/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 299 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:

   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   +(x, +(y, z)) -> +(+(x, y), z)
   f(g(f(x))) -> f(h(s(0), x))
   f(g(h(x, y))) -> f(h(s(x), y))
   f(h(x, h(y, z))) -> f(h(+(x, y), z))

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:

   +'(x, 0') -> x
   +'(x, s(y)) -> s(+'(x, y))
   +'(0', y) -> y
   +'(s(x), y) -> s(+'(x, y))
   +'(x, +'(y, z)) -> +'(+'(x, y), z)
   f(g(f(x))) -> f(h(s(0'), x))
   f(g(h(x, y))) -> f(h(s(x), y))
   f(h(x, h(y, z))) -> f(h(+'(x, y), z))

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:
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
+'(x, +'(y, z)) -> +'(+'(x, y), z)
f(g(f(x))) -> f(h(s(0'), x))
f(g(h(x, y))) -> f(h(s(x), y))
f(h(x, h(y, z))) -> f(h(+'(x, y), z))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
f :: g:h -> g:h
g :: g:h -> g:h
h :: 0':s -> g:h -> g:h
hole_0':s1_0 :: 0':s
hole_g:h2_0 :: g:h
gen_0':s3_0 :: Nat -> 0':s
gen_g:h4_0 :: Nat -> g:h

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

They will be analysed ascendingly in the following order:
+' < f

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(6)
Obligation:
TRS:
Rules:
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
+'(x, +'(y, z)) -> +'(+'(x, y), z)
f(g(f(x))) -> f(h(s(0'), x))
f(g(h(x, y))) -> f(h(s(x), y))
f(h(x, h(y, z))) -> f(h(+'(x, y), z))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
f :: g:h -> g:h
g :: g:h -> g:h
h :: 0':s -> g:h -> g:h
hole_0':s1_0 :: 0':s
hole_g:h2_0 :: g:h
gen_0':s3_0 :: Nat -> 0':s
gen_g:h4_0 :: Nat -> g:h


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
gen_g:h4_0(0) <=> hole_g:h2_0
gen_g:h4_0(+(x, 1)) <=> g(gen_g:h4_0(x))


The following defined symbols remain to be analysed:
+', f

They will be analysed ascendingly in the following order:
+' < f

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':s3_0(a), gen_0':s3_0(n6_0)) -> gen_0':s3_0(+(n6_0, a)), rt in Omega(1 + n6_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(+(n6_0, 1))) ->_R^Omega(1)
s(+'(gen_0':s3_0(a), gen_0':s3_0(n6_0))) ->_IH
s(gen_0':s3_0(+(a, 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:

TRS:
Rules:
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
+'(x, +'(y, z)) -> +'(+'(x, y), z)
f(g(f(x))) -> f(h(s(0'), x))
f(g(h(x, y))) -> f(h(s(x), y))
f(h(x, h(y, z))) -> f(h(+'(x, y), z))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
f :: g:h -> g:h
g :: g:h -> g:h
h :: 0':s -> g:h -> g:h
hole_0':s1_0 :: 0':s
hole_g:h2_0 :: g:h
gen_0':s3_0 :: Nat -> 0':s
gen_g:h4_0 :: Nat -> g:h


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))
gen_g:h4_0(0) <=> hole_g:h2_0
gen_g:h4_0(+(x, 1)) <=> g(gen_g:h4_0(x))


The following defined symbols remain to be analysed:
+', f

They will be analysed ascendingly in the following order:
+' < f

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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:
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
+'(x, +'(y, z)) -> +'(+'(x, y), z)
f(g(f(x))) -> f(h(s(0'), x))
f(g(h(x, y))) -> f(h(s(x), y))
f(h(x, h(y, z))) -> f(h(+'(x, y), z))

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
f :: g:h -> g:h
g :: g:h -> g:h
h :: 0':s -> g:h -> g:h
hole_0':s1_0 :: 0':s
hole_g:h2_0 :: g:h
gen_0':s3_0 :: Nat -> 0':s
gen_g:h4_0 :: Nat -> g:h


Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n6_0)) -> gen_0':s3_0(+(n6_0, a)), 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_g:h4_0(0) <=> hole_g:h2_0
gen_g:h4_0(+(x, 1)) <=> g(gen_g:h4_0(x))


The following defined symbols remain to be analysed:
f