/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) NestedDefinedSymbolProof [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), 438 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), 119 ms]
            (20) BOUNDS(1, INF)


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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:

   w(r(x)) -> r(w(x))
   b(r(x)) -> r(b(x))
   b(w(x)) -> w(b(x))

S is empty.
Rewrite Strategy: FULL
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(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
b(w(x)) -> w(b(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:

   w(r(x)) -> r(w(x))
   b(r(x)) -> r(b(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:

   w(r(x)) -> r(w(x))
   b(r(x)) -> r(b(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.

"[9, 10, 11, 12]
{(9,10,[w_1|0, b_1|0]), (9,11,[r_1|1]), (9,12,[r_1|1]), (10,10,[r_1|0]), (11,10,[w_1|1]), (11,11,[r_1|1]), (12,10,[b_1|1]), (12,12,[r_1|1])}"
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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:

   w(r(x)) -> r(w(x))
   b(r(x)) -> r(b(x))
   b(w(x)) -> w(b(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:
w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))

Types:
w :: r -> r
r :: r -> r
b :: r -> r
hole_r1_0 :: r
gen_r2_0 :: Nat -> r

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

They will be analysed ascendingly in the following order:
w < b

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(12)
Obligation:
TRS:
Rules:
w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))

Types:
w :: r -> r
r :: r -> r
b :: r -> r
hole_r1_0 :: r
gen_r2_0 :: Nat -> r


Generator Equations:
gen_r2_0(0) <=> hole_r1_0
gen_r2_0(+(x, 1)) <=> r(gen_r2_0(x))


The following defined symbols remain to be analysed:
w, b

They will be analysed ascendingly in the following order:
w < b

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
w(gen_r2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
w(gen_r2_0(+(1, 0)))

Induction Step:
w(gen_r2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
r(w(gen_r2_0(+(1, n4_0)))) ->_IH
r(*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:

TRS:
Rules:
w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))

Types:
w :: r -> r
r :: r -> r
b :: r -> r
hole_r1_0 :: r
gen_r2_0 :: Nat -> r


Generator Equations:
gen_r2_0(0) <=> hole_r1_0
gen_r2_0(+(x, 1)) <=> r(gen_r2_0(x))


The following defined symbols remain to be analysed:
w, b

They will be analysed ascendingly in the following order:
w < b

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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:
w(r(x)) -> r(w(x))
b(r(x)) -> r(b(x))
b(w(x)) -> w(b(x))

Types:
w :: r -> r
r :: r -> r
b :: r -> r
hole_r1_0 :: r
gen_r2_0 :: Nat -> r


Lemmas:
w(gen_r2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_r2_0(0) <=> hole_r1_0
gen_r2_0(+(x, 1)) <=> r(gen_r2_0(x))


The following defined symbols remain to be analysed:
b
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(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
b(gen_r2_0(+(1, n130_0))) -> *3_0, rt in Omega(n130_0)

Induction Base:
b(gen_r2_0(+(1, 0)))

Induction Step:
b(gen_r2_0(+(1, +(n130_0, 1)))) ->_R^Omega(1)
r(b(gen_r2_0(+(1, n130_0)))) ->_IH
r(*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)
BOUNDS(1, INF)