/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (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), 256 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 (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   le(0, Y) -> true
   le(s(X), 0) -> false
   le(s(X), s(Y)) -> le(X, Y)
   minus(0, Y) -> 0
   minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
   ifMinus(true, s(X), Y) -> 0
   ifMinus(false, s(X), Y) -> s(minus(X, Y))
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

S is empty.
Rewrite Strategy: FULL
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(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:

   le(0', Y) -> true
   le(s(X), 0') -> false
   le(s(X), s(Y)) -> le(X, Y)
   minus(0', Y) -> 0'
   minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
   ifMinus(true, s(X), Y) -> 0'
   ifMinus(false, s(X), Y) -> s(minus(X, Y))
   quot(0', s(Y)) -> 0'
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

S is empty.
Rewrite Strategy: FULL
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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(4)
Obligation:
TRS:
Rules:
le(0', Y) -> true
le(s(X), 0') -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0', Y) -> 0'
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0'
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0', s(Y)) -> 0'
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
minus :: 0':s -> 0':s -> 0':s
ifMinus :: true:false -> 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
le, minus, quot

They will be analysed ascendingly in the following order:
le < minus
minus < quot

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(6)
Obligation:
TRS:
Rules:
le(0', Y) -> true
le(s(X), 0') -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0', Y) -> 0'
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0'
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0', s(Y)) -> 0'
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
minus :: 0':s -> 0':s -> 0':s
ifMinus :: true:false -> 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
le, minus, quot

They will be analysed ascendingly in the following order:
le < minus
minus < quot

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)

Induction Base:
le(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
true

Induction Step:
le(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
true

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:

TRS:
Rules:
le(0', Y) -> true
le(s(X), 0') -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0', Y) -> 0'
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0'
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0', s(Y)) -> 0'
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
minus :: 0':s -> 0':s -> 0':s
ifMinus :: true:false -> 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
le, minus, quot

They will be analysed ascendingly in the following order:
le < minus
minus < quot

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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:
le(0', Y) -> true
le(s(X), 0') -> false
le(s(X), s(Y)) -> le(X, Y)
minus(0', Y) -> 0'
minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
ifMinus(true, s(X), Y) -> 0'
ifMinus(false, s(X), Y) -> s(minus(X, Y))
quot(0', s(Y)) -> 0'
quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
minus :: 0':s -> 0':s -> 0':s
ifMinus :: true:false -> 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
minus, quot

They will be analysed ascendingly in the following order:
minus < quot