/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 439 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:

   f(X) -> g(n__h(n__f(X)))
   h(X) -> n__h(X)
   f(X) -> n__f(X)
   activate(n__h(X)) -> h(activate(X))
   activate(n__f(X)) -> f(activate(X))
   activate(X) -> 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:

   f(X) -> g(n__h(n__f(X)))
   h(X) -> n__h(X)
   f(X) -> n__f(X)
   activate(n__h(X)) -> h(activate(X))
   activate(n__f(X)) -> f(activate(X))
   activate(X) -> 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, 6, 7, 8, 9, 10, 11]
{(1,2,[f_1|0, h_1|0, activate_1|0, n__f_1|1, n__h_1|1, g_1|1]), (1,6,[g_1|1]), (1,8,[h_1|1, n__h_1|2]), (1,9,[f_1|1, n__f_1|2]), (1,10,[g_1|2]), (2,2,[g_1|0, n__h_1|0, n__f_1|0]), (6,7,[n__h_1|1]), (7,2,[n__f_1|1]), (8,2,[activate_1|1, n__h_1|1, n__f_1|1, g_1|1]), (8,8,[h_1|1, n__h_1|2]), (8,9,[f_1|1, n__f_1|2]), (8,10,[g_1|2]), (9,2,[activate_1|1, n__h_1|1, n__f_1|1, g_1|1]), (9,8,[h_1|1, n__h_1|2]), (9,9,[f_1|1, n__f_1|2]), (9,10,[g_1|2]), (10,11,[n__h_1|2]), (11,9,[n__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:

   f(X) -> g(n__h(n__f(X)))
   h(X) -> n__h(X)
   f(X) -> n__f(X)
   activate(n__h(X)) -> h(activate(X))
   activate(n__f(X)) -> f(activate(X))
   activate(X) -> 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:

   f(X) -> g
   h(X) -> n__h(X)
   f(X) -> n__f(X)
   activate(n__h(X)) -> h(activate(X))
   activate(n__f(X)) -> f(activate(X))
   activate(X) -> 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:
f(X) -> g
h(X) -> n__h(X)
f(X) -> n__f(X)
activate(n__h(X)) -> h(activate(X))
activate(n__f(X)) -> f(activate(X))
activate(X) -> X

Types:
f :: g:n__h:n__f -> g:n__h:n__f
g :: g:n__h:n__f
h :: g:n__h:n__f -> g:n__h:n__f
n__h :: g:n__h:n__f -> g:n__h:n__f
n__f :: g:n__h:n__f -> g:n__h:n__f
activate :: g:n__h:n__f -> g:n__h:n__f
hole_g:n__h:n__f1_0 :: g:n__h:n__f
gen_g:n__h:n__f2_0 :: Nat -> g:n__h:n__f

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

Types:
f :: g:n__h:n__f -> g:n__h:n__f
g :: g:n__h:n__f
h :: g:n__h:n__f -> g:n__h:n__f
n__h :: g:n__h:n__f -> g:n__h:n__f
n__f :: g:n__h:n__f -> g:n__h:n__f
activate :: g:n__h:n__f -> g:n__h:n__f
hole_g:n__h:n__f1_0 :: g:n__h:n__f
gen_g:n__h:n__f2_0 :: Nat -> g:n__h:n__f


Generator Equations:
gen_g:n__h:n__f2_0(0) <=> g
gen_g:n__h:n__f2_0(+(x, 1)) <=> n__h(gen_g:n__h:n__f2_0(x))


The following defined symbols remain to be analysed:
activate
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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
activate(gen_g:n__h:n__f2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
activate(gen_g:n__h:n__f2_0(+(1, 0)))

Induction Step:
activate(gen_g:n__h:n__f2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
h(activate(gen_g:n__h:n__f2_0(+(1, n4_0)))) ->_IH
h(*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)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
f(X) -> g
h(X) -> n__h(X)
f(X) -> n__f(X)
activate(n__h(X)) -> h(activate(X))
activate(n__f(X)) -> f(activate(X))
activate(X) -> X

Types:
f :: g:n__h:n__f -> g:n__h:n__f
g :: g:n__h:n__f
h :: g:n__h:n__f -> g:n__h:n__f
n__h :: g:n__h:n__f -> g:n__h:n__f
n__f :: g:n__h:n__f -> g:n__h:n__f
activate :: g:n__h:n__f -> g:n__h:n__f
hole_g:n__h:n__f1_0 :: g:n__h:n__f
gen_g:n__h:n__f2_0 :: Nat -> g:n__h:n__f


Generator Equations:
gen_g:n__h:n__f2_0(0) <=> g
gen_g:n__h:n__f2_0(+(x, 1)) <=> n__h(gen_g:n__h:n__f2_0(x))


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