/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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 (full) 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), 305 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs
        (13) RewriteLemmaProof [LOWER BOUND(ID), 134 ms]
        (14) typed CpxTrs


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(0)
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(true, x, y) -> f(gt(x, y), trunc(x), s(y))
   trunc(0) -> 0
   trunc(s(0)) -> 0
   trunc(s(s(x))) -> s(s(trunc(x)))
   gt(0, v) -> false
   gt(s(u), 0) -> true
   gt(s(u), s(v)) -> gt(u, v)

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
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(true, x, y) -> f(gt(x, y), trunc(x), s(y))
   trunc(0') -> 0'
   trunc(s(0')) -> 0'
   trunc(s(s(x))) -> s(s(trunc(x)))
   gt(0', v) -> false
   gt(s(u), 0') -> true
   gt(s(u), s(v)) -> gt(u, v)

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
TRS:
Rules:
f(true, x, y) -> f(gt(x, y), trunc(x), s(y))
trunc(0') -> 0'
trunc(s(0')) -> 0'
trunc(s(s(x))) -> s(s(trunc(x)))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)

Types:
f :: true:false -> s:0' -> s:0' -> f
true :: true:false
gt :: s:0' -> s:0' -> true:false
trunc :: s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat -> s:0'

----------------------------------------

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
f, gt, trunc

They will be analysed ascendingly in the following order:
gt < f
trunc < f

----------------------------------------

(6)
Obligation:
TRS:
Rules:
f(true, x, y) -> f(gt(x, y), trunc(x), s(y))
trunc(0') -> 0'
trunc(s(0')) -> 0'
trunc(s(s(x))) -> s(s(trunc(x)))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)

Types:
f :: true:false -> s:0' -> s:0' -> f
true :: true:false
gt :: s:0' -> s:0' -> true:false
trunc :: s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
gt, f, trunc

They will be analysed ascendingly in the following order:
gt < f
trunc < f

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0)

Induction Base:
gt(gen_s:0'4_0(0), gen_s:0'4_0(0)) ->_R^Omega(1)
false

Induction Step:
gt(gen_s:0'4_0(+(n6_0, 1)), gen_s:0'4_0(+(n6_0, 1))) ->_R^Omega(1)
gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) ->_IH
false

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(8)
Complex Obligation (BEST)

----------------------------------------

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

TRS:
Rules:
f(true, x, y) -> f(gt(x, y), trunc(x), s(y))
trunc(0') -> 0'
trunc(s(0')) -> 0'
trunc(s(s(x))) -> s(s(trunc(x)))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)

Types:
f :: true:false -> s:0' -> s:0' -> f
true :: true:false
gt :: s:0' -> s:0' -> true:false
trunc :: s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
gt, f, trunc

They will be analysed ascendingly in the following order:
gt < f
trunc < f

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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:
TRS:
Rules:
f(true, x, y) -> f(gt(x, y), trunc(x), s(y))
trunc(0') -> 0'
trunc(s(0')) -> 0'
trunc(s(s(x))) -> s(s(trunc(x)))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)

Types:
f :: true:false -> s:0' -> s:0' -> f
true :: true:false
gt :: s:0' -> s:0' -> true:false
trunc :: s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat -> s:0'


Lemmas:
gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0)


Generator Equations:
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
trunc, f

They will be analysed ascendingly in the following order:
trunc < f

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
trunc(gen_s:0'4_0(*(2, n235_0))) -> gen_s:0'4_0(*(2, n235_0)), rt in Omega(1 + n235_0)

Induction Base:
trunc(gen_s:0'4_0(*(2, 0))) ->_R^Omega(1)
0'

Induction Step:
trunc(gen_s:0'4_0(*(2, +(n235_0, 1)))) ->_R^Omega(1)
s(s(trunc(gen_s:0'4_0(*(2, n235_0))))) ->_IH
s(s(gen_s:0'4_0(*(2, c236_0))))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(14)
Obligation:
TRS:
Rules:
f(true, x, y) -> f(gt(x, y), trunc(x), s(y))
trunc(0') -> 0'
trunc(s(0')) -> 0'
trunc(s(s(x))) -> s(s(trunc(x)))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)

Types:
f :: true:false -> s:0' -> s:0' -> f
true :: true:false
gt :: s:0' -> s:0' -> true:false
trunc :: s:0' -> s:0'
s :: s:0' -> s:0'
0' :: s:0'
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_s:0'3_0 :: s:0'
gen_s:0'4_0 :: Nat -> s:0'


Lemmas:
gt(gen_s:0'4_0(n6_0), gen_s:0'4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
trunc(gen_s:0'4_0(*(2, n235_0))) -> gen_s:0'4_0(*(2, n235_0)), rt in Omega(1 + n235_0)


Generator Equations:
gen_s:0'4_0(0) <=> 0'
gen_s:0'4_0(+(x, 1)) <=> s(gen_s:0'4_0(x))


The following defined symbols remain to be analysed:
f