil:cons -> undefined
undefined :: undefined
tail :: nil:cons -> nil:cons
reverse :: nil:cons -> nil:cons
rev :: 0':s -> nil:cons -> nil:cons -> nil:cons -> nil:cons
if :: false:true -> 0':s -> nil:cons -> nil:cons -> nil:cons -> nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_undefined3_0 :: undefined
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:cons6_0 :: Nat -> nil:cons


Lemmas:
length(gen_nil:cons6_0(n8_0)) -> gen_0':s5_0(n8_0), rt in Omega(1 + n8_0)


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:cons6_0(0) <=> nil
gen_nil:cons6_0(+(x, 1)) <=> cons(undefined, gen_nil:cons6_0(x))


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

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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lt(gen_0':s5_0(n232_0), gen_0':s5_0(n232_0)) -> false, rt in Omega(1 + n232_0)

Induction Base:
lt(gen_0':s5_0(0), gen_0':s5_0(0)) ->_R^Omega(1)
false

Induction Step:
lt(gen_0':s5_0(+(n232_0, 1)), gen_0':s5_0(+(n232_0, 1))) ->_R^Omega(1)
lt(gen_0':s5_0(n232_0), gen_0':s5_0(n232_0)) ->_IH
false

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

(14)
Obligation:
TRS:
Rules:
length(nil) -> 0'
length(cons(x, l)) -> s(length(l))
lt(x, 0') -> false
lt(0', s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
head(cons(x, l)) -> x
head(nil) -> undefined
tail(nil) -> nil
tail(cons(x, l)) -> l
reverse(l) -> rev(0', l, nil, l)
rev(x, l, accu, orig) -> if(lt(x, length(orig)), x, l, accu, orig)
if(true, x, l, accu, orig) -> rev(s(x), tail(l), cons(head(l), accu), orig)
if(false, x, l, accu, orig) -> accu

Types:
length :: nil:cons -> 0':s
nil :: nil:cons
0' :: 0':s
cons :: undefined -> nil:cons -> nil:cons
s :: 0':s -> 0':s
lt :: 0':s -> 0':s -> false:true
false :: false:true
true :: false:true
head :: nil:cons -> undefined
undefined :: undefined
tail :: nil:cons -> nil:cons
reverse :: nil:cons -> nil:cons
rev :: 0':s -> nil:cons -> nil:cons -> nil:cons -> nil:cons
if :: false:true -> 0':s -> nil:cons -> nil:cons -> nil:cons -> nil:cons
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_undefined3_0 :: undefined
hole_false:true4_0 :: false:true
gen_0':s5_0 :: Nat -> 0':s
gen_nil:cons6_0 :: Nat -> nil:cons


Lemmas:
length(gen_nil:cons6_0(n8_0)) -> gen_0':s5_0(n8_0), rt in Omega(1 + n8_0)
lt(gen_0':s5_0(n232_0), gen_0':s5_0(n232_0)) -> false, rt in Omega(1 + n232_0)


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))
gen_nil:cons6_0(0) <=> nil
gen_nil:cons6_0(+(x, 1)) <=> cons(undefined, gen_nil:cons6_0(x))


The following defined symbols remain to be analysed:
rev