/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 301 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), 50 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 43 ms]
        (16) typed CpxTrs


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

(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)
   inc(0) -> 0
   inc(s(x)) -> s(inc(x))
   minus(0, y) -> 0
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   log(x) -> log2(x, 0)
   log2(x, y) -> if(le(x, 0), le(x, s(0)), x, inc(y))
   if(true, b, x, y) -> log_undefined
   if(false, b, x, y) -> if2(b, x, y)
   if2(true, x, s(y)) -> y
   if2(false, x, y) -> log2(quot(x, s(s(0))), y)

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:

   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)
   inc(0') -> 0'
   inc(s(x)) -> s(inc(x))
   minus(0', y) -> 0'
   minus(x, 0') -> x
   minus(s(x), s(y)) -> minus(x, y)
   quot(0', s(y)) -> 0'
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   log(x) -> log2(x, 0')
   log2(x, y) -> if(le(x, 0'), le(x, s(0')), x, inc(y))
   if(true, b, x, y) -> log_undefined
   if(false, b, x, y) -> if2(b, x, y)
   if2(true, x, s(y)) -> y
   if2(false, x, y) -> log2(quot(x, s(s(0'))), y)

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

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

(4)
Obligation:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
inc(0') -> 0'
inc(s(x)) -> s(inc(x))
minus(0', y) -> 0'
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
log(x) -> log2(x, 0')
log2(x, y) -> if(le(x, 0'), le(x, s(0')), x, inc(y))
if(true, b, x, y) -> log_undefined
if(false, b, x, y) -> if2(b, x, y)
if2(true, x, s(y)) -> y
if2(false, x, y) -> log2(quot(x, s(s(0'))), y)

Types:
le :: 0':s:log_undefined -> 0':s:log_undefined -> true:false
0' :: 0':s:log_undefined
true :: true:false
s :: 0':s:log_undefined -> 0':s:log_undefined
false :: true:false
inc :: 0':s:log_undefined -> 0':s:log_undefined
minus :: 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
quot :: 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
log :: 0':s:log_undefined -> 0':s:log_undefined
log2 :: 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
if :: true:false -> true:false -> 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
log_undefined :: 0':s:log_undefined
if2 :: true:false -> 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
hole_true:false1_0 :: true:false
hole_0':s:log_undefined2_0 :: 0':s:log_undefined
gen_0':s:log_undefined3_0 :: Nat -> 0':s:log_undefined

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
le, inc, minus, quot, log2

They will be analysed ascendingly in the following order:
le < log2
inc < log2
minus < quot
quot < log2

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

(6)
Obligation:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
inc(0') -> 0'
inc(s(x)) -> s(inc(x))
minus(0', y) -> 0'
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
log(x) -> log2(x, 0')
log2(x, y) -> if(le(x, 0'), le(x, s(0')), x, inc(y))
if(true, b, x, y) -> log_undefined
if(false, b, x, y) -> if2(b, x, y)
if2(true, x, s(y)) -> y
if2(false, x, y) -> log2(quot(x, s(s(0'))), y)

Types:
le :: 0':s:log_undefined -> 0':s:log_undefined -> true:false
0' :: 0':s:log_undefined
true :: true:false
s :: 0':s:log_undefined -> 0':s:log_undefined
false :: true:false
inc :: 0':s:log_undefined -> 0':s:log_undefined
minus :: 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
quot :: 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
log :: 0':s:log_undefined -> 0':s:log_undefined
log2 :: 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
if :: true:false -> true:false -> 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
log_undefined :: 0':s:log_undefined
if2 :: true:false -> 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
hole_true:false1_0 :: true:false
hole_0':s:log_undefined2_0 :: 0':s:log_undefined
gen_0':s:log_undefined3_0 :: Nat -> 0':s:log_undefined


Generator Equations:
gen_0':s:log_undefined3_0(0) <=> 0'
gen_0':s:log_undefined3_0(+(x, 1)) <=> s(gen_0':s:log_undefined3_0(x))


The following defined symbols remain to be analysed:
le, inc, minus, quot, log2

They will be analysed ascendingly in the following order:
le < log2
inc < log2
minus < quot
quot < log2

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s:log_undefined3_0(n5_0), gen_0':s:log_undefined3_0(n5_0)) -> true, rt in Omega(1 + n5_0)

Induction Base:
le(gen_0':s:log_undefined3_0(0), gen_0':s:log_undefined3_0(0)) ->_R^Omega(1)
true

Induction Step:
le(gen_0':s:log_undefined3_0(+(n5_0, 1)), gen_0':s:log_undefined3_0(+(n5_0, 1))) ->_R^Omega(1)
le(gen_0':s:log_undefined3_0(n5_0), gen_0':s:log_undefined3_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).
----------------------------------------

(8)
Complex Obligation (BEST)

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

(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)
inc(0') -> 0'
inc(s(x)) -> s(inc(x))
minus(0', y) -> 0'
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
log(x) -> log2(x, 0')
log2(x, y) -> if(le(x, 0'), le(x, s(0')), x, inc(y))
if(true, b, x, y) -> log_undefined
if(false, b, x, y) -> if2(b, x, y)
if2(true, x, s(y)) -> y
if2(false, x, y) -> log2(quot(x, s(s(0'))), y)

Types:
le :: 0':s:log_undefined -> 0':s:log_undefined -> true:false
0' :: 0':s:log_undefined
true :: true:false
s :: 0':s:log_undefined -> 0':s:log_undefined
false :: true:false
inc :: 0':s:log_undefined -> 0':s:log_undefined
minus :: 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
quot :: 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
log :: 0':s:log_undefined -> 0':s:log_undefined
log2 :: 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
if :: true:false -> true:false -> 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
log_undefined :: 0':s:log_undefined
if2 :: true:false -> 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
hole_true:false1_0 :: true:false
hole_0':s:log_undefined2_0 :: 0':s:log_undefined
gen_0':s:log_undefined3_0 :: Nat -> 0':s:log_undefined


Generator Equations:
gen_0':s:log_undefined3_0(0) <=> 0'
gen_0':s:log_undefined3_0(+(x, 1)) <=> s(gen_0':s:log_undefined3_0(x))


The following defined symbols remain to be analysed:
le, inc, minus, quot, log2

They will be analysed ascendingly in the following order:
le < log2
inc < log2
minus < quot
quot < log2

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

(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
inc(0') -> 0'
inc(s(x)) -> s(inc(x))
minus(0', y) -> 0'
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
log(x) -> log2(x, 0')
log2(x, y) -> if(le(x, 0'), le(x, s(0')), x, inc(y))
if(true, b, x, y) -> log_undefined
if(false, b, x, y) -> if2(b, x, y)
if2(true, x, s(y)) -> y
if2(false, x, y) -> log2(quot(x, s(s(0'))), y)

Types:
le :: 0':s:log_undefined -> 0':s:log_undefined -> true:false
0' :: 0':s:log_undefined
true :: true:false
s :: 0':s:log_undefined -> 0':s:log_undefined
false :: true:false
inc :: 0':s:log_undefined -> 0':s:log_undefined
minus :: 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
quot :: 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
log :: 0':s:log_undefined -> 0':s:log_undefined
log2 :: 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
if :: true:false -> true:false -> 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
log_undefined :: 0':s:log_undefined
if2 :: true:false -> 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
hole_true:false1_0 :: true:false
hole_0':s:log_undefined2_0 :: 0':s:log_undefined
gen_0':s:log_undefined3_0 :: Nat -> 0':s:log_undefined


Lemmas:
le(gen_0':s:log_undefined3_0(n5_0), gen_0':s:log_undefined3_0(n5_0)) -> true, rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s:log_undefined3_0(0) <=> 0'
gen_0':s:log_undefined3_0(+(x, 1)) <=> s(gen_0':s:log_undefined3_0(x))


The following defined symbols remain to be analysed:
inc, minus, quot, log2

They will be analysed ascendingly in the following order:
inc < log2
minus < quot
quot < log2

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
inc(gen_0':s:log_undefined3_0(n288_0)) -> gen_0':s:log_undefined3_0(n288_0), rt in Omega(1 + n288_0)

Induction Base:
inc(gen_0':s:log_undefined3_0(0)) ->_R^Omega(1)
0'

Induction Step:
inc(gen_0':s:log_undefined3_0(+(n288_0, 1))) ->_R^Omega(1)
s(inc(gen_0':s:log_undefined3_0(n288_0))) ->_IH
s(gen_0':s:log_undefined3_0(c289_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:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
inc(0') -> 0'
inc(s(x)) -> s(inc(x))
minus(0', y) -> 0'
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
log(x) -> log2(x, 0')
log2(x, y) -> if(le(x, 0'), le(x, s(0')), x, inc(y))
if(true, b, x, y) -> log_undefined
if(false, b, x, y) -> if2(b, x, y)
if2(true, x, s(y)) -> y
if2(false, x, y) -> log2(quot(x, s(s(0'))), y)

Types:
le :: 0':s:log_undefined -> 0':s:log_undefined -> true:false
0' :: 0':s:log_undefined
true :: true:false
s :: 0':s:log_undefined -> 0':s:log_undefined
false :: true:false
inc :: 0':s:log_undefined -> 0':s:log_undefined
minus :: 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
quot :: 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
log :: 0':s:log_undefined -> 0':s:log_undefined
log2 :: 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
if :: true:false -> true:false -> 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
log_undefined :: 0':s:log_undefined
if2 :: true:false -> 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
hole_true:false1_0 :: true:false
hole_0':s:log_undefined2_0 :: 0':s:log_undefined
gen_0':s:log_undefined3_0 :: Nat -> 0':s:log_undefined


Lemmas:
le(gen_0':s:log_undefined3_0(n5_0), gen_0':s:log_undefined3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
inc(gen_0':s:log_undefined3_0(n288_0)) -> gen_0':s:log_undefined3_0(n288_0), rt in Omega(1 + n288_0)


Generator Equations:
gen_0':s:log_undefined3_0(0) <=> 0'
gen_0':s:log_undefined3_0(+(x, 1)) <=> s(gen_0':s:log_undefined3_0(x))


The following defined symbols remain to be analysed:
minus, quot, log2

They will be analysed ascendingly in the following order:
minus < quot
quot < log2

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s:log_undefined3_0(n502_0), gen_0':s:log_undefined3_0(n502_0)) -> gen_0':s:log_undefined3_0(0), rt in Omega(1 + n502_0)

Induction Base:
minus(gen_0':s:log_undefined3_0(0), gen_0':s:log_undefined3_0(0)) ->_R^Omega(1)
0'

Induction Step:
minus(gen_0':s:log_undefined3_0(+(n502_0, 1)), gen_0':s:log_undefined3_0(+(n502_0, 1))) ->_R^Omega(1)
minus(gen_0':s:log_undefined3_0(n502_0), gen_0':s:log_undefined3_0(n502_0)) ->_IH
gen_0':s:log_undefined3_0(0)

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

(16)
Obligation:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
inc(0') -> 0'
inc(s(x)) -> s(inc(x))
minus(0', y) -> 0'
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
quot(0', s(y)) -> 0'
quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
log(x) -> log2(x, 0')
log2(x, y) -> if(le(x, 0'), le(x, s(0')), x, inc(y))
if(true, b, x, y) -> log_undefined
if(false, b, x, y) -> if2(b, x, y)
if2(true, x, s(y)) -> y
if2(false, x, y) -> log2(quot(x, s(s(0'))), y)

Types:
le :: 0':s:log_undefined -> 0':s:log_undefined -> true:false
0' :: 0':s:log_undefined
true :: true:false
s :: 0':s:log_undefined -> 0':s:log_undefined
false :: true:false
inc :: 0':s:log_undefined -> 0':s:log_undefined
minus :: 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
quot :: 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
log :: 0':s:log_undefined -> 0':s:log_undefined
log2 :: 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
if :: true:false -> true:false -> 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
log_undefined :: 0':s:log_undefined
if2 :: true:false -> 0':s:log_undefined -> 0':s:log_undefined -> 0':s:log_undefined
hole_true:false1_0 :: true:false
hole_0':s:log_undefined2_0 :: 0':s:log_undefined
gen_0':s:log_undefined3_0 :: Nat -> 0':s:log_undefined


Lemmas:
le(gen_0':s:log_undefined3_0(n5_0), gen_0':s:log_undefined3_0(n5_0)) -> true, rt in Omega(1 + n5_0)
inc(gen_0':s:log_undefined3_0(n288_0)) -> gen_0':s:log_undefined3_0(n288_0), rt in Omega(1 + n288_0)
minus(gen_0':s:log_undefined3_0(n502_0), gen_0':s:log_undefined3_0(n502_0)) -> gen_0':s:log_undefined3_0(0), rt in Omega(1 + n502_0)


Generator Equations:
gen_0':s:log_undefined3_0(0) <=> 0'
gen_0':s:log_undefined3_0(+(x, 1)) <=> s(gen_0':s:log_undefined3_0(x))


The following defined symbols remain to be analysed:
quot, log2

They will be analysed ascendingly in the following order:
quot < log2