/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), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 0 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 232 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs


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(0)
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__f(a, b, X) -> a__f(X, X, mark(X))
   a__c -> a
   a__c -> b
   mark(f(X1, X2, X3)) -> a__f(X1, X2, mark(X3))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__c -> c

S is empty.
Rewrite Strategy: INNERMOST
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(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(2)
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__f(a, b, X) -> a__f(X, X, mark(X))
   a__c -> a
   a__c -> b
   mark(f(X1, X2, X3)) -> a__f(X1, X2, mark(X3))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__c -> c

S is empty.
Rewrite Strategy: INNERMOST
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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(4)
Obligation:
Innermost TRS:
Rules:
a__f(a, b, X) -> a__f(X, X, mark(X))
a__c -> a
a__c -> b
mark(f(X1, X2, X3)) -> a__f(X1, X2, mark(X3))
mark(c) -> a__c
mark(a) -> a
mark(b) -> b
a__f(X1, X2, X3) -> f(X1, X2, X3)
a__c -> c

Types:
a__f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
a :: a:b:f:c
b :: a:b:f:c
mark :: a:b:f:c -> a:b:f:c
a__c :: a:b:f:c
f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
c :: a:b:f:c
hole_a:b:f:c1_0 :: a:b:f:c
gen_a:b:f:c2_0 :: Nat -> a:b:f:c

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
a__f, mark

They will be analysed ascendingly in the following order:
a__f = mark

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(6)
Obligation:
Innermost TRS:
Rules:
a__f(a, b, X) -> a__f(X, X, mark(X))
a__c -> a
a__c -> b
mark(f(X1, X2, X3)) -> a__f(X1, X2, mark(X3))
mark(c) -> a__c
mark(a) -> a
mark(b) -> b
a__f(X1, X2, X3) -> f(X1, X2, X3)
a__c -> c

Types:
a__f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
a :: a:b:f:c
b :: a:b:f:c
mark :: a:b:f:c -> a:b:f:c
a__c :: a:b:f:c
f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
c :: a:b:f:c
hole_a:b:f:c1_0 :: a:b:f:c
gen_a:b:f:c2_0 :: Nat -> a:b:f:c


Generator Equations:
gen_a:b:f:c2_0(0) <=> a
gen_a:b:f:c2_0(+(x, 1)) <=> f(a, a, gen_a:b:f:c2_0(x))


The following defined symbols remain to be analysed:
mark, a__f

They will be analysed ascendingly in the following order:
a__f = mark

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_a:b:f:c2_0(n4_0)) -> gen_a:b:f:c2_0(n4_0), rt in Omega(1 + n4_0)

Induction Base:
mark(gen_a:b:f:c2_0(0)) ->_R^Omega(1)
a

Induction Step:
mark(gen_a:b:f:c2_0(+(n4_0, 1))) ->_R^Omega(1)
a__f(a, a, mark(gen_a:b:f:c2_0(n4_0))) ->_IH
a__f(a, a, gen_a:b:f:c2_0(c5_0)) ->_R^Omega(1)
f(a, a, gen_a:b:f:c2_0(n4_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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(8)
Complex Obligation (BEST)

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(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
a__f(a, b, X) -> a__f(X, X, mark(X))
a__c -> a
a__c -> b
mark(f(X1, X2, X3)) -> a__f(X1, X2, mark(X3))
mark(c) -> a__c
mark(a) -> a
mark(b) -> b
a__f(X1, X2, X3) -> f(X1, X2, X3)
a__c -> c

Types:
a__f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
a :: a:b:f:c
b :: a:b:f:c
mark :: a:b:f:c -> a:b:f:c
a__c :: a:b:f:c
f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
c :: a:b:f:c
hole_a:b:f:c1_0 :: a:b:f:c
gen_a:b:f:c2_0 :: Nat -> a:b:f:c


Generator Equations:
gen_a:b:f:c2_0(0) <=> a
gen_a:b:f:c2_0(+(x, 1)) <=> f(a, a, gen_a:b:f:c2_0(x))


The following defined symbols remain to be analysed:
mark, a__f

They will be analysed ascendingly in the following order:
a__f = mark

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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(11)
BOUNDS(n^1, INF)

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(12)
Obligation:
Innermost TRS:
Rules:
a__f(a, b, X) -> a__f(X, X, mark(X))
a__c -> a
a__c -> b
mark(f(X1, X2, X3)) -> a__f(X1, X2, mark(X3))
mark(c) -> a__c
mark(a) -> a
mark(b) -> b
a__f(X1, X2, X3) -> f(X1, X2, X3)
a__c -> c

Types:
a__f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
a :: a:b:f:c
b :: a:b:f:c
mark :: a:b:f:c -> a:b:f:c
a__c :: a:b:f:c
f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
c :: a:b:f:c
hole_a:b:f:c1_0 :: a:b:f:c
gen_a:b:f:c2_0 :: Nat -> a:b:f:c


Lemmas:
mark(gen_a:b:f:c2_0(n4_0)) -> gen_a:b:f:c2_0(n4_0), rt in Omega(1 + n4_0)


Generator Equations:
gen_a:b:f:c2_0(0) <=> a
gen_a:b:f:c2_0(+(x, 1)) <=> f(a, a, gen_a:b:f:c2_0(x))


The following defined symbols remain to be analysed:
a__f

They will be analysed ascendingly in the following order:
a__f = mark