/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 296 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), 57 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 44 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:

   min(x, 0) -> 0
   min(0, y) -> 0
   min(s(x), s(y)) -> s(min(x, y))
   max(x, 0) -> x
   max(0, y) -> y
   max(s(x), s(y)) -> s(max(x, y))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
   gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
   gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
   gcd(x, 0, 0) -> x
   gcd(0, y, 0) -> y
   gcd(0, 0, z) -> z

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:

   min(x, 0') -> 0'
   min(0', y) -> 0'
   min(s(x), s(y)) -> s(min(x, y))
   max(x, 0') -> x
   max(0', y) -> y
   max(s(x), s(y)) -> s(max(x, y))
   -(x, 0') -> x
   -(s(x), s(y)) -> -(x, y)
   gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
   gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
   gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
   gcd(x, 0', 0') -> x
   gcd(0', y, 0') -> y
   gcd(0', 0', z) -> z

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

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

(4)
Obligation:
TRS:
Rules:
min(x, 0') -> 0'
min(0', y) -> 0'
min(s(x), s(y)) -> s(min(x, y))
max(x, 0') -> x
max(0', y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0', 0') -> x
gcd(0', y, 0') -> y
gcd(0', 0', z) -> z

Types:
min :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
gcd :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
min, max, -, gcd

They will be analysed ascendingly in the following order:
min < gcd
max < gcd
- < gcd

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

(6)
Obligation:
TRS:
Rules:
min(x, 0') -> 0'
min(0', y) -> 0'
min(s(x), s(y)) -> s(min(x, y))
max(x, 0') -> x
max(0', y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0', 0') -> x
gcd(0', y, 0') -> y
gcd(0', 0', z) -> z

Types:
min :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
gcd :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
min, max, -, gcd

They will be analysed ascendingly in the following order:
min < gcd
max < gcd
- < gcd

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)

Induction Base:
min(gen_0':s2_0(0), gen_0':s2_0(0)) ->_R^Omega(1)
0'

Induction Step:
min(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(+(n4_0, 1))) ->_R^Omega(1)
s(min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0))) ->_IH
s(gen_0':s2_0(c5_0))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(8)
Complex Obligation (BEST)

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

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
min(x, 0') -> 0'
min(0', y) -> 0'
min(s(x), s(y)) -> s(min(x, y))
max(x, 0') -> x
max(0', y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0', 0') -> x
gcd(0', y, 0') -> y
gcd(0', 0', z) -> z

Types:
min :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
gcd :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
min, max, -, gcd

They will be analysed ascendingly in the following order:
min < gcd
max < gcd
- < gcd

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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
min(x, 0') -> 0'
min(0', y) -> 0'
min(s(x), s(y)) -> s(min(x, y))
max(x, 0') -> x
max(0', y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0', 0') -> x
gcd(0', y, 0') -> y
gcd(0', 0', z) -> z

Types:
min :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
gcd :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
max, -, gcd

They will be analysed ascendingly in the following order:
max < gcd
- < gcd

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
max(gen_0':s2_0(n321_0), gen_0':s2_0(n321_0)) -> gen_0':s2_0(n321_0), rt in Omega(1 + n321_0)

Induction Base:
max(gen_0':s2_0(0), gen_0':s2_0(0)) ->_R^Omega(1)
gen_0':s2_0(0)

Induction Step:
max(gen_0':s2_0(+(n321_0, 1)), gen_0':s2_0(+(n321_0, 1))) ->_R^Omega(1)
s(max(gen_0':s2_0(n321_0), gen_0':s2_0(n321_0))) ->_IH
s(gen_0':s2_0(c322_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:
min(x, 0') -> 0'
min(0', y) -> 0'
min(s(x), s(y)) -> s(min(x, y))
max(x, 0') -> x
max(0', y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0', 0') -> x
gcd(0', y, 0') -> y
gcd(0', 0', z) -> z

Types:
min :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
gcd :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)
max(gen_0':s2_0(n321_0), gen_0':s2_0(n321_0)) -> gen_0':s2_0(n321_0), rt in Omega(1 + n321_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
-, gcd

They will be analysed ascendingly in the following order:
- < gcd

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(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
-(gen_0':s2_0(n724_0), gen_0':s2_0(n724_0)) -> gen_0':s2_0(0), rt in Omega(1 + n724_0)

Induction Base:
-(gen_0':s2_0(0), gen_0':s2_0(0)) ->_R^Omega(1)
gen_0':s2_0(0)

Induction Step:
-(gen_0':s2_0(+(n724_0, 1)), gen_0':s2_0(+(n724_0, 1))) ->_R^Omega(1)
-(gen_0':s2_0(n724_0), gen_0':s2_0(n724_0)) ->_IH
gen_0':s2_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:
min(x, 0') -> 0'
min(0', y) -> 0'
min(s(x), s(y)) -> s(min(x, y))
max(x, 0') -> x
max(0', y) -> y
max(s(x), s(y)) -> s(max(x, y))
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
gcd(x, 0', 0') -> x
gcd(0', y, 0') -> y
gcd(0', 0', z) -> z

Types:
min :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
max :: 0':s -> 0':s -> 0':s
- :: 0':s -> 0':s -> 0':s
gcd :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
min(gen_0':s2_0(n4_0), gen_0':s2_0(n4_0)) -> gen_0':s2_0(n4_0), rt in Omega(1 + n4_0)
max(gen_0':s2_0(n321_0), gen_0':s2_0(n321_0)) -> gen_0':s2_0(n321_0), rt in Omega(1 + n321_0)
-(gen_0':s2_0(n724_0), gen_0':s2_0(n724_0)) -> gen_0':s2_0(0), rt in Omega(1 + n724_0)


Generator Equations:
gen_0':s2_0(0) <=> 0'
gen_0':s2_0(+(x, 1)) <=> s(gen_0':s2_0(x))


The following defined symbols remain to be analysed:
gcd