/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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), 1069 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), 45 ms]
        (14) BOUNDS(1, INF)


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(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:

   fib(0) -> 0
   fib(s(0)) -> s(0)
   fib(s(s(0))) -> s(0)
   fib(s(s(x))) -> sp(g(x))
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(x)) -> np(g(x))
   sp(pair(x, y)) -> +(x, y)
   np(pair(x, y)) -> pair(+(x, y), x)
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))

S is empty.
Rewrite Strategy: INNERMOST
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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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   fib(0') -> 0'
   fib(s(0')) -> s(0')
   fib(s(s(0'))) -> s(0')
   fib(s(s(x))) -> sp(g(x))
   g(0') -> pair(s(0'), 0')
   g(s(0')) -> pair(s(0'), s(0'))
   g(s(x)) -> np(g(x))
   sp(pair(x, y)) -> +'(x, y)
   np(pair(x, y)) -> pair(+'(x, y), x)
   +'(x, 0') -> x
   +'(x, s(y)) -> s(+'(x, y))

S is empty.
Rewrite Strategy: INNERMOST
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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(4)
Obligation:
Innermost TRS:
Rules:
fib(0') -> 0'
fib(s(0')) -> s(0')
fib(s(s(0'))) -> s(0')
fib(s(s(x))) -> sp(g(x))
g(0') -> pair(s(0'), 0')
g(s(0')) -> pair(s(0'), s(0'))
g(s(x)) -> np(g(x))
sp(pair(x, y)) -> +'(x, y)
np(pair(x, y)) -> pair(+'(x, y), x)
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))

Types:
fib :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
sp :: pair -> 0':s
g :: 0':s -> pair
pair :: 0':s -> 0':s -> pair
np :: pair -> pair
+' :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_pair2_0 :: pair
gen_0':s3_0 :: Nat -> 0':s

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
g, +'
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(6)
Obligation:
Innermost TRS:
Rules:
fib(0') -> 0'
fib(s(0')) -> s(0')
fib(s(s(0'))) -> s(0')
fib(s(s(x))) -> sp(g(x))
g(0') -> pair(s(0'), 0')
g(s(0')) -> pair(s(0'), s(0'))
g(s(x)) -> np(g(x))
sp(pair(x, y)) -> +'(x, y)
np(pair(x, y)) -> pair(+'(x, y), x)
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))

Types:
fib :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
sp :: pair -> 0':s
g :: 0':s -> pair
pair :: 0':s -> 0':s -> pair
np :: pair -> pair
+' :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_pair2_0 :: pair
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
g, +'
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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(gen_0':s3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)

Induction Base:
g(gen_0':s3_0(+(1, 0)))

Induction Step:
g(gen_0':s3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1)
np(g(gen_0':s3_0(+(1, n5_0)))) ->_IH
np(*4_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:
fib(0') -> 0'
fib(s(0')) -> s(0')
fib(s(s(0'))) -> s(0')
fib(s(s(x))) -> sp(g(x))
g(0') -> pair(s(0'), 0')
g(s(0')) -> pair(s(0'), s(0'))
g(s(x)) -> np(g(x))
sp(pair(x, y)) -> +'(x, y)
np(pair(x, y)) -> pair(+'(x, y), x)
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))

Types:
fib :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
sp :: pair -> 0':s
g :: 0':s -> pair
pair :: 0':s -> 0':s -> pair
np :: pair -> pair
+' :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_pair2_0 :: pair
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
g, +'
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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:
Innermost TRS:
Rules:
fib(0') -> 0'
fib(s(0')) -> s(0')
fib(s(s(0'))) -> s(0')
fib(s(s(x))) -> sp(g(x))
g(0') -> pair(s(0'), 0')
g(s(0')) -> pair(s(0'), s(0'))
g(s(x)) -> np(g(x))
sp(pair(x, y)) -> +'(x, y)
np(pair(x, y)) -> pair(+'(x, y), x)
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))

Types:
fib :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
sp :: pair -> 0':s
g :: 0':s -> pair
pair :: 0':s -> 0':s -> pair
np :: pair -> pair
+' :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_pair2_0 :: pair
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
g(gen_0':s3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
+'
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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':s3_0(a), gen_0':s3_0(n4883_0)) -> gen_0':s3_0(+(n4883_0, a)), rt in Omega(1 + n4883_0)

Induction Base:
+'(gen_0':s3_0(a), gen_0':s3_0(0)) ->_R^Omega(1)
gen_0':s3_0(a)

Induction Step:
+'(gen_0':s3_0(a), gen_0':s3_0(+(n4883_0, 1))) ->_R^Omega(1)
s(+'(gen_0':s3_0(a), gen_0':s3_0(n4883_0))) ->_IH
s(gen_0':s3_0(+(a, c4884_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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(14)
BOUNDS(1, INF)