/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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), 1038 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(X) -> a__if(mark(X), c, f(true))
   a__if(true, X, Y) -> mark(X)
   a__if(false, X, Y) -> mark(Y)
   mark(f(X)) -> a__f(mark(X))
   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
   mark(c) -> c
   mark(true) -> true
   mark(false) -> false
   a__f(X) -> f(X)
   a__if(X1, X2, X3) -> if(X1, X2, X3)

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(X) -> a__if(mark(X), c, f(true))
   a__if(true, X, Y) -> mark(X)
   a__if(false, X, Y) -> mark(Y)
   mark(f(X)) -> a__f(mark(X))
   mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
   mark(c) -> c
   mark(true) -> true
   mark(false) -> false
   a__f(X) -> f(X)
   a__if(X1, X2, X3) -> if(X1, X2, X3)

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(X) -> a__if(mark(X), c, f(true))
a__if(true, X, Y) -> mark(X)
a__if(false, X, Y) -> mark(Y)
mark(f(X)) -> a__f(mark(X))
mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
mark(c) -> c
mark(true) -> true
mark(false) -> false
a__f(X) -> f(X)
a__if(X1, X2, X3) -> if(X1, X2, X3)

Types:
a__f :: c:true:f:false:if -> c:true:f:false:if
a__if :: c:true:f:false:if -> c:true:f:false:if -> c:true:f:false:if -> c:true:f:false:if
mark :: c:true:f:false:if -> c:true:f:false:if
c :: c:true:f:false:if
f :: c:true:f:false:if -> c:true:f:false:if
true :: c:true:f:false:if
false :: c:true:f:false:if
if :: c:true:f:false:if -> c:true:f:false:if -> c:true:f:false:if -> c:true:f:false:if
hole_c:true:f:false:if1_0 :: c:true:f:false:if
gen_c:true:f:false:if2_0 :: Nat -> c:true:f:false:if

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

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

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(6)
Obligation:
Innermost TRS:
Rules:
a__f(X) -> a__if(mark(X), c, f(true))
a__if(true, X, Y) -> mark(X)
a__if(false, X, Y) -> mark(Y)
mark(f(X)) -> a__f(mark(X))
mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
mark(c) -> c
mark(true) -> true
mark(false) -> false
a__f(X) -> f(X)
a__if(X1, X2, X3) -> if(X1, X2, X3)

Types:
a__f :: c:true:f:false:if -> c:true:f:false:if
a__if :: c:true:f:false:if -> c:true:f:false:if -> c:true:f:false:if -> c:true:f:false:if
mark :: c:true:f:false:if -> c:true:f:false:if
c :: c:true:f:false:if
f :: c:true:f:false:if -> c:true:f:false:if
true :: c:true:f:false:if
false :: c:true:f:false:if
if :: c:true:f:false:if -> c:true:f:false:if -> c:true:f:false:if -> c:true:f:false:if
hole_c:true:f:false:if1_0 :: c:true:f:false:if
gen_c:true:f:false:if2_0 :: Nat -> c:true:f:false:if


Generator Equations:
gen_c:true:f:false:if2_0(0) <=> c
gen_c:true:f:false:if2_0(+(x, 1)) <=> f(gen_c:true:f:false:if2_0(x))


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

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

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_c:true:f:false:if2_0(+(1, n26_0))) -> *3_0, rt in Omega(n26_0)

Induction Base:
mark(gen_c:true:f:false:if2_0(+(1, 0)))

Induction Step:
mark(gen_c:true:f:false:if2_0(+(1, +(n26_0, 1)))) ->_R^Omega(1)
a__f(mark(gen_c:true:f:false:if2_0(+(1, n26_0)))) ->_IH
a__f(*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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(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(X) -> a__if(mark(X), c, f(true))
a__if(true, X, Y) -> mark(X)
a__if(false, X, Y) -> mark(Y)
mark(f(X)) -> a__f(mark(X))
mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
mark(c) -> c
mark(true) -> true
mark(false) -> false
a__f(X) -> f(X)
a__if(X1, X2, X3) -> if(X1, X2, X3)

Types:
a__f :: c:true:f:false:if -> c:true:f:false:if
a__if :: c:true:f:false:if -> c:true:f:false:if -> c:true:f:false:if -> c:true:f:false:if
mark :: c:true:f:false:if -> c:true:f:false:if
c :: c:true:f:false:if
f :: c:true:f:false:if -> c:true:f:false:if
true :: c:true:f:false:if
false :: c:true:f:false:if
if :: c:true:f:false:if -> c:true:f:false:if -> c:true:f:false:if -> c:true:f:false:if
hole_c:true:f:false:if1_0 :: c:true:f:false:if
gen_c:true:f:false:if2_0 :: Nat -> c:true:f:false:if


Generator Equations:
gen_c:true:f:false:if2_0(0) <=> c
gen_c:true:f:false:if2_0(+(x, 1)) <=> f(gen_c:true:f:false:if2_0(x))


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

They will be analysed ascendingly in the following order:
a__f = a__if
a__f = mark
a__if = 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(X) -> a__if(mark(X), c, f(true))
a__if(true, X, Y) -> mark(X)
a__if(false, X, Y) -> mark(Y)
mark(f(X)) -> a__f(mark(X))
mark(if(X1, X2, X3)) -> a__if(mark(X1), mark(X2), X3)
mark(c) -> c
mark(true) -> true
mark(false) -> false
a__f(X) -> f(X)
a__if(X1, X2, X3) -> if(X1, X2, X3)

Types:
a__f :: c:true:f:false:if -> c:true:f:false:if
a__if :: c:true:f:false:if -> c:true:f:false:if -> c:true:f:false:if -> c:true:f:false:if
mark :: c:true:f:false:if -> c:true:f:false:if
c :: c:true:f:false:if
f :: c:true:f:false:if -> c:true:f:false:if
true :: c:true:f:false:if
false :: c:true:f:false:if
if :: c:true:f:false:if -> c:true:f:false:if -> c:true:f:false:if -> c:true:f:false:if
hole_c:true:f:false:if1_0 :: c:true:f:false:if
gen_c:true:f:false:if2_0 :: Nat -> c:true:f:false:if


Lemmas:
mark(gen_c:true:f:false:if2_0(+(1, n26_0))) -> *3_0, rt in Omega(n26_0)


Generator Equations:
gen_c:true:f:false:if2_0(0) <=> c
gen_c:true:f:false:if2_0(+(x, 1)) <=> f(gen_c:true:f:false:if2_0(x))


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

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