/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) SlicingProof [LOWER BOUND(ID), 0 ms]
(4) CpxTRS
(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) typed CpxTrs
(7) OrderProof [LOWER BOUND(ID), 0 ms]
(8) typed CpxTrs
(9) RewriteLemmaProof [LOWER BOUND(ID), 327 ms]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 63 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 46 ms]
        (18) 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:

   gcd(x, y) -> gcd2(x, y, 0)
   gcd2(x, y, i) -> if1(le(x, 0), le(y, 0), le(x, y), le(y, x), x, y, inc(i))
   if1(true, b1, b2, b3, x, y, i) -> pair(result(y), neededIterations(i))
   if1(false, b1, b2, b3, x, y, i) -> if2(b1, b2, b3, x, y, i)
   if2(true, b2, b3, x, y, i) -> pair(result(x), neededIterations(i))
   if2(false, b2, b3, x, y, i) -> if3(b2, b3, x, y, i)
   if3(false, b3, x, y, i) -> gcd2(minus(x, y), y, i)
   if3(true, b3, x, y, i) -> if4(b3, x, y, i)
   if4(false, x, y, i) -> gcd2(x, minus(y, x), i)
   if4(true, x, y, i) -> pair(result(x), neededIterations(i))
   inc(0) -> 0
   inc(s(i)) -> s(inc(i))
   le(s(x), 0) -> false
   le(0, y) -> true
   le(s(x), s(y)) -> le(x, y)
   minus(x, 0) -> x
   minus(0, y) -> 0
   minus(s(x), s(y)) -> minus(x, y)
   a -> b
   a -> c

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:

   gcd(x, y) -> gcd2(x, y, 0')
   gcd2(x, y, i) -> if1(le(x, 0'), le(y, 0'), le(x, y), le(y, x), x, y, inc(i))
   if1(true, b1, b2, b3, x, y, i) -> pair(result(y), neededIterations(i))
   if1(false, b1, b2, b3, x, y, i) -> if2(b1, b2, b3, x, y, i)
   if2(true, b2, b3, x, y, i) -> pair(result(x), neededIterations(i))
   if2(false, b2, b3, x, y, i) -> if3(b2, b3, x, y, i)
   if3(false, b3, x, y, i) -> gcd2(minus(x, y), y, i)
   if3(true, b3, x, y, i) -> if4(b3, x, y, i)
   if4(false, x, y, i) -> gcd2(x, minus(y, x), i)
   if4(true, x, y, i) -> pair(result(x), neededIterations(i))
   inc(0') -> 0'
   inc(s(i)) -> s(inc(i))
   le(s(x), 0') -> false
   le(0', y) -> true
   le(s(x), s(y)) -> le(x, y)
   minus(x, 0') -> x
   minus(0', y) -> 0'
   minus(s(x), s(y)) -> minus(x, y)
   a -> b
   a -> c

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

(3) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
pair/0
pair/1
result/0
neededIterations/0

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

(4)
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:

   gcd(x, y) -> gcd2(x, y, 0')
   gcd2(x, y, i) -> if1(le(x, 0'), le(y, 0'), le(x, y), le(y, x), x, y, inc(i))
   if1(true, b1, b2, b3, x, y, i) -> pair
   if1(false, b1, b2, b3, x, y, i) -> if2(b1, b2, b3, x, y, i)
   if2(true, b2, b3, x, y, i) -> pair
   if2(false, b2, b3, x, y, i) -> if3(b2, b3, x, y, i)
   if3(false, b3, x, y, i) -> gcd2(minus(x, y), y, i)
   if3(true, b3, x, y, i) -> if4(b3, x, y, i)
   if4(false, x, y, i) -> gcd2(x, minus(y, x), i)
   if4(true, x, y, i) -> pair
   inc(0') -> 0'
   inc(s(i)) -> s(inc(i))
   le(s(x), 0') -> false
   le(0', y) -> true
   le(s(x), s(y)) -> le(x, y)
   minus(x, 0') -> x
   minus(0', y) -> 0'
   minus(s(x), s(y)) -> minus(x, y)
   a -> b
   a -> c

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

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

(6)
Obligation:
TRS:
Rules:
gcd(x, y) -> gcd2(x, y, 0')
gcd2(x, y, i) -> if1(le(x, 0'), le(y, 0'), le(x, y), le(y, x), x, y, inc(i))
if1(true, b1, b2, b3, x, y, i) -> pair
if1(false, b1, b2, b3, x, y, i) -> if2(b1, b2, b3, x, y, i)
if2(true, b2, b3, x, y, i) -> pair
if2(false, b2, b3, x, y, i) -> if3(b2, b3, x, y, i)
if3(false, b3, x, y, i) -> gcd2(minus(x, y), y, i)
if3(true, b3, x, y, i) -> if4(b3, x, y, i)
if4(false, x, y, i) -> gcd2(x, minus(y, x), i)
if4(true, x, y, i) -> pair
inc(0') -> 0'
inc(s(i)) -> s(inc(i))
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(0', y) -> 0'
minus(s(x), s(y)) -> minus(x, y)
a -> b
a -> c

Types:
gcd :: 0':s -> 0':s -> pair
gcd2 :: 0':s -> 0':s -> 0':s -> pair
0' :: 0':s
if1 :: true:false -> true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s -> pair
le :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
true :: true:false
pair :: pair
false :: true:false
if2 :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s -> pair
if3 :: true:false -> true:false -> 0':s -> 0':s -> 0':s -> pair
minus :: 0':s -> 0':s -> 0':s
if4 :: true:false -> 0':s -> 0':s -> 0':s -> pair
s :: 0':s -> 0':s
a :: b:c
b :: b:c
c :: b:c
hole_pair1_0 :: pair
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_b:c4_0 :: b:c
gen_0':s5_0 :: Nat -> 0':s

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

(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
gcd2, le, inc, minus

They will be analysed ascendingly in the following order:
le < gcd2
inc < gcd2
minus < gcd2

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

(8)
Obligation:
TRS:
Rules:
gcd(x, y) -> gcd2(x, y, 0')
gcd2(x, y, i) -> if1(le(x, 0'), le(y, 0'), le(x, y), le(y, x), x, y, inc(i))
if1(true, b1, b2, b3, x, y, i) -> pair
if1(false, b1, b2, b3, x, y, i) -> if2(b1, b2, b3, x, y, i)
if2(true, b2, b3, x, y, i) -> pair
if2(false, b2, b3, x, y, i) -> if3(b2, b3, x, y, i)
if3(false, b3, x, y, i) -> gcd2(minus(x, y), y, i)
if3(true, b3, x, y, i) -> if4(b3, x, y, i)
if4(false, x, y, i) -> gcd2(x, minus(y, x), i)
if4(true, x, y, i) -> pair
inc(0') -> 0'
inc(s(i)) -> s(inc(i))
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(0', y) -> 0'
minus(s(x), s(y)) -> minus(x, y)
a -> b
a -> c

Types:
gcd :: 0':s -> 0':s -> pair
gcd2 :: 0':s -> 0':s -> 0':s -> pair
0' :: 0':s
if1 :: true:false -> true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s -> pair
le :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
true :: true:false
pair :: pair
false :: true:false
if2 :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s -> pair
if3 :: true:false -> true:false -> 0':s -> 0':s -> 0':s -> pair
minus :: 0':s -> 0':s -> 0':s
if4 :: true:false -> 0':s -> 0':s -> 0':s -> pair
s :: 0':s -> 0':s
a :: b:c
b :: b:c
c :: b:c
hole_pair1_0 :: pair
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_b:c4_0 :: b:c
gen_0':s5_0 :: Nat -> 0':s


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


The following defined symbols remain to be analysed:
le, gcd2, inc, minus

They will be analysed ascendingly in the following order:
le < gcd2
inc < gcd2
minus < gcd2

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

(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s5_0(+(1, n7_0)), gen_0':s5_0(n7_0)) -> false, rt in Omega(1 + n7_0)

Induction Base:
le(gen_0':s5_0(+(1, 0)), gen_0':s5_0(0)) ->_R^Omega(1)
false

Induction Step:
le(gen_0':s5_0(+(1, +(n7_0, 1))), gen_0':s5_0(+(n7_0, 1))) ->_R^Omega(1)
le(gen_0':s5_0(+(1, n7_0)), gen_0':s5_0(n7_0)) ->_IH
false

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

(10)
Complex Obligation (BEST)

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

(11)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
gcd(x, y) -> gcd2(x, y, 0')
gcd2(x, y, i) -> if1(le(x, 0'), le(y, 0'), le(x, y), le(y, x), x, y, inc(i))
if1(true, b1, b2, b3, x, y, i) -> pair
if1(false, b1, b2, b3, x, y, i) -> if2(b1, b2, b3, x, y, i)
if2(true, b2, b3, x, y, i) -> pair
if2(false, b2, b3, x, y, i) -> if3(b2, b3, x, y, i)
if3(false, b3, x, y, i) -> gcd2(minus(x, y), y, i)
if3(true, b3, x, y, i) -> if4(b3, x, y, i)
if4(false, x, y, i) -> gcd2(x, minus(y, x), i)
if4(true, x, y, i) -> pair
inc(0') -> 0'
inc(s(i)) -> s(inc(i))
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(0', y) -> 0'
minus(s(x), s(y)) -> minus(x, y)
a -> b
a -> c

Types:
gcd :: 0':s -> 0':s -> pair
gcd2 :: 0':s -> 0':s -> 0':s -> pair
0' :: 0':s
if1 :: true:false -> true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s -> pair
le :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
true :: true:false
pair :: pair
false :: true:false
if2 :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s -> pair
if3 :: true:false -> true:false -> 0':s -> 0':s -> 0':s -> pair
minus :: 0':s -> 0':s -> 0':s
if4 :: true:false -> 0':s -> 0':s -> 0':s -> pair
s :: 0':s -> 0':s
a :: b:c
b :: b:c
c :: b:c
hole_pair1_0 :: pair
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_b:c4_0 :: b:c
gen_0':s5_0 :: Nat -> 0':s


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


The following defined symbols remain to be analysed:
le, gcd2, inc, minus

They will be analysed ascendingly in the following order:
le < gcd2
inc < gcd2
minus < gcd2

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

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

(13)
BOUNDS(n^1, INF)

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

(14)
Obligation:
TRS:
Rules:
gcd(x, y) -> gcd2(x, y, 0')
gcd2(x, y, i) -> if1(le(x, 0'), le(y, 0'), le(x, y), le(y, x), x, y, inc(i))
if1(true, b1, b2, b3, x, y, i) -> pair
if1(false, b1, b2, b3, x, y, i) -> if2(b1, b2, b3, x, y, i)
if2(true, b2, b3, x, y, i) -> pair
if2(false, b2, b3, x, y, i) -> if3(b2, b3, x, y, i)
if3(false, b3, x, y, i) -> gcd2(minus(x, y), y, i)
if3(true, b3, x, y, i) -> if4(b3, x, y, i)
if4(false, x, y, i) -> gcd2(x, minus(y, x), i)
if4(true, x, y, i) -> pair
inc(0') -> 0'
inc(s(i)) -> s(inc(i))
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(0', y) -> 0'
minus(s(x), s(y)) -> minus(x, y)
a -> b
a -> c

Types:
gcd :: 0':s -> 0':s -> pair
gcd2 :: 0':s -> 0':s -> 0':s -> pair
0' :: 0':s
if1 :: true:false -> true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s -> pair
le :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
true :: true:false
pair :: pair
false :: true:false
if2 :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s -> pair
if3 :: true:false -> true:false -> 0':s -> 0':s -> 0':s -> pair
minus :: 0':s -> 0':s -> 0':s
if4 :: true:false -> 0':s -> 0':s -> 0':s -> pair
s :: 0':s -> 0':s
a :: b:c
b :: b:c
c :: b:c
hole_pair1_0 :: pair
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_b:c4_0 :: b:c
gen_0':s5_0 :: Nat -> 0':s


Lemmas:
le(gen_0':s5_0(+(1, n7_0)), gen_0':s5_0(n7_0)) -> false, rt in Omega(1 + n7_0)


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


The following defined symbols remain to be analysed:
inc, gcd2, minus

They will be analysed ascendingly in the following order:
inc < gcd2
minus < gcd2

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
inc(gen_0':s5_0(n314_0)) -> gen_0':s5_0(n314_0), rt in Omega(1 + n314_0)

Induction Base:
inc(gen_0':s5_0(0)) ->_R^Omega(1)
0'

Induction Step:
inc(gen_0':s5_0(+(n314_0, 1))) ->_R^Omega(1)
s(inc(gen_0':s5_0(n314_0))) ->_IH
s(gen_0':s5_0(c315_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:
gcd(x, y) -> gcd2(x, y, 0')
gcd2(x, y, i) -> if1(le(x, 0'), le(y, 0'), le(x, y), le(y, x), x, y, inc(i))
if1(true, b1, b2, b3, x, y, i) -> pair
if1(false, b1, b2, b3, x, y, i) -> if2(b1, b2, b3, x, y, i)
if2(true, b2, b3, x, y, i) -> pair
if2(false, b2, b3, x, y, i) -> if3(b2, b3, x, y, i)
if3(false, b3, x, y, i) -> gcd2(minus(x, y), y, i)
if3(true, b3, x, y, i) -> if4(b3, x, y, i)
if4(false, x, y, i) -> gcd2(x, minus(y, x), i)
if4(true, x, y, i) -> pair
inc(0') -> 0'
inc(s(i)) -> s(inc(i))
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(0', y) -> 0'
minus(s(x), s(y)) -> minus(x, y)
a -> b
a -> c

Types:
gcd :: 0':s -> 0':s -> pair
gcd2 :: 0':s -> 0':s -> 0':s -> pair
0' :: 0':s
if1 :: true:false -> true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s -> pair
le :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
true :: true:false
pair :: pair
false :: true:false
if2 :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s -> pair
if3 :: true:false -> true:false -> 0':s -> 0':s -> 0':s -> pair
minus :: 0':s -> 0':s -> 0':s
if4 :: true:false -> 0':s -> 0':s -> 0':s -> pair
s :: 0':s -> 0':s
a :: b:c
b :: b:c
c :: b:c
hole_pair1_0 :: pair
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_b:c4_0 :: b:c
gen_0':s5_0 :: Nat -> 0':s


Lemmas:
le(gen_0':s5_0(+(1, n7_0)), gen_0':s5_0(n7_0)) -> false, rt in Omega(1 + n7_0)
inc(gen_0':s5_0(n314_0)) -> gen_0':s5_0(n314_0), rt in Omega(1 + n314_0)


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


The following defined symbols remain to be analysed:
minus, gcd2

They will be analysed ascendingly in the following order:
minus < gcd2

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s5_0(n544_0), gen_0':s5_0(n544_0)) -> gen_0':s5_0(0), rt in Omega(1 + n544_0)

Induction Base:
minus(gen_0':s5_0(0), gen_0':s5_0(0)) ->_R^Omega(1)
gen_0':s5_0(0)

Induction Step:
minus(gen_0':s5_0(+(n544_0, 1)), gen_0':s5_0(+(n544_0, 1))) ->_R^Omega(1)
minus(gen_0':s5_0(n544_0), gen_0':s5_0(n544_0)) ->_IH
gen_0':s5_0(0)

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

(18)
Obligation:
TRS:
Rules:
gcd(x, y) -> gcd2(x, y, 0')
gcd2(x, y, i) -> if1(le(x, 0'), le(y, 0'), le(x, y), le(y, x), x, y, inc(i))
if1(true, b1, b2, b3, x, y, i) -> pair
if1(false, b1, b2, b3, x, y, i) -> if2(b1, b2, b3, x, y, i)
if2(true, b2, b3, x, y, i) -> pair
if2(false, b2, b3, x, y, i) -> if3(b2, b3, x, y, i)
if3(false, b3, x, y, i) -> gcd2(minus(x, y), y, i)
if3(true, b3, x, y, i) -> if4(b3, x, y, i)
if4(false, x, y, i) -> gcd2(x, minus(y, x), i)
if4(true, x, y, i) -> pair
inc(0') -> 0'
inc(s(i)) -> s(inc(i))
le(s(x), 0') -> false
le(0', y) -> true
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(0', y) -> 0'
minus(s(x), s(y)) -> minus(x, y)
a -> b
a -> c

Types:
gcd :: 0':s -> 0':s -> pair
gcd2 :: 0':s -> 0':s -> 0':s -> pair
0' :: 0':s
if1 :: true:false -> true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s -> pair
le :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
true :: true:false
pair :: pair
false :: true:false
if2 :: true:false -> true:false -> true:false -> 0':s -> 0':s -> 0':s -> pair
if3 :: true:false -> true:false -> 0':s -> 0':s -> 0':s -> pair
minus :: 0':s -> 0':s -> 0':s
if4 :: true:false -> 0':s -> 0':s -> 0':s -> pair
s :: 0':s -> 0':s
a :: b:c
b :: b:c
c :: b:c
hole_pair1_0 :: pair
hole_0':s2_0 :: 0':s
hole_true:false3_0 :: true:false
hole_b:c4_0 :: b:c
gen_0':s5_0 :: Nat -> 0':s


Lemmas:
le(gen_0':s5_0(+(1, n7_0)), gen_0':s5_0(n7_0)) -> false, rt in Omega(1 + n7_0)
inc(gen_0':s5_0(n314_0)) -> gen_0':s5_0(n314_0), rt in Omega(1 + n314_0)
minus(gen_0':s5_0(n544_0), gen_0':s5_0(n544_0)) -> gen_0':s5_0(0), rt in Omega(1 + n544_0)


Generator Equations:
gen_0':s5_0(0) <=> 0'
gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x))


The following defined symbols remain to be analysed:
gcd2