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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) CpxTrsMatchBoundsTAProof [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), 289 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:

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

   f(x, a) -> x
   f(x, g(y)) -> f(g(x), 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 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1]
transitions: 
a0() -> 0
g0(0) -> 0
f0(0, 0) -> 1
g1(0) -> 2
f1(2, 0) -> 1
g1(2) -> 2
0 -> 1
2 -> 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:

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

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

Types:
f :: a:g -> a:g -> a:g
a :: a:g
g :: a:g -> a:g
hole_a:g1_0 :: a:g
gen_a:g2_0 :: Nat -> a:g

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

Types:
f :: a:g -> a:g -> a:g
a :: a:g
g :: a:g -> a:g
hole_a:g1_0 :: a:g
gen_a:g2_0 :: Nat -> a:g


Generator Equations:
gen_a:g2_0(0) <=> a
gen_a:g2_0(+(x, 1)) <=> g(gen_a:g2_0(x))


The following defined symbols remain to be analysed:
f
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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_a:g2_0(a), gen_a:g2_0(n4_0)) -> gen_a:g2_0(+(n4_0, a)), rt in Omega(1 + n4_0)

Induction Base:
f(gen_a:g2_0(a), gen_a:g2_0(0)) ->_R^Omega(1)
gen_a:g2_0(a)

Induction Step:
f(gen_a:g2_0(a), gen_a:g2_0(+(n4_0, 1))) ->_R^Omega(1)
f(g(gen_a:g2_0(a)), gen_a:g2_0(n4_0)) ->_IH
gen_a:g2_0(+(+(a, 1), c5_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:
f(x, a) -> x
f(x, g(y)) -> f(g(x), y)

Types:
f :: a:g -> a:g -> a:g
a :: a:g
g :: a:g -> a:g
hole_a:g1_0 :: a:g
gen_a:g2_0 :: Nat -> a:g


Generator Equations:
gen_a:g2_0(0) <=> a
gen_a:g2_0(+(x, 1)) <=> g(gen_a:g2_0(x))


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