/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^2), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2, 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
        (13) RewriteLemmaProof [LOWER BOUND(ID), 18 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 48 ms]
        (16) BEST
            (17) proven lower bound
                (18) LowerBoundPropagationProof [FINISHED, 0 ms]
                (19) BOUNDS(n^2, INF)
            (20) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   lt(0, s(x)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(x, y))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   fac(x) -> loop(x, s(0), s(0))
   loop(x, c, y) -> if(lt(x, c), x, c, y)
   if(false, x, c, y) -> loop(x, s(c), times(y, s(c)))
   if(true, x, c, y) -> 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^2, INF).


The TRS R consists of the following rules:

   lt(0', s(x)) -> true
   lt(x, 0') -> false
   lt(s(x), s(y)) -> lt(x, y)
   times(0', y) -> 0'
   times(s(x), y) -> plus(y, times(x, y))
   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   fac(x) -> loop(x, s(0'), s(0'))
   loop(x, c, y) -> if(lt(x, c), x, c, y)
   if(false, x, c, y) -> loop(x, s(c), times(y, s(c)))
   if(true, x, c, y) -> y

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

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

(4)
Obligation:
TRS:
Rules:
lt(0', s(x)) -> true
lt(x, 0') -> false
lt(s(x), s(y)) -> lt(x, y)
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
fac(x) -> loop(x, s(0'), s(0'))
loop(x, c, y) -> if(lt(x, c), x, c, y)
if(false, x, c, y) -> loop(x, s(c), times(y, s(c)))
if(true, x, c, y) -> y

Types:
lt :: 0':s -> 0':s -> true:false
0' :: 0':s
s :: 0':s -> 0':s
true :: true:false
false :: true:false
times :: 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
fac :: 0':s -> 0':s
loop :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 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:
lt, times, plus, loop

They will be analysed ascendingly in the following order:
lt < loop
plus < times
times < loop

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

(6)
Obligation:
TRS:
Rules:
lt(0', s(x)) -> true
lt(x, 0') -> false
lt(s(x), s(y)) -> lt(x, y)
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
fac(x) -> loop(x, s(0'), s(0'))
loop(x, c, y) -> if(lt(x, c), x, c, y)
if(false, x, c, y) -> loop(x, s(c), times(y, s(c)))
if(true, x, c, y) -> y

Types:
lt :: 0':s -> 0':s -> true:false
0' :: 0':s
s :: 0':s -> 0':s
true :: true:false
false :: true:false
times :: 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
fac :: 0':s -> 0':s
loop :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 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:
lt, times, plus, loop

They will be analysed ascendingly in the following order:
lt < loop
plus < times
times < loop

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

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

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

Induction Step:
lt(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1)
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, 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:
lt(0', s(x)) -> true
lt(x, 0') -> false
lt(s(x), s(y)) -> lt(x, y)
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
fac(x) -> loop(x, s(0'), s(0'))
loop(x, c, y) -> if(lt(x, c), x, c, y)
if(false, x, c, y) -> loop(x, s(c), times(y, s(c)))
if(true, x, c, y) -> y

Types:
lt :: 0':s -> 0':s -> true:false
0' :: 0':s
s :: 0':s -> 0':s
true :: true:false
false :: true:false
times :: 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
fac :: 0':s -> 0':s
loop :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 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:
lt, times, plus, loop

They will be analysed ascendingly in the following order:
lt < loop
plus < times
times < loop

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
lt(0', s(x)) -> true
lt(x, 0') -> false
lt(s(x), s(y)) -> lt(x, y)
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
fac(x) -> loop(x, s(0'), s(0'))
loop(x, c, y) -> if(lt(x, c), x, c, y)
if(false, x, c, y) -> loop(x, s(c), times(y, s(c)))
if(true, x, c, y) -> y

Types:
lt :: 0':s -> 0':s -> true:false
0' :: 0':s
s :: 0':s -> 0':s
true :: true:false
false :: true:false
times :: 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
fac :: 0':s -> 0':s
loop :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 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:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, 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:
plus, times, loop

They will be analysed ascendingly in the following order:
plus < times
times < loop

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s3_0(n257_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n257_0, b)), rt in Omega(1 + n257_0)

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

Induction Step:
plus(gen_0':s3_0(+(n257_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
s(plus(gen_0':s3_0(n257_0), gen_0':s3_0(b))) ->_IH
s(gen_0':s3_0(+(b, c258_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:
lt(0', s(x)) -> true
lt(x, 0') -> false
lt(s(x), s(y)) -> lt(x, y)
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
fac(x) -> loop(x, s(0'), s(0'))
loop(x, c, y) -> if(lt(x, c), x, c, y)
if(false, x, c, y) -> loop(x, s(c), times(y, s(c)))
if(true, x, c, y) -> y

Types:
lt :: 0':s -> 0':s -> true:false
0' :: 0':s
s :: 0':s -> 0':s
true :: true:false
false :: true:false
times :: 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
fac :: 0':s -> 0':s
loop :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 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:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) -> true, rt in Omega(1 + n5_0)
plus(gen_0':s3_0(n257_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n257_0, b)), rt in Omega(1 + n257_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:
times, loop

They will be analysed ascendingly in the following order:
times < loop

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
times(gen_0':s3_0(n800_0), gen_0':s3_0(b)) -> gen_0':s3_0(*(n800_0, b)), rt in Omega(1 + b*n800_0 + n800_0)

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

Induction Step:
times(gen_0':s3_0(+(n800_0, 1)), gen_0':s3_0(b)) ->_R^Omega(1)
plus(gen_0':s3_0(b), times(gen_0':s3_0(n800_0), gen_0':s3_0(b))) ->_IH
plus(gen_0':s3_0(b), gen_0':s3_0(*(c801_0, b))) ->_L^Omega(1 + b)
gen_0':s3_0(+(b, *(n800_0, b)))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(16)
Complex Obligation (BEST)

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

(17)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
lt(0', s(x)) -> true
lt(x, 0') -> false
lt(s(x), s(y)) -> lt(x, y)
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
fac(x) -> loop(x, s(0'), s(0'))
loop(x, c, y) -> if(lt(x, c), x, c, y)
if(false, x, c, y) -> loop(x, s(c), times(y, s(c)))
if(true, x, c, y) -> y

Types:
lt :: 0':s -> 0':s -> true:false
0' :: 0':s
s :: 0':s -> 0':s
true :: true:false
false :: true:false
times :: 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
fac :: 0':s -> 0':s
loop :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 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:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) -> true, rt in Omega(1 + n5_0)
plus(gen_0':s3_0(n257_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n257_0, b)), rt in Omega(1 + n257_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:
times, loop

They will be analysed ascendingly in the following order:
times < loop

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

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

(19)
BOUNDS(n^2, INF)

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

(20)
Obligation:
TRS:
Rules:
lt(0', s(x)) -> true
lt(x, 0') -> false
lt(s(x), s(y)) -> lt(x, y)
times(0', y) -> 0'
times(s(x), y) -> plus(y, times(x, y))
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
fac(x) -> loop(x, s(0'), s(0'))
loop(x, c, y) -> if(lt(x, c), x, c, y)
if(false, x, c, y) -> loop(x, s(c), times(y, s(c)))
if(true, x, c, y) -> y

Types:
lt :: 0':s -> 0':s -> true:false
0' :: 0':s
s :: 0':s -> 0':s
true :: true:false
false :: true:false
times :: 0':s -> 0':s -> 0':s
plus :: 0':s -> 0':s -> 0':s
fac :: 0':s -> 0':s
loop :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 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:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(+(1, n5_0))) -> true, rt in Omega(1 + n5_0)
plus(gen_0':s3_0(n257_0), gen_0':s3_0(b)) -> gen_0':s3_0(+(n257_0, b)), rt in Omega(1 + n257_0)
times(gen_0':s3_0(n800_0), gen_0':s3_0(b)) -> gen_0':s3_0(*(n800_0, b)), rt in Omega(1 + b*n800_0 + n800_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:
loop