/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The Runtime Complexity (innermost) 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), 288 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), 171 ms]
        (14) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h), l))
   a(h, h, x) -> s(x)
   a(x, s(y), h) -> a(x, y, s(h))
   a(x, s(y), s(z)) -> a(x, y, a(x, s(y), z))
   a(s(x), h, z) -> a(x, z, z)

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

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h), l))
   a(h, h, x) -> s(x)
   a(x, s(y), h) -> a(x, y, s(h))
   a(x, s(y), s(z)) -> a(x, y, a(x, s(y), z))
   a(s(x), h, z) -> a(x, z, z)

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

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

(4)
Obligation:
Innermost TRS:
Rules:
app(nil, k) -> k
app(l, nil) -> l
app(cons(x, l), k) -> cons(x, app(l, k))
sum(cons(x, nil)) -> cons(x, nil)
sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h), l))
a(h, h, x) -> s(x)
a(x, s(y), h) -> a(x, y, s(h))
a(x, s(y), s(z)) -> a(x, y, a(x, s(y), z))
a(s(x), h, z) -> a(x, z, z)

Types:
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: h:s -> nil:cons -> nil:cons
sum :: nil:cons -> nil:cons
a :: h:s -> h:s -> h:s -> h:s
h :: h:s
s :: h:s -> h:s
hole_nil:cons1_0 :: nil:cons
hole_h:s2_0 :: h:s
gen_nil:cons3_0 :: Nat -> nil:cons
gen_h:s4_0 :: Nat -> h:s

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
app, sum, a

They will be analysed ascendingly in the following order:
a < sum

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

(6)
Obligation:
Innermost TRS:
Rules:
app(nil, k) -> k
app(l, nil) -> l
app(cons(x, l), k) -> cons(x, app(l, k))
sum(cons(x, nil)) -> cons(x, nil)
sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h), l))
a(h, h, x) -> s(x)
a(x, s(y), h) -> a(x, y, s(h))
a(x, s(y), s(z)) -> a(x, y, a(x, s(y), z))
a(s(x), h, z) -> a(x, z, z)

Types:
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: h:s -> nil:cons -> nil:cons
sum :: nil:cons -> nil:cons
a :: h:s -> h:s -> h:s -> h:s
h :: h:s
s :: h:s -> h:s
hole_nil:cons1_0 :: nil:cons
hole_h:s2_0 :: h:s
gen_nil:cons3_0 :: Nat -> nil:cons
gen_h:s4_0 :: Nat -> h:s


Generator Equations:
gen_nil:cons3_0(0) <=> nil
gen_nil:cons3_0(+(x, 1)) <=> cons(h, gen_nil:cons3_0(x))
gen_h:s4_0(0) <=> h
gen_h:s4_0(+(x, 1)) <=> s(gen_h:s4_0(x))


The following defined symbols remain to be analysed:
app, sum, a

They will be analysed ascendingly in the following order:
a < sum

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

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

Induction Base:
app(gen_nil:cons3_0(0), gen_nil:cons3_0(b)) ->_R^Omega(1)
gen_nil:cons3_0(b)

Induction Step:
app(gen_nil:cons3_0(+(n6_0, 1)), gen_nil:cons3_0(b)) ->_R^Omega(1)
cons(h, app(gen_nil:cons3_0(n6_0), gen_nil:cons3_0(b))) ->_IH
cons(h, gen_nil:cons3_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)
Complex Obligation (BEST)

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

Innermost TRS:
Rules:
app(nil, k) -> k
app(l, nil) -> l
app(cons(x, l), k) -> cons(x, app(l, k))
sum(cons(x, nil)) -> cons(x, nil)
sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h), l))
a(h, h, x) -> s(x)
a(x, s(y), h) -> a(x, y, s(h))
a(x, s(y), s(z)) -> a(x, y, a(x, s(y), z))
a(s(x), h, z) -> a(x, z, z)

Types:
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: h:s -> nil:cons -> nil:cons
sum :: nil:cons -> nil:cons
a :: h:s -> h:s -> h:s -> h:s
h :: h:s
s :: h:s -> h:s
hole_nil:cons1_0 :: nil:cons
hole_h:s2_0 :: h:s
gen_nil:cons3_0 :: Nat -> nil:cons
gen_h:s4_0 :: Nat -> h:s


Generator Equations:
gen_nil:cons3_0(0) <=> nil
gen_nil:cons3_0(+(x, 1)) <=> cons(h, gen_nil:cons3_0(x))
gen_h:s4_0(0) <=> h
gen_h:s4_0(+(x, 1)) <=> s(gen_h:s4_0(x))


The following defined symbols remain to be analysed:
app, sum, a

They will be analysed ascendingly in the following order:
a < sum

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

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

(12)
Obligation:
Innermost TRS:
Rules:
app(nil, k) -> k
app(l, nil) -> l
app(cons(x, l), k) -> cons(x, app(l, k))
sum(cons(x, nil)) -> cons(x, nil)
sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h), l))
a(h, h, x) -> s(x)
a(x, s(y), h) -> a(x, y, s(h))
a(x, s(y), s(z)) -> a(x, y, a(x, s(y), z))
a(s(x), h, z) -> a(x, z, z)

Types:
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: h:s -> nil:cons -> nil:cons
sum :: nil:cons -> nil:cons
a :: h:s -> h:s -> h:s -> h:s
h :: h:s
s :: h:s -> h:s
hole_nil:cons1_0 :: nil:cons
hole_h:s2_0 :: h:s
gen_nil:cons3_0 :: Nat -> nil:cons
gen_h:s4_0 :: Nat -> h:s


Lemmas:
app(gen_nil:cons3_0(n6_0), gen_nil:cons3_0(b)) -> gen_nil:cons3_0(+(n6_0, b)), rt in Omega(1 + n6_0)


Generator Equations:
gen_nil:cons3_0(0) <=> nil
gen_nil:cons3_0(+(x, 1)) <=> cons(h, gen_nil:cons3_0(x))
gen_h:s4_0(0) <=> h
gen_h:s4_0(+(x, 1)) <=> s(gen_h:s4_0(x))


The following defined symbols remain to be analysed:
a, sum

They will be analysed ascendingly in the following order:
a < sum

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
a(gen_h:s4_0(n694_0), gen_h:s4_0(0), gen_h:s4_0(0)) -> gen_h:s4_0(1), rt in Omega(1 + n694_0)

Induction Base:
a(gen_h:s4_0(0), gen_h:s4_0(0), gen_h:s4_0(0)) ->_R^Omega(1)
s(gen_h:s4_0(0))

Induction Step:
a(gen_h:s4_0(+(n694_0, 1)), gen_h:s4_0(0), gen_h:s4_0(0)) ->_R^Omega(1)
a(gen_h:s4_0(n694_0), gen_h:s4_0(0), gen_h:s4_0(0)) ->_IH
gen_h:s4_0(1)

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

(14)
Obligation:
Innermost TRS:
Rules:
app(nil, k) -> k
app(l, nil) -> l
app(cons(x, l), k) -> cons(x, app(l, k))
sum(cons(x, nil)) -> cons(x, nil)
sum(cons(x, cons(y, l))) -> sum(cons(a(x, y, h), l))
a(h, h, x) -> s(x)
a(x, s(y), h) -> a(x, y, s(h))
a(x, s(y), s(z)) -> a(x, y, a(x, s(y), z))
a(s(x), h, z) -> a(x, z, z)

Types:
app :: nil:cons -> nil:cons -> nil:cons
nil :: nil:cons
cons :: h:s -> nil:cons -> nil:cons
sum :: nil:cons -> nil:cons
a :: h:s -> h:s -> h:s -> h:s
h :: h:s
s :: h:s -> h:s
hole_nil:cons1_0 :: nil:cons
hole_h:s2_0 :: h:s
gen_nil:cons3_0 :: Nat -> nil:cons
gen_h:s4_0 :: Nat -> h:s


Lemmas:
app(gen_nil:cons3_0(n6_0), gen_nil:cons3_0(b)) -> gen_nil:cons3_0(+(n6_0, b)), rt in Omega(1 + n6_0)
a(gen_h:s4_0(n694_0), gen_h:s4_0(0), gen_h:s4_0(0)) -> gen_h:s4_0(1), rt in Omega(1 + n694_0)


Generator Equations:
gen_nil:cons3_0(0) <=> nil
gen_nil:cons3_0(+(x, 1)) <=> cons(h, gen_nil:cons3_0(x))
gen_h:s4_0(0) <=> h
gen_h:s4_0(+(x, 1)) <=> s(gen_h:s4_0(x))


The following defined symbols remain to be analysed:
sum