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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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

(0) CpxTRS
(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsMatchBoundsTAProof [FINISHED, 0 ms]
    (4) BOUNDS(1, n^1)
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxTRS
    (7) SlicingProof [LOWER BOUND(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), 432 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) typed CpxTrs
            (19) RewriteLemmaProof [LOWER BOUND(ID), 100 ms]
            (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 90 ms]
            (22) BOUNDS(1, INF)


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


The TRS R consists of the following rules:

   g(f(x), y) -> f(h(x, y))
   h(x, y) -> g(x, f(y))

S is empty.
Rewrite Strategy: INNERMOST
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(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   g(f(x), y) -> f(h(x, y))
   h(x, y) -> g(x, f(y))

S is empty.
Rewrite Strategy: INNERMOST
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(3) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2]
transitions: 
f0(0) -> 0
g0(0, 0) -> 1
h0(0, 0) -> 2
h1(0, 0) -> 3
f1(3) -> 1
f1(0) -> 4
g1(0, 4) -> 2
h1(0, 4) -> 3
f1(3) -> 2
f2(0) -> 5
g2(0, 5) -> 3
f2(4) -> 5
h1(0, 5) -> 3
f1(3) -> 3
f2(5) -> 5

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(4)
BOUNDS(1, n^1)

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(5) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(6)
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:

   g(f(x), y) -> f(h(x, y))
   h(x, y) -> g(x, f(y))

S is empty.
Rewrite Strategy: INNERMOST
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(7) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
g/1
h/1

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(8)
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:

   g(f(x)) -> f(h(x))
   h(x) -> g(x)

S is empty.
Rewrite Strategy: INNERMOST
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(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(10)
Obligation:
Innermost TRS:
Rules:
g(f(x)) -> f(h(x))
h(x) -> g(x)

Types:
g :: f -> f
f :: f -> f
h :: f -> f
hole_f1_0 :: f
gen_f2_0 :: Nat -> f

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

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

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(12)
Obligation:
Innermost TRS:
Rules:
g(f(x)) -> f(h(x))
h(x) -> g(x)

Types:
g :: f -> f
f :: f -> f
h :: f -> f
hole_f1_0 :: f
gen_f2_0 :: Nat -> f


Generator Equations:
gen_f2_0(0) <=> hole_f1_0
gen_f2_0(+(x, 1)) <=> f(gen_f2_0(x))


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

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

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
h(gen_f2_0(n4_0)) -> *3_0, rt in Omega(n4_0)

Induction Base:
h(gen_f2_0(0))

Induction Step:
h(gen_f2_0(+(n4_0, 1))) ->_R^Omega(1)
g(gen_f2_0(+(n4_0, 1))) ->_R^Omega(1)
f(h(gen_f2_0(n4_0))) ->_IH
f(*3_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:

Innermost TRS:
Rules:
g(f(x)) -> f(h(x))
h(x) -> g(x)

Types:
g :: f -> f
f :: f -> f
h :: f -> f
hole_f1_0 :: f
gen_f2_0 :: Nat -> f


Generator Equations:
gen_f2_0(0) <=> hole_f1_0
gen_f2_0(+(x, 1)) <=> f(gen_f2_0(x))


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

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

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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:
Innermost TRS:
Rules:
g(f(x)) -> f(h(x))
h(x) -> g(x)

Types:
g :: f -> f
f :: f -> f
h :: f -> f
hole_f1_0 :: f
gen_f2_0 :: Nat -> f


Lemmas:
h(gen_f2_0(n4_0)) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_f2_0(0) <=> hole_f1_0
gen_f2_0(+(x, 1)) <=> f(gen_f2_0(x))


The following defined symbols remain to be analysed:
g

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

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(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(gen_f2_0(+(1, n121_0))) -> *3_0, rt in Omega(n121_0)

Induction Base:
g(gen_f2_0(+(1, 0)))

Induction Step:
g(gen_f2_0(+(1, +(n121_0, 1)))) ->_R^Omega(1)
f(h(gen_f2_0(+(1, n121_0)))) ->_R^Omega(1)
f(g(gen_f2_0(+(1, n121_0)))) ->_IH
f(*3_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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(20)
Obligation:
Innermost TRS:
Rules:
g(f(x)) -> f(h(x))
h(x) -> g(x)

Types:
g :: f -> f
f :: f -> f
h :: f -> f
hole_f1_0 :: f
gen_f2_0 :: Nat -> f


Lemmas:
h(gen_f2_0(n4_0)) -> *3_0, rt in Omega(n4_0)
g(gen_f2_0(+(1, n121_0))) -> *3_0, rt in Omega(n121_0)


Generator Equations:
gen_f2_0(0) <=> hole_f1_0
gen_f2_0(+(x, 1)) <=> f(gen_f2_0(x))


The following defined symbols remain to be analysed:
h

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

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(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
h(gen_f2_0(n355_0)) -> *3_0, rt in Omega(n355_0)

Induction Base:
h(gen_f2_0(0))

Induction Step:
h(gen_f2_0(+(n355_0, 1))) ->_R^Omega(1)
g(gen_f2_0(+(n355_0, 1))) ->_R^Omega(1)
f(h(gen_f2_0(n355_0))) ->_IH
f(*3_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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(22)
BOUNDS(1, INF)