/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), O(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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, n^1).

(0) CpxTRS
(1) DependencyGraphProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
    (6) BOUNDS(1, n^1)
(7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTRS
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) typed CpxTrs
    (11) OrderProof [LOWER BOUND(ID), 0 ms]
    (12) typed CpxTrs
    (13) RewriteLemmaProof [LOWER BOUND(ID), 408 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) typed CpxTrs


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(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   f(s(0), g(x)) -> f(x, g(x))
   g(s(x)) -> g(x)

S is empty.
Rewrite Strategy: FULL
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(1) DependencyGraphProof (UPPER BOUND(ID))
The following rules are not reachable from basic terms in the dependency graph and can be removed:

f(s(0), g(x)) -> f(x, g(x))

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(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   g(s(x)) -> g(x)

S is empty.
Rewrite Strategy: FULL
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(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(4)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   g(s(x)) -> g(x)

S is empty.
Rewrite Strategy: FULL
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(5) CpxTrsMatchBoundsProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. 
The certificate found is represented by the following graph.

"[1, 2]
{(1,2,[g_1|0, g_1|1]), (2,2,[s_1|0])}"
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(6)
BOUNDS(1, n^1)

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(7) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(8)
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(0'), g(x)) -> f(x, g(x))
   g(s(x)) -> g(x)

S is empty.
Rewrite Strategy: FULL
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(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(10)
Obligation:
TRS:
Rules:
f(s(0'), g(x)) -> f(x, g(x))
g(s(x)) -> g(x)

Types:
f :: 0':s -> g -> f
s :: 0':s -> 0':s
0' :: 0':s
g :: 0':s -> g
hole_f1_0 :: f
hole_0':s2_0 :: 0':s
hole_g3_0 :: g
gen_0':s4_0 :: Nat -> 0':s

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

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

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(12)
Obligation:
TRS:
Rules:
f(s(0'), g(x)) -> f(x, g(x))
g(s(x)) -> g(x)

Types:
f :: 0':s -> g -> f
s :: 0':s -> 0':s
0' :: 0':s
g :: 0':s -> g
hole_f1_0 :: f
hole_0':s2_0 :: 0':s
hole_g3_0 :: g
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


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

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

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(gen_0':s4_0(+(1, n6_0))) -> *5_0, rt in Omega(n6_0)

Induction Base:
g(gen_0':s4_0(+(1, 0)))

Induction Step:
g(gen_0':s4_0(+(1, +(n6_0, 1)))) ->_R^Omega(1)
g(gen_0':s4_0(+(1, n6_0))) ->_IH
*5_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)
Complex Obligation (BEST)

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(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
f(s(0'), g(x)) -> f(x, g(x))
g(s(x)) -> g(x)

Types:
f :: 0':s -> g -> f
s :: 0':s -> 0':s
0' :: 0':s
g :: 0':s -> g
hole_f1_0 :: f
hole_0':s2_0 :: 0':s
hole_g3_0 :: g
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


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

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

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(16) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(17)
BOUNDS(n^1, INF)

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(18)
Obligation:
TRS:
Rules:
f(s(0'), g(x)) -> f(x, g(x))
g(s(x)) -> g(x)

Types:
f :: 0':s -> g -> f
s :: 0':s -> 0':s
0' :: 0':s
g :: 0':s -> g
hole_f1_0 :: f
hole_0':s2_0 :: 0':s
hole_g3_0 :: g
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
g(gen_0':s4_0(+(1, n6_0))) -> *5_0, rt in Omega(n6_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
f