/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 267 ms]
(8) proven lower bound
(9) LowerBoundPropagationProof [FINISHED, 0 ms]
(10) BOUNDS(n^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:

   is_empty(nil) -> true
   is_empty(cons(x, l)) -> false
   hd(cons(x, l)) -> x
   tl(cons(x, l)) -> l
   append(l1, l2) -> ifappend(l1, l2, is_empty(l1))
   ifappend(l1, l2, true) -> l2
   ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2))

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:

   is_empty(nil) -> true
   is_empty(cons(x, l)) -> false
   hd(cons(x, l)) -> x
   tl(cons(x, l)) -> l
   append(l1, l2) -> ifappend(l1, l2, is_empty(l1))
   ifappend(l1, l2, true) -> l2
   ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2))

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:
is_empty(nil) -> true
is_empty(cons(x, l)) -> false
hd(cons(x, l)) -> x
tl(cons(x, l)) -> l
append(l1, l2) -> ifappend(l1, l2, is_empty(l1))
ifappend(l1, l2, true) -> l2
ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2))

Types:
is_empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: hd -> nil:cons -> nil:cons
false :: true:false
hd :: nil:cons -> hd
tl :: nil:cons -> nil:cons
append :: nil:cons -> nil:cons -> nil:cons
ifappend :: nil:cons -> nil:cons -> true:false -> nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_hd3_0 :: hd
gen_nil:cons4_0 :: Nat -> nil:cons

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
append
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(6)
Obligation:
TRS:
Rules:
is_empty(nil) -> true
is_empty(cons(x, l)) -> false
hd(cons(x, l)) -> x
tl(cons(x, l)) -> l
append(l1, l2) -> ifappend(l1, l2, is_empty(l1))
ifappend(l1, l2, true) -> l2
ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2))

Types:
is_empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: hd -> nil:cons -> nil:cons
false :: true:false
hd :: nil:cons -> hd
tl :: nil:cons -> nil:cons
append :: nil:cons -> nil:cons -> nil:cons
ifappend :: nil:cons -> nil:cons -> true:false -> nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_hd3_0 :: hd
gen_nil:cons4_0 :: Nat -> nil:cons


Generator Equations:
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(hole_hd3_0, gen_nil:cons4_0(x))


The following defined symbols remain to be analysed:
append
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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
append(gen_nil:cons4_0(n6_0), gen_nil:cons4_0(b)) -> gen_nil:cons4_0(+(n6_0, b)), rt in Omega(1 + n6_0)

Induction Base:
append(gen_nil:cons4_0(0), gen_nil:cons4_0(b)) ->_R^Omega(1)
ifappend(gen_nil:cons4_0(0), gen_nil:cons4_0(b), is_empty(gen_nil:cons4_0(0))) ->_R^Omega(1)
ifappend(gen_nil:cons4_0(0), gen_nil:cons4_0(b), true) ->_R^Omega(1)
gen_nil:cons4_0(b)

Induction Step:
append(gen_nil:cons4_0(+(n6_0, 1)), gen_nil:cons4_0(b)) ->_R^Omega(1)
ifappend(gen_nil:cons4_0(+(n6_0, 1)), gen_nil:cons4_0(b), is_empty(gen_nil:cons4_0(+(n6_0, 1)))) ->_R^Omega(1)
ifappend(gen_nil:cons4_0(+(1, n6_0)), gen_nil:cons4_0(b), false) ->_R^Omega(1)
cons(hd(gen_nil:cons4_0(+(1, n6_0))), append(tl(gen_nil:cons4_0(+(1, n6_0))), gen_nil:cons4_0(b))) ->_R^Omega(1)
cons(hole_hd3_0, append(tl(gen_nil:cons4_0(+(1, n6_0))), gen_nil:cons4_0(b))) ->_R^Omega(1)
cons(hole_hd3_0, append(gen_nil:cons4_0(n6_0), gen_nil:cons4_0(b))) ->_IH
cons(hole_hd3_0, gen_nil:cons4_0(+(b, c7_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)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
is_empty(nil) -> true
is_empty(cons(x, l)) -> false
hd(cons(x, l)) -> x
tl(cons(x, l)) -> l
append(l1, l2) -> ifappend(l1, l2, is_empty(l1))
ifappend(l1, l2, true) -> l2
ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2))

Types:
is_empty :: nil:cons -> true:false
nil :: nil:cons
true :: true:false
cons :: hd -> nil:cons -> nil:cons
false :: true:false
hd :: nil:cons -> hd
tl :: nil:cons -> nil:cons
append :: nil:cons -> nil:cons -> nil:cons
ifappend :: nil:cons -> nil:cons -> true:false -> nil:cons
hole_true:false1_0 :: true:false
hole_nil:cons2_0 :: nil:cons
hole_hd3_0 :: hd
gen_nil:cons4_0 :: Nat -> nil:cons


Generator Equations:
gen_nil:cons4_0(0) <=> nil
gen_nil:cons4_0(+(x, 1)) <=> cons(hole_hd3_0, gen_nil:cons4_0(x))


The following defined symbols remain to be analysed:
append
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(9) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(10)
BOUNDS(n^1, INF)