/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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), 443 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(a) -> f(c(a))
   f(c(X)) -> X
   f(c(a)) -> f(d(b))
   f(a) -> f(d(a))
   f(d(X)) -> X
   f(c(b)) -> f(d(a))
   e(g(X)) -> e(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:

   f(a) -> f(c(a))
   f(c(X)) -> X
   f(c(a)) -> f(d(b))
   f(a) -> f(d(a))
   f(d(X)) -> X
   f(c(b)) -> f(d(a))
   e(g(X)) -> e(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 3. 
The certificate found is represented by the following graph.

"[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
{(1,2,[f_1|0, e_1|0, a|1, c_1|1, d_1|1, b|1, g_1|1, e_1|1, a|2, b|2, b|3]), (1,3,[f_1|1]), (1,5,[f_1|1]), (1,7,[f_1|1]), (1,9,[f_1|2]), (2,2,[a|0, c_1|0, d_1|0, b|0, g_1|0]), (3,4,[c_1|1]), (4,2,[a|1]), (5,6,[d_1|1]), (6,2,[a|1]), (7,8,[d_1|1]), (8,2,[b|1]), (9,10,[d_1|2]), (10,2,[b|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 (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   f(a) -> f(c(a))
   f(c(X)) -> X
   f(c(a)) -> f(d(b))
   f(a) -> f(d(a))
   f(d(X)) -> X
   f(c(b)) -> f(d(a))
   e(g(X)) -> e(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:
f(a) -> f(c(a))
f(c(X)) -> X
f(c(a)) -> f(d(b))
f(a) -> f(d(a))
f(d(X)) -> X
f(c(b)) -> f(d(a))
e(g(X)) -> e(X)

Types:
f :: a:c:b:d -> a:c:b:d
a :: a:c:b:d
c :: a:c:b:d -> a:c:b:d
d :: a:c:b:d -> a:c:b:d
b :: a:c:b:d
e :: g -> e
g :: g -> g
hole_a:c:b:d1_0 :: a:c:b:d
hole_e2_0 :: e
hole_g3_0 :: g
gen_a:c:b:d4_0 :: Nat -> a:c:b:d
gen_g5_0 :: Nat -> g

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(9) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
f, e
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(10)
Obligation:
Innermost TRS:
Rules:
f(a) -> f(c(a))
f(c(X)) -> X
f(c(a)) -> f(d(b))
f(a) -> f(d(a))
f(d(X)) -> X
f(c(b)) -> f(d(a))
e(g(X)) -> e(X)

Types:
f :: a:c:b:d -> a:c:b:d
a :: a:c:b:d
c :: a:c:b:d -> a:c:b:d
d :: a:c:b:d -> a:c:b:d
b :: a:c:b:d
e :: g -> e
g :: g -> g
hole_a:c:b:d1_0 :: a:c:b:d
hole_e2_0 :: e
hole_g3_0 :: g
gen_a:c:b:d4_0 :: Nat -> a:c:b:d
gen_g5_0 :: Nat -> g


Generator Equations:
gen_a:c:b:d4_0(0) <=> b
gen_a:c:b:d4_0(+(x, 1)) <=> c(gen_a:c:b:d4_0(x))
gen_g5_0(0) <=> hole_g3_0
gen_g5_0(+(x, 1)) <=> g(gen_g5_0(x))


The following defined symbols remain to be analysed:
f, e
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(11) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
e(gen_g5_0(+(1, n43_0))) -> *6_0, rt in Omega(n43_0)

Induction Base:
e(gen_g5_0(+(1, 0)))

Induction Step:
e(gen_g5_0(+(1, +(n43_0, 1)))) ->_R^Omega(1)
e(gen_g5_0(+(1, n43_0))) ->_IH
*6_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(a) -> f(c(a))
f(c(X)) -> X
f(c(a)) -> f(d(b))
f(a) -> f(d(a))
f(d(X)) -> X
f(c(b)) -> f(d(a))
e(g(X)) -> e(X)

Types:
f :: a:c:b:d -> a:c:b:d
a :: a:c:b:d
c :: a:c:b:d -> a:c:b:d
d :: a:c:b:d -> a:c:b:d
b :: a:c:b:d
e :: g -> e
g :: g -> g
hole_a:c:b:d1_0 :: a:c:b:d
hole_e2_0 :: e
hole_g3_0 :: g
gen_a:c:b:d4_0 :: Nat -> a:c:b:d
gen_g5_0 :: Nat -> g


Generator Equations:
gen_a:c:b:d4_0(0) <=> b
gen_a:c:b:d4_0(+(x, 1)) <=> c(gen_a:c:b:d4_0(x))
gen_g5_0(0) <=> hole_g3_0
gen_g5_0(+(x, 1)) <=> g(gen_g5_0(x))


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