/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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) CpxTrsMatchBoundsTAProof [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), 529 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:

   a(a(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
   f(a(x), a(y)) -> a(f(x, y))
   f(b(x), b(y)) -> b(f(x, y))

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:
a(a(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
f(a(x), a(y)) -> a(f(x, y))

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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:

   f(b(x), b(y)) -> b(f(x, y))

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:

   f(b(x), b(y)) -> b(f(x, y))

S is empty.
Rewrite Strategy: FULL
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(5) 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 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1]
transitions: 
b0(0) -> 0
f0(0, 0) -> 1
f1(0, 0) -> 2
b1(2) -> 1
b1(2) -> 2

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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:

   a(a(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
   f(a(x), a(y)) -> a(f(x, y))
   f(b(x), b(y)) -> b(f(x, y))

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:
a(a(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
f(a(x), a(y)) -> a(f(x, y))
f(b(x), b(y)) -> b(f(x, y))

Types:
a :: b -> b
f :: b -> b -> b
b :: b -> b
hole_b1_0 :: b
gen_b2_0 :: Nat -> b

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

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

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(12)
Obligation:
TRS:
Rules:
a(a(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
f(a(x), a(y)) -> a(f(x, y))
f(b(x), b(y)) -> b(f(x, y))

Types:
a :: b -> b
f :: b -> b -> b
b :: b -> b
hole_b1_0 :: b
gen_b2_0 :: Nat -> b


Generator Equations:
gen_b2_0(0) <=> hole_b1_0
gen_b2_0(+(x, 1)) <=> b(gen_b2_0(x))


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

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

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_b2_0(+(1, n4_0)), gen_b2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
f(gen_b2_0(+(1, 0)), gen_b2_0(+(1, 0)))

Induction Step:
f(gen_b2_0(+(1, +(n4_0, 1))), gen_b2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
b(f(gen_b2_0(+(1, n4_0)), gen_b2_0(+(1, n4_0)))) ->_IH
b(*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:
a(a(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
f(a(x), a(y)) -> a(f(x, y))
f(b(x), b(y)) -> b(f(x, y))

Types:
a :: b -> b
f :: b -> b -> b
b :: b -> b
hole_b1_0 :: b
gen_b2_0 :: Nat -> b


Generator Equations:
gen_b2_0(0) <=> hole_b1_0
gen_b2_0(+(x, 1)) <=> b(gen_b2_0(x))


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

They will be analysed ascendingly in the following order:
a = 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:
a(a(f(x, y))) -> f(a(b(a(b(a(x))))), a(b(a(b(a(y))))))
f(a(x), a(y)) -> a(f(x, y))
f(b(x), b(y)) -> b(f(x, y))

Types:
a :: b -> b
f :: b -> b -> b
b :: b -> b
hole_b1_0 :: b
gen_b2_0 :: Nat -> b


Lemmas:
f(gen_b2_0(+(1, n4_0)), gen_b2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_b2_0(0) <=> hole_b1_0
gen_b2_0(+(x, 1)) <=> b(gen_b2_0(x))


The following defined symbols remain to be analysed:
a

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