/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 280 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), 37 ms]
        (14) 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:

   minus(0, y) -> 0
   minus(s(x), y) -> if(gt(s(x), y), x, y)
   if(true, x, y) -> s(minus(x, y))
   if(false, x, y) -> 0
   mod(x, 0) -> 0
   mod(x, s(y)) -> if1(lt(x, s(y)), x, s(y))
   if1(true, x, y) -> x
   if1(false, x, y) -> mod(minus(x, y), y)
   gt(0, y) -> false
   gt(s(x), 0) -> true
   gt(s(x), s(y)) -> gt(x, y)
   lt(x, 0) -> false
   lt(0, s(x)) -> true
   lt(s(x), s(y)) -> lt(x, 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:

   minus(0', y) -> 0'
   minus(s(x), y) -> if(gt(s(x), y), x, y)
   if(true, x, y) -> s(minus(x, y))
   if(false, x, y) -> 0'
   mod(x, 0') -> 0'
   mod(x, s(y)) -> if1(lt(x, s(y)), x, s(y))
   if1(true, x, y) -> x
   if1(false, x, y) -> mod(minus(x, y), y)
   gt(0', y) -> false
   gt(s(x), 0') -> true
   gt(s(x), s(y)) -> gt(x, y)
   lt(x, 0') -> false
   lt(0', s(x)) -> true
   lt(s(x), s(y)) -> lt(x, y)

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

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

(4)
Obligation:
TRS:
Rules:
minus(0', y) -> 0'
minus(s(x), y) -> if(gt(s(x), y), x, y)
if(true, x, y) -> s(minus(x, y))
if(false, x, y) -> 0'
mod(x, 0') -> 0'
mod(x, s(y)) -> if1(lt(x, s(y)), x, s(y))
if1(true, x, y) -> x
if1(false, x, y) -> mod(minus(x, y), y)
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
lt(x, 0') -> false
lt(0', s(x)) -> true
lt(s(x), s(y)) -> lt(x, y)

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
gt :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
mod :: 0':s -> 0':s -> 0':s
if1 :: true:false -> 0':s -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s

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

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

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

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

(6)
Obligation:
TRS:
Rules:
minus(0', y) -> 0'
minus(s(x), y) -> if(gt(s(x), y), x, y)
if(true, x, y) -> s(minus(x, y))
if(false, x, y) -> 0'
mod(x, 0') -> 0'
mod(x, s(y)) -> if1(lt(x, s(y)), x, s(y))
if1(true, x, y) -> x
if1(false, x, y) -> mod(minus(x, y), y)
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
lt(x, 0') -> false
lt(0', s(x)) -> true
lt(s(x), s(y)) -> lt(x, y)

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
gt :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
mod :: 0':s -> 0':s -> 0':s
if1 :: true:false -> 0':s -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
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:
gt, minus, mod, lt

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

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

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

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

Induction Step:
gt(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_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:
minus(0', y) -> 0'
minus(s(x), y) -> if(gt(s(x), y), x, y)
if(true, x, y) -> s(minus(x, y))
if(false, x, y) -> 0'
mod(x, 0') -> 0'
mod(x, s(y)) -> if1(lt(x, s(y)), x, s(y))
if1(true, x, y) -> x
if1(false, x, y) -> mod(minus(x, y), y)
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
lt(x, 0') -> false
lt(0', s(x)) -> true
lt(s(x), s(y)) -> lt(x, y)

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
gt :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
mod :: 0':s -> 0':s -> 0':s
if1 :: true:false -> 0':s -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
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:
gt, minus, mod, lt

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

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
minus(0', y) -> 0'
minus(s(x), y) -> if(gt(s(x), y), x, y)
if(true, x, y) -> s(minus(x, y))
if(false, x, y) -> 0'
mod(x, 0') -> 0'
mod(x, s(y)) -> if1(lt(x, s(y)), x, s(y))
if1(true, x, y) -> x
if1(false, x, y) -> mod(minus(x, y), y)
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
lt(x, 0') -> false
lt(0', s(x)) -> true
lt(s(x), s(y)) -> lt(x, y)

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
gt :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
mod :: 0':s -> 0':s -> 0':s
if1 :: true:false -> 0':s -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> false, 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, mod, lt

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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lt(gen_0':s3_0(n359_0), gen_0':s3_0(n359_0)) -> false, rt in Omega(1 + n359_0)

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

Induction Step:
lt(gen_0':s3_0(+(n359_0, 1)), gen_0':s3_0(+(n359_0, 1))) ->_R^Omega(1)
lt(gen_0':s3_0(n359_0), gen_0':s3_0(n359_0)) ->_IH
false

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

(14)
Obligation:
TRS:
Rules:
minus(0', y) -> 0'
minus(s(x), y) -> if(gt(s(x), y), x, y)
if(true, x, y) -> s(minus(x, y))
if(false, x, y) -> 0'
mod(x, 0') -> 0'
mod(x, s(y)) -> if1(lt(x, s(y)), x, s(y))
if1(true, x, y) -> x
if1(false, x, y) -> mod(minus(x, y), y)
gt(0', y) -> false
gt(s(x), 0') -> true
gt(s(x), s(y)) -> gt(x, y)
lt(x, 0') -> false
lt(0', s(x)) -> true
lt(s(x), s(y)) -> lt(x, y)

Types:
minus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
gt :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
mod :: 0':s -> 0':s -> 0':s
if1 :: true:false -> 0':s -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
gt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> false, rt in Omega(1 + n5_0)
lt(gen_0':s3_0(n359_0), gen_0':s3_0(n359_0)) -> false, rt in Omega(1 + n359_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