/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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), 293 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), 55 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 20 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)
   zero(0) -> true
   zero(s(x)) -> false
   id(0) -> 0
   id(s(x)) -> s(id(x))
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
   if_mod(true, b1, b2, x, y) -> 0
   if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y)
   if2(true, b2, x, y) -> 0
   if2(false, b2, x, y) -> if3(b2, x, y)
   if3(true, x, y) -> mod(minus(x, y), s(y))
   if3(false, x, y) -> x

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)
   zero(0') -> true
   zero(s(x)) -> false
   id(0') -> 0'
   id(s(x)) -> s(id(x))
   minus(x, 0') -> x
   minus(s(x), s(y)) -> minus(x, y)
   mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
   if_mod(true, b1, b2, x, y) -> 0'
   if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y)
   if2(true, b2, x, y) -> 0'
   if2(false, b2, x, y) -> if3(b2, x, y)
   if3(true, x, y) -> mod(minus(x, y), s(y))
   if3(false, x, y) -> x

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)
zero(0') -> true
zero(s(x)) -> false
id(0') -> 0'
id(s(x)) -> s(id(x))
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) -> 0'
if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y)
if2(true, b2, x, y) -> 0'
if2(false, b2, x, y) -> if3(b2, x, y)
if3(true, x, y) -> mod(minus(x, y), s(y))
if3(false, x, y) -> x

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
zero :: 0':s -> true:false
id :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s
if2 :: true:false -> true:false -> 0':s -> 0':s -> 0':s
if3 :: true:false -> 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

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

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

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

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

(6)
Obligation:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
zero(0') -> true
zero(s(x)) -> false
id(0') -> 0'
id(s(x)) -> s(id(x))
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) -> 0'
if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y)
if2(true, b2, x, y) -> 0'
if2(false, b2, x, y) -> if3(b2, x, y)
if3(true, x, y) -> mod(minus(x, y), s(y))
if3(false, x, y) -> x

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
zero :: 0':s -> true:false
id :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s
if2 :: true:false -> true:false -> 0':s -> 0':s -> 0':s
if3 :: true:false -> 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, id, minus, mod

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

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

(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).
----------------------------------------

(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)
zero(0') -> true
zero(s(x)) -> false
id(0') -> 0'
id(s(x)) -> s(id(x))
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) -> 0'
if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y)
if2(true, b2, x, y) -> 0'
if2(false, b2, x, y) -> if3(b2, x, y)
if3(true, x, y) -> mod(minus(x, y), s(y))
if3(false, x, y) -> x

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
zero :: 0':s -> true:false
id :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s
if2 :: true:false -> true:false -> 0':s -> 0':s -> 0':s
if3 :: true:false -> 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, id, minus, mod

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

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

(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)
zero(0') -> true
zero(s(x)) -> false
id(0') -> 0'
id(s(x)) -> s(id(x))
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) -> 0'
if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y)
if2(true, b2, x, y) -> 0'
if2(false, b2, x, y) -> if3(b2, x, y)
if3(true, x, y) -> mod(minus(x, y), s(y))
if3(false, x, y) -> x

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
zero :: 0':s -> true:false
id :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s
if2 :: true:false -> true:false -> 0':s -> 0':s -> 0':s
if3 :: true:false -> 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:
id, minus, mod

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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
id(gen_0':s3_0(n288_0)) -> gen_0':s3_0(n288_0), rt in Omega(1 + n288_0)

Induction Base:
id(gen_0':s3_0(0)) ->_R^Omega(1)
0'

Induction Step:
id(gen_0':s3_0(+(n288_0, 1))) ->_R^Omega(1)
s(id(gen_0':s3_0(n288_0))) ->_IH
s(gen_0':s3_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)
zero(0') -> true
zero(s(x)) -> false
id(0') -> 0'
id(s(x)) -> s(id(x))
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) -> 0'
if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y)
if2(true, b2, x, y) -> 0'
if2(false, b2, x, y) -> if3(b2, x, y)
if3(true, x, y) -> mod(minus(x, y), s(y))
if3(false, x, y) -> x

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
zero :: 0':s -> true:false
id :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s
if2 :: true:false -> true:false -> 0':s -> 0':s -> 0':s
if3 :: true:false -> 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)
id(gen_0':s3_0(n288_0)) -> gen_0':s3_0(n288_0), rt in Omega(1 + n288_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, mod

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

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s3_0(n502_0), gen_0':s3_0(n502_0)) -> gen_0':s3_0(0), rt in Omega(1 + n502_0)

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

Induction Step:
minus(gen_0':s3_0(+(n502_0, 1)), gen_0':s3_0(+(n502_0, 1))) ->_R^Omega(1)
minus(gen_0':s3_0(n502_0), gen_0':s3_0(n502_0)) ->_IH
gen_0':s3_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)
zero(0') -> true
zero(s(x)) -> false
id(0') -> 0'
id(s(x)) -> s(id(x))
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
mod(x, y) -> if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) -> 0'
if_mod(false, b1, b2, x, y) -> if2(b1, b2, x, y)
if2(true, b2, x, y) -> 0'
if2(false, b2, x, y) -> if3(b2, x, y)
if3(true, x, y) -> mod(minus(x, y), s(y))
if3(false, x, y) -> x

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
zero :: 0':s -> true:false
id :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
mod :: 0':s -> 0':s -> 0':s
if_mod :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s
if2 :: true:false -> true:false -> 0':s -> 0':s -> 0':s
if3 :: true:false -> 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)
id(gen_0':s3_0(n288_0)) -> gen_0':s3_0(n288_0), rt in Omega(1 + n288_0)
minus(gen_0':s3_0(n502_0), gen_0':s3_0(n502_0)) -> gen_0':s3_0(0), rt in Omega(1 + n502_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:
mod