/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 281 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs


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

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.
----------------------------------------

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

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

Types:
fib :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
sel :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
fib1 :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
s :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
0' :: 0':s:n__add:n__fib1:cons
cons :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
n__fib1 :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
n__add :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
add :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
activate :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
hole_0':s:n__add:n__fib1:cons1_0 :: 0':s:n__add:n__fib1:cons
gen_0':s:n__add:n__fib1:cons2_0 :: Nat -> 0':s:n__add:n__fib1:cons

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
sel, add, activate

They will be analysed ascendingly in the following order:
activate < sel
add < activate

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(6)
Obligation:
Innermost TRS:
Rules:
fib(N) -> sel(N, fib1(s(0'), s(0')))
fib1(X, Y) -> cons(X, n__fib1(Y, n__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)
add(X1, X2) -> n__add(X1, X2)
activate(n__fib1(X1, X2)) -> fib1(activate(X1), activate(X2))
activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
activate(X) -> X

Types:
fib :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
sel :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
fib1 :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
s :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
0' :: 0':s:n__add:n__fib1:cons
cons :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
n__fib1 :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
n__add :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
add :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
activate :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
hole_0':s:n__add:n__fib1:cons1_0 :: 0':s:n__add:n__fib1:cons
gen_0':s:n__add:n__fib1:cons2_0 :: Nat -> 0':s:n__add:n__fib1:cons


Generator Equations:
gen_0':s:n__add:n__fib1:cons2_0(0) <=> 0'
gen_0':s:n__add:n__fib1:cons2_0(+(x, 1)) <=> s(gen_0':s:n__add:n__fib1:cons2_0(x))


The following defined symbols remain to be analysed:
add, sel, activate

They will be analysed ascendingly in the following order:
activate < sel
add < activate

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
add(gen_0':s:n__add:n__fib1:cons2_0(n4_0), gen_0':s:n__add:n__fib1:cons2_0(b)) -> gen_0':s:n__add:n__fib1:cons2_0(+(n4_0, b)), rt in Omega(1 + n4_0)

Induction Base:
add(gen_0':s:n__add:n__fib1:cons2_0(0), gen_0':s:n__add:n__fib1:cons2_0(b)) ->_R^Omega(1)
gen_0':s:n__add:n__fib1:cons2_0(b)

Induction Step:
add(gen_0':s:n__add:n__fib1:cons2_0(+(n4_0, 1)), gen_0':s:n__add:n__fib1:cons2_0(b)) ->_R^Omega(1)
s(add(gen_0':s:n__add:n__fib1:cons2_0(n4_0), gen_0':s:n__add:n__fib1:cons2_0(b))) ->_IH
s(gen_0':s:n__add:n__fib1:cons2_0(+(b, c5_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(N) -> sel(N, fib1(s(0'), s(0')))
fib1(X, Y) -> cons(X, n__fib1(Y, n__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)
add(X1, X2) -> n__add(X1, X2)
activate(n__fib1(X1, X2)) -> fib1(activate(X1), activate(X2))
activate(n__add(X1, X2)) -> add(activate(X1), activate(X2))
activate(X) -> X

Types:
fib :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
sel :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
fib1 :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
s :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
0' :: 0':s:n__add:n__fib1:cons
cons :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
n__fib1 :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
n__add :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
add :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
activate :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
hole_0':s:n__add:n__fib1:cons1_0 :: 0':s:n__add:n__fib1:cons
gen_0':s:n__add:n__fib1:cons2_0 :: Nat -> 0':s:n__add:n__fib1:cons


Generator Equations:
gen_0':s:n__add:n__fib1:cons2_0(0) <=> 0'
gen_0':s:n__add:n__fib1:cons2_0(+(x, 1)) <=> s(gen_0':s:n__add:n__fib1:cons2_0(x))


The following defined symbols remain to be analysed:
add, sel, activate

They will be analysed ascendingly in the following order:
activate < sel
add < activate

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

Types:
fib :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
sel :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
fib1 :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
s :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
0' :: 0':s:n__add:n__fib1:cons
cons :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
n__fib1 :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
n__add :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
add :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
activate :: 0':s:n__add:n__fib1:cons -> 0':s:n__add:n__fib1:cons
hole_0':s:n__add:n__fib1:cons1_0 :: 0':s:n__add:n__fib1:cons
gen_0':s:n__add:n__fib1:cons2_0 :: Nat -> 0':s:n__add:n__fib1:cons


Lemmas:
add(gen_0':s:n__add:n__fib1:cons2_0(n4_0), gen_0':s:n__add:n__fib1:cons2_0(b)) -> gen_0':s:n__add:n__fib1:cons2_0(+(n4_0, b)), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s:n__add:n__fib1:cons2_0(0) <=> 0'
gen_0':s:n__add:n__fib1:cons2_0(+(x, 1)) <=> s(gen_0':s:n__add:n__fib1:cons2_0(x))


The following defined symbols remain to be analysed:
activate, sel

They will be analysed ascendingly in the following order:
activate < sel