/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) 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) 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), 192 ms]
    (14) proven lower bound
    (15) LowerBoundPropagationProof [FINISHED, 0 ms]
    (16) BOUNDS(n^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:

   a__f(X) -> g(h(f(X)))
   mark(f(X)) -> a__f(mark(X))
   mark(g(X)) -> g(X)
   mark(h(X)) -> h(mark(X))
   a__f(X) -> f(X)

S is empty.
Rewrite Strategy: FULL
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(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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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:

   a__f(X) -> g(h(f(X)))
   mark(f(X)) -> a__f(mark(X))
   mark(g(X)) -> g(X)
   mark(h(X)) -> h(mark(X))
   a__f(X) -> f(X)

S is empty.
Rewrite Strategy: FULL
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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 2. 
The certificate found is represented by the following graph.

"[1, 2, 3, 4, 5, 6, 7, 8]
{(1,2,[a__f_1|0, mark_1|0, f_1|1, g_1|1]), (1,3,[g_1|1]), (1,5,[a__f_1|1, f_1|2]), (1,6,[h_1|1]), (1,7,[g_1|2]), (2,2,[g_1|0, h_1|0, f_1|0]), (3,4,[h_1|1]), (4,2,[f_1|1]), (5,2,[mark_1|1, g_1|1]), (5,5,[a__f_1|1, f_1|2]), (5,6,[h_1|1]), (5,7,[g_1|2]), (6,2,[mark_1|1, g_1|1]), (6,5,[a__f_1|1, f_1|2]), (6,6,[h_1|1]), (6,7,[g_1|2]), (7,8,[h_1|2]), (8,5,[f_1|2])}"
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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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a__f(X) -> g(h(f(X)))
   mark(f(X)) -> a__f(mark(X))
   mark(g(X)) -> g(X)
   mark(h(X)) -> h(mark(X))
   a__f(X) -> f(X)

S is empty.
Rewrite Strategy: FULL
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(7) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
g/0

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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__f(X) -> g
   mark(f(X)) -> a__f(mark(X))
   mark(g) -> g
   mark(h(X)) -> h(mark(X))
   a__f(X) -> f(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:
a__f(X) -> g
mark(f(X)) -> a__f(mark(X))
mark(g) -> g
mark(h(X)) -> h(mark(X))
a__f(X) -> f(X)

Types:
a__f :: g:f:h -> g:f:h
g :: g:f:h
mark :: g:f:h -> g:f:h
f :: g:f:h -> g:f:h
h :: g:f:h -> g:f:h
hole_g:f:h1_0 :: g:f:h
gen_g:f:h2_0 :: Nat -> g:f:h

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(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
mark
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(12)
Obligation:
TRS:
Rules:
a__f(X) -> g
mark(f(X)) -> a__f(mark(X))
mark(g) -> g
mark(h(X)) -> h(mark(X))
a__f(X) -> f(X)

Types:
a__f :: g:f:h -> g:f:h
g :: g:f:h
mark :: g:f:h -> g:f:h
f :: g:f:h -> g:f:h
h :: g:f:h -> g:f:h
hole_g:f:h1_0 :: g:f:h
gen_g:f:h2_0 :: Nat -> g:f:h


Generator Equations:
gen_g:f:h2_0(0) <=> g
gen_g:f:h2_0(+(x, 1)) <=> f(gen_g:f:h2_0(x))


The following defined symbols remain to be analysed:
mark
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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_g:f:h2_0(n4_0)) -> gen_g:f:h2_0(0), rt in Omega(1 + n4_0)

Induction Base:
mark(gen_g:f:h2_0(0)) ->_R^Omega(1)
g

Induction Step:
mark(gen_g:f:h2_0(+(n4_0, 1))) ->_R^Omega(1)
a__f(mark(gen_g:f:h2_0(n4_0))) ->_IH
a__f(gen_g:f:h2_0(0)) ->_R^Omega(1)
g

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)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
a__f(X) -> g
mark(f(X)) -> a__f(mark(X))
mark(g) -> g
mark(h(X)) -> h(mark(X))
a__f(X) -> f(X)

Types:
a__f :: g:f:h -> g:f:h
g :: g:f:h
mark :: g:f:h -> g:f:h
f :: g:f:h -> g:f:h
h :: g:f:h -> g:f:h
hole_g:f:h1_0 :: g:f:h
gen_g:f:h2_0 :: Nat -> g:f:h


Generator Equations:
gen_g:f:h2_0(0) <=> g
gen_g:f:h2_0(+(x, 1)) <=> f(gen_g:f:h2_0(x))


The following defined symbols remain to be analysed:
mark
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(15) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(16)
BOUNDS(n^1, INF)