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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 25.4 s]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs
        (13) RewriteLemmaProof [LOWER BOUND(ID), 61 ms]
        (14) BOUNDS(1, INF)


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(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:

   intlist(nil) -> nil
   int(s(x), 0) -> nil
   int(x, x) -> cons(x, nil)
   intlist(cons(x, y)) -> cons(s(x), intlist(y))
   int(s(x), s(y)) -> intlist(int(x, y))
   int(0, s(y)) -> cons(0, int(s(0), s(y)))
   intlist(cons(x, nil)) -> cons(s(x), nil)

S is empty.
Rewrite Strategy: FULL
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(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(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:

   intlist(nil) -> nil
   int(s(x), 0') -> nil
   int(x, x) -> cons(x, nil)
   intlist(cons(x, y)) -> cons(s(x), intlist(y))
   int(s(x), s(y)) -> intlist(int(x, y))
   int(0', s(y)) -> cons(0', int(s(0'), s(y)))
   intlist(cons(x, nil)) -> cons(s(x), nil)

S is empty.
Rewrite Strategy: FULL
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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(4)
Obligation:
TRS:
Rules:
intlist(nil) -> nil
int(s(x), 0') -> nil
int(x, x) -> cons(x, nil)
intlist(cons(x, y)) -> cons(s(x), intlist(y))
int(s(x), s(y)) -> intlist(int(x, y))
int(0', s(y)) -> cons(0', int(s(0'), s(y)))
intlist(cons(x, nil)) -> cons(s(x), nil)

Types:
intlist :: nil:cons -> nil:cons
nil :: nil:cons
int :: s:0' -> s:0' -> nil:cons
s :: s:0' -> s:0'
0' :: s:0'
cons :: s:0' -> nil:cons -> nil:cons
hole_nil:cons1_0 :: nil:cons
hole_s:0'2_0 :: s:0'
gen_nil:cons3_0 :: Nat -> nil:cons
gen_s:0'4_0 :: Nat -> s:0'

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
intlist, int

They will be analysed ascendingly in the following order:
intlist < int

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(6)
Obligation:
TRS:
Rules:
intlist(nil) -> nil
int(s(x), 0') -> nil
int(x, x) -> cons(x, nil)
intlist(cons(x, y)) -> cons(s(x), intlist(y))
int(s(x), s(y)) -> intlist(int(x, y))
int(0', s(y)) -> cons(0', int(s(0'), s(y)))
intlist(cons(x, nil)) -> cons(s(x), nil)

Types:
intlist :: nil:cons -> nil:cons
nil :: nil:cons
int :: s:0' -> s:0' -> nil:cons
s :: s:0' -> s:0'
0' :: s:0'
cons :: s:0' -> nil:cons -> nil:cons
hole_nil:cons1_0 :: nil:cons
hole_s:0'2_0 :: s:0'
gen_nil:cons3_0 :: Nat -> nil:cons
gen_s:0'4_0 :: Nat -> s:0'


Generator Equations:
gen_nil:cons3_0(0) <=> nil
gen_nil:cons3_0(+(x, 1)) <=> cons(0', gen_nil:cons3_0(x))
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
intlist, int

They will be analysed ascendingly in the following order:
intlist < int

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
intlist(gen_nil:cons3_0(+(1, n6_0))) -> *5_0, rt in Omega(n6_0)

Induction Base:
intlist(gen_nil:cons3_0(+(1, 0)))

Induction Step:
intlist(gen_nil:cons3_0(+(1, +(n6_0, 1)))) ->_R^Omega(1)
cons(s(0'), intlist(gen_nil:cons3_0(+(1, n6_0)))) ->_IH
cons(s(0'), *5_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(8)
Complex Obligation (BEST)

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(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
intlist(nil) -> nil
int(s(x), 0') -> nil
int(x, x) -> cons(x, nil)
intlist(cons(x, y)) -> cons(s(x), intlist(y))
int(s(x), s(y)) -> intlist(int(x, y))
int(0', s(y)) -> cons(0', int(s(0'), s(y)))
intlist(cons(x, nil)) -> cons(s(x), nil)

Types:
intlist :: nil:cons -> nil:cons
nil :: nil:cons
int :: s:0' -> s:0' -> nil:cons
s :: s:0' -> s:0'
0' :: s:0'
cons :: s:0' -> nil:cons -> nil:cons
hole_nil:cons1_0 :: nil:cons
hole_s:0'2_0 :: s:0'
gen_nil:cons3_0 :: Nat -> nil:cons
gen_s:0'4_0 :: Nat -> s:0'


Generator Equations:
gen_nil:cons3_0(0) <=> nil
gen_nil:cons3_0(+(x, 1)) <=> cons(0', gen_nil:cons3_0(x))
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
intlist, int

They will be analysed ascendingly in the following order:
intlist < int

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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(11)
BOUNDS(n^1, INF)

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(12)
Obligation:
TRS:
Rules:
intlist(nil) -> nil
int(s(x), 0') -> nil
int(x, x) -> cons(x, nil)
intlist(cons(x, y)) -> cons(s(x), intlist(y))
int(s(x), s(y)) -> intlist(int(x, y))
int(0', s(y)) -> cons(0', int(s(0'), s(y)))
intlist(cons(x, nil)) -> cons(s(x), nil)

Types:
intlist :: nil:cons -> nil:cons
nil :: nil:cons
int :: s:0' -> s:0' -> nil:cons
s :: s:0' -> s:0'
0' :: s:0'
cons :: s:0' -> nil:cons -> nil:cons
hole_nil:cons1_0 :: nil:cons
hole_s:0'2_0 :: s:0'
gen_nil:cons3_0 :: Nat -> nil:cons
gen_s:0'4_0 :: Nat -> s:0'


Lemmas:
intlist(gen_nil:cons3_0(+(1, n6_0))) -> *5_0, rt in Omega(n6_0)


Generator Equations:
gen_nil:cons3_0(0) <=> nil
gen_nil:cons3_0(+(x, 1)) <=> cons(0', gen_nil:cons3_0(x))
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
int
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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
int(gen_s:0'4_0(+(1, n389747_0)), gen_s:0'4_0(n389747_0)) -> gen_nil:cons3_0(0), rt in Omega(1 + n389747_0)

Induction Base:
int(gen_s:0'4_0(+(1, 0)), gen_s:0'4_0(0)) ->_R^Omega(1)
nil

Induction Step:
int(gen_s:0'4_0(+(1, +(n389747_0, 1))), gen_s:0'4_0(+(n389747_0, 1))) ->_R^Omega(1)
intlist(int(gen_s:0'4_0(+(1, n389747_0)), gen_s:0'4_0(n389747_0))) ->_IH
intlist(gen_nil:cons3_0(0)) ->_R^Omega(1)
nil

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(14)
BOUNDS(1, INF)