/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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), 279 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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   f(s(x)) -> f(id_inc(c(x, x)))
   f(c(s(x), y)) -> g(c(x, y))
   g(c(s(x), y)) -> g(c(y, x))
   g(c(x, s(y))) -> g(c(y, x))
   g(c(x, x)) -> f(x)
   id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
   id_inc(s(x)) -> s(id_inc(x))
   id_inc(0) -> 0
   id_inc(0) -> s(0)

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:

   f(s(x)) -> f(id_inc(c(x, x)))
   f(c(s(x), y)) -> g(c(x, y))
   g(c(s(x), y)) -> g(c(y, x))
   g(c(x, s(y))) -> g(c(y, x))
   g(c(x, x)) -> f(x)
   id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
   id_inc(s(x)) -> s(id_inc(x))
   id_inc(0') -> 0'
   id_inc(0') -> s(0')

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:
f(s(x)) -> f(id_inc(c(x, x)))
f(c(s(x), y)) -> g(c(x, y))
g(c(s(x), y)) -> g(c(y, x))
g(c(x, s(y))) -> g(c(y, x))
g(c(x, x)) -> f(x)
id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
id_inc(s(x)) -> s(id_inc(x))
id_inc(0') -> 0'
id_inc(0') -> s(0')

Types:
f :: s:c:0' -> f:g
s :: s:c:0' -> s:c:0'
id_inc :: s:c:0' -> s:c:0'
c :: s:c:0' -> s:c:0' -> s:c:0'
g :: s:c:0' -> f:g
0' :: s:c:0'
hole_f:g1_0 :: f:g
hole_s:c:0'2_0 :: s:c:0'
gen_s:c:0'3_0 :: Nat -> s:c:0'

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
f, id_inc, g

They will be analysed ascendingly in the following order:
id_inc < f
f = g

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(6)
Obligation:
TRS:
Rules:
f(s(x)) -> f(id_inc(c(x, x)))
f(c(s(x), y)) -> g(c(x, y))
g(c(s(x), y)) -> g(c(y, x))
g(c(x, s(y))) -> g(c(y, x))
g(c(x, x)) -> f(x)
id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
id_inc(s(x)) -> s(id_inc(x))
id_inc(0') -> 0'
id_inc(0') -> s(0')

Types:
f :: s:c:0' -> f:g
s :: s:c:0' -> s:c:0'
id_inc :: s:c:0' -> s:c:0'
c :: s:c:0' -> s:c:0' -> s:c:0'
g :: s:c:0' -> f:g
0' :: s:c:0'
hole_f:g1_0 :: f:g
hole_s:c:0'2_0 :: s:c:0'
gen_s:c:0'3_0 :: Nat -> s:c:0'


Generator Equations:
gen_s:c:0'3_0(0) <=> 0'
gen_s:c:0'3_0(+(x, 1)) <=> s(gen_s:c:0'3_0(x))


The following defined symbols remain to be analysed:
id_inc, f, g

They will be analysed ascendingly in the following order:
id_inc < f
f = g

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
id_inc(gen_s:c:0'3_0(n5_0)) -> gen_s:c:0'3_0(n5_0), rt in Omega(1 + n5_0)

Induction Base:
id_inc(gen_s:c:0'3_0(0)) ->_R^Omega(1)
0'

Induction Step:
id_inc(gen_s:c:0'3_0(+(n5_0, 1))) ->_R^Omega(1)
s(id_inc(gen_s:c:0'3_0(n5_0))) ->_IH
s(gen_s:c:0'3_0(c6_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:
f(s(x)) -> f(id_inc(c(x, x)))
f(c(s(x), y)) -> g(c(x, y))
g(c(s(x), y)) -> g(c(y, x))
g(c(x, s(y))) -> g(c(y, x))
g(c(x, x)) -> f(x)
id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
id_inc(s(x)) -> s(id_inc(x))
id_inc(0') -> 0'
id_inc(0') -> s(0')

Types:
f :: s:c:0' -> f:g
s :: s:c:0' -> s:c:0'
id_inc :: s:c:0' -> s:c:0'
c :: s:c:0' -> s:c:0' -> s:c:0'
g :: s:c:0' -> f:g
0' :: s:c:0'
hole_f:g1_0 :: f:g
hole_s:c:0'2_0 :: s:c:0'
gen_s:c:0'3_0 :: Nat -> s:c:0'


Generator Equations:
gen_s:c:0'3_0(0) <=> 0'
gen_s:c:0'3_0(+(x, 1)) <=> s(gen_s:c:0'3_0(x))


The following defined symbols remain to be analysed:
id_inc, f, g

They will be analysed ascendingly in the following order:
id_inc < f
f = 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:
TRS:
Rules:
f(s(x)) -> f(id_inc(c(x, x)))
f(c(s(x), y)) -> g(c(x, y))
g(c(s(x), y)) -> g(c(y, x))
g(c(x, s(y))) -> g(c(y, x))
g(c(x, x)) -> f(x)
id_inc(c(x, y)) -> c(id_inc(x), id_inc(y))
id_inc(s(x)) -> s(id_inc(x))
id_inc(0') -> 0'
id_inc(0') -> s(0')

Types:
f :: s:c:0' -> f:g
s :: s:c:0' -> s:c:0'
id_inc :: s:c:0' -> s:c:0'
c :: s:c:0' -> s:c:0' -> s:c:0'
g :: s:c:0' -> f:g
0' :: s:c:0'
hole_f:g1_0 :: f:g
hole_s:c:0'2_0 :: s:c:0'
gen_s:c:0'3_0 :: Nat -> s:c:0'


Lemmas:
id_inc(gen_s:c:0'3_0(n5_0)) -> gen_s:c:0'3_0(n5_0), rt in Omega(1 + n5_0)


Generator Equations:
gen_s:c:0'3_0(0) <=> 0'
gen_s:c:0'3_0(+(x, 1)) <=> s(gen_s:c:0'3_0(x))


The following defined symbols remain to be analysed:
g, f

They will be analysed ascendingly in the following order:
f = g