/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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) CpxTrsMatchBoundsProof [FINISHED, 0 ms]
    (4) BOUNDS(1, n^1)
(5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxTRS
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) typed CpxTrs
    (9) OrderProof [LOWER BOUND(ID), 0 ms]
    (10) typed CpxTrs
    (11) RewriteLemmaProof [LOWER BOUND(ID), 436 ms]
    (12) proven lower bound
    (13) LowerBoundPropagationProof [FINISHED, 0 ms]
    (14) BOUNDS(n^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:

   a(b(x)) -> b(a(x))
   a(c(x)) -> x

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:

   a(b(x)) -> b(a(x))
   a(c(x)) -> x

S is empty.
Rewrite Strategy: INNERMOST
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(3) 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, 3]
{(1,2,[a_1|0, b_1|1, c_1|1]), (1,3,[b_1|1]), (2,2,[b_1|0, c_1|0]), (3,2,[a_1|1, b_1|1, c_1|1]), (3,3,[b_1|1])}"
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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:

   a(b(x)) -> b(a(x))
   a(c(x)) -> x

S is empty.
Rewrite Strategy: INNERMOST
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(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(8)
Obligation:
Innermost TRS:
Rules:
a(b(x)) -> b(a(x))
a(c(x)) -> x

Types:
a :: b:c -> b:c
b :: b:c -> b:c
c :: b:c -> b:c
hole_b:c1_0 :: b:c
gen_b:c2_0 :: Nat -> b:c

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(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
a
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(10)
Obligation:
Innermost TRS:
Rules:
a(b(x)) -> b(a(x))
a(c(x)) -> x

Types:
a :: b:c -> b:c
b :: b:c -> b:c
c :: b:c -> b:c
hole_b:c1_0 :: b:c
gen_b:c2_0 :: Nat -> b:c


Generator Equations:
gen_b:c2_0(0) <=> hole_b:c1_0
gen_b:c2_0(+(x, 1)) <=> b(gen_b:c2_0(x))


The following defined symbols remain to be analysed:
a
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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
a(gen_b:c2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
a(gen_b:c2_0(+(1, 0)))

Induction Step:
a(gen_b:c2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
b(a(gen_b:c2_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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(12)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
a(b(x)) -> b(a(x))
a(c(x)) -> x

Types:
a :: b:c -> b:c
b :: b:c -> b:c
c :: b:c -> b:c
hole_b:c1_0 :: b:c
gen_b:c2_0 :: Nat -> b:c


Generator Equations:
gen_b:c2_0(0) <=> hole_b:c1_0
gen_b:c2_0(+(x, 1)) <=> b(gen_b:c2_0(x))


The following defined symbols remain to be analysed:
a
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(13) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(14)
BOUNDS(n^1, INF)