/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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 (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), 210 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), 52 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:

   fib(N) -> sel(N, fib1(s(0), s(0)))
   fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
   fib1(X1, X2) -> n__fib1(X1, X2)
   activate(n__fib1(X1, X2)) -> fib1(X1, X2)
   activate(X) -> X

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:

   fib(N) -> sel(N, fib1(s(0'), s(0')))
   fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y)))
   add(0', X) -> X
   add(s(X), Y) -> s(add(X, Y))
   sel(0', cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
   fib1(X1, X2) -> n__fib1(X1, X2)
   activate(n__fib1(X1, X2)) -> fib1(X1, X2)
   activate(X) -> X

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:
fib(N) -> sel(N, fib1(s(0'), s(0')))
fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y)))
add(0', X) -> X
add(s(X), Y) -> s(add(X, Y))
sel(0', cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
fib1(X1, X2) -> n__fib1(X1, X2)
activate(n__fib1(X1, X2)) -> fib1(X1, X2)
activate(X) -> X

Types:
fib :: 0':s -> 0':s
sel :: 0':s -> n__fib1:cons -> 0':s
fib1 :: 0':s -> 0':s -> n__fib1:cons
s :: 0':s -> 0':s
0' :: 0':s
cons :: 0':s -> n__fib1:cons -> n__fib1:cons
n__fib1 :: 0':s -> 0':s -> n__fib1:cons
add :: 0':s -> 0':s -> 0':s
activate :: n__fib1:cons -> n__fib1:cons
hole_0':s1_0 :: 0':s
hole_n__fib1:cons2_0 :: n__fib1:cons
gen_0':s3_0 :: Nat -> 0':s
gen_n__fib1:cons4_0 :: Nat -> n__fib1:cons

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
sel, add
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(6)
Obligation:
TRS:
Rules:
fib(N) -> sel(N, fib1(s(0'), s(0')))
fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y)))
add(0', X) -> X
add(s(X), Y) -> s(add(X, Y))
sel(0', cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
fib1(X1, X2) -> n__fib1(X1, X2)
activate(n__fib1(X1, X2)) -> fib1(X1, X2)
activate(X) -> X

Types:
fib :: 0':s -> 0':s
sel :: 0':s -> n__fib1:cons -> 0':s
fib1 :: 0':s -> 0':s -> n__fib1:cons
s :: 0':s -> 0':s
0' :: 0':s
cons :: 0':s -> n__fib1:cons -> n__fib1:cons
n__fib1 :: 0':s -> 0':s -> n__fib1:cons
add :: 0':s -> 0':s -> 0':s
activate :: n__fib1:cons -> n__fib1:cons
hole_0':s1_0 :: 0':s
hole_n__fib1:cons2_0 :: n__fib1:cons
gen_0':s3_0 :: Nat -> 0':s
gen_n__fib1:cons4_0 :: Nat -> n__fib1:cons


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
gen_n__fib1:cons4_0(0) <=> n__fib1(0', 0')
gen_n__fib1:cons4_0(+(x, 1)) <=> cons(0', gen_n__fib1:cons4_0(x))


The following defined symbols remain to be analysed:
sel, add
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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sel(gen_0':s3_0(n6_0), gen_n__fib1:cons4_0(1)) -> gen_0':s3_0(0), rt in Omega(1 + n6_0)

Induction Base:
sel(gen_0':s3_0(0), gen_n__fib1:cons4_0(1)) ->_R^Omega(1)
0'

Induction Step:
sel(gen_0':s3_0(+(n6_0, 1)), gen_n__fib1:cons4_0(1)) ->_R^Omega(1)
sel(gen_0':s3_0(n6_0), activate(gen_n__fib1:cons4_0(0))) ->_R^Omega(1)
sel(gen_0':s3_0(n6_0), fib1(0', 0')) ->_R^Omega(1)
sel(gen_0':s3_0(n6_0), cons(0', n__fib1(0', add(0', 0')))) ->_R^Omega(1)
sel(gen_0':s3_0(n6_0), cons(0', n__fib1(0', 0'))) ->_IH
gen_0':s3_0(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:
fib(N) -> sel(N, fib1(s(0'), s(0')))
fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y)))
add(0', X) -> X
add(s(X), Y) -> s(add(X, Y))
sel(0', cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
fib1(X1, X2) -> n__fib1(X1, X2)
activate(n__fib1(X1, X2)) -> fib1(X1, X2)
activate(X) -> X

Types:
fib :: 0':s -> 0':s
sel :: 0':s -> n__fib1:cons -> 0':s
fib1 :: 0':s -> 0':s -> n__fib1:cons
s :: 0':s -> 0':s
0' :: 0':s
cons :: 0':s -> n__fib1:cons -> n__fib1:cons
n__fib1 :: 0':s -> 0':s -> n__fib1:cons
add :: 0':s -> 0':s -> 0':s
activate :: n__fib1:cons -> n__fib1:cons
hole_0':s1_0 :: 0':s
hole_n__fib1:cons2_0 :: n__fib1:cons
gen_0':s3_0 :: Nat -> 0':s
gen_n__fib1:cons4_0 :: Nat -> n__fib1:cons


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
gen_n__fib1:cons4_0(0) <=> n__fib1(0', 0')
gen_n__fib1:cons4_0(+(x, 1)) <=> cons(0', gen_n__fib1:cons4_0(x))


The following defined symbols remain to be analysed:
sel, add
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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:
fib(N) -> sel(N, fib1(s(0'), s(0')))
fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y)))
add(0', X) -> X
add(s(X), Y) -> s(add(X, Y))
sel(0', cons(X, XS)) -> X
sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
fib1(X1, X2) -> n__fib1(X1, X2)
activate(n__fib1(X1, X2)) -> fib1(X1, X2)
activate(X) -> X

Types:
fib :: 0':s -> 0':s
sel :: 0':s -> n__fib1:cons -> 0':s
fib1 :: 0':s -> 0':s -> n__fib1:cons
s :: 0':s -> 0':s
0' :: 0':s
cons :: 0':s -> n__fib1:cons -> n__fib1:cons
n__fib1 :: 0':s -> 0':s -> n__fib1:cons
add :: 0':s -> 0':s -> 0':s
activate :: n__fib1:cons -> n__fib1:cons
hole_0':s1_0 :: 0':s
hole_n__fib1:cons2_0 :: n__fib1:cons
gen_0':s3_0 :: Nat -> 0':s
gen_n__fib1:cons4_0 :: Nat -> n__fib1:cons


Lemmas:
sel(gen_0':s3_0(n6_0), gen_n__fib1:cons4_0(1)) -> gen_0':s3_0(0), rt in Omega(1 + n6_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
gen_n__fib1:cons4_0(0) <=> n__fib1(0', 0')
gen_n__fib1:cons4_0(+(x, 1)) <=> cons(0', gen_n__fib1:cons4_0(x))


The following defined symbols remain to be analysed:
add
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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
add(gen_0':s3_0(n315_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n315_0, b)), rt in Omega(1 + n315_0)

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

Induction Step:
add(gen_0':s3_0(+(n315_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
s(add(gen_0':s3_0(n315_0), gen_0':s3_0(b))) ->_IH
s(gen_0':s3_0(+(b, c316_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)