/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 933 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), 0 ms]
        (14) typed CpxTrs


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(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(X, s(Y)) -> pred(minus(X, Y))
   minus(X, 0) -> X
   pred(s(X)) -> X
   le(s(X), s(Y)) -> le(X, Y)
   le(s(X), 0) -> false
   le(0, Y) -> true
   gcd(0, Y) -> 0
   gcd(s(X), 0) -> s(X)
   gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
   if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
   if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))

S is empty.
Rewrite Strategy: FULL
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(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(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(X, s(Y)) -> pred(minus(X, Y))
   minus(X, 0') -> X
   pred(s(X)) -> X
   le(s(X), s(Y)) -> le(X, Y)
   le(s(X), 0') -> false
   le(0', Y) -> true
   gcd(0', Y) -> 0'
   gcd(s(X), 0') -> s(X)
   gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
   if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
   if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))

S is empty.
Rewrite Strategy: FULL
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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(4)
Obligation:
TRS:
Rules:
minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0') -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0') -> false
le(0', Y) -> true
gcd(0', Y) -> 0'
gcd(s(X), 0') -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))

Types:
minus :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
pred :: s:0' -> s:0'
0' :: s:0'
le :: s:0' -> s:0' -> false:true
false :: false:true
true :: false:true
gcd :: s:0' -> s:0' -> s:0'
if :: false:true -> s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
gen_s:0'3_0 :: Nat -> s:0'

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

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

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(6)
Obligation:
TRS:
Rules:
minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0') -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0') -> false
le(0', Y) -> true
gcd(0', Y) -> 0'
gcd(s(X), 0') -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))

Types:
minus :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
pred :: s:0' -> s:0'
0' :: s:0'
le :: s:0' -> s:0' -> false:true
false :: false:true
true :: false:true
gcd :: s:0' -> s:0' -> s:0'
if :: false:true -> s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
gen_s:0'3_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


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

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

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)

Induction Base:
minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, 0)))

Induction Step:
minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, +(n5_0, 1)))) ->_R^Omega(1)
pred(minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, n5_0)))) ->_IH
pred(*4_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)

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

TRS:
Rules:
minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0') -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0') -> false
le(0', Y) -> true
gcd(0', Y) -> 0'
gcd(s(X), 0') -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))

Types:
minus :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
pred :: s:0' -> s:0'
0' :: s:0'
le :: s:0' -> s:0' -> false:true
false :: false:true
true :: false:true
gcd :: s:0' -> s:0' -> s:0'
if :: false:true -> s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
gen_s:0'3_0 :: Nat -> s:0'


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


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

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

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

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(12)
Obligation:
TRS:
Rules:
minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0') -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0') -> false
le(0', Y) -> true
gcd(0', Y) -> 0'
gcd(s(X), 0') -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))

Types:
minus :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
pred :: s:0' -> s:0'
0' :: s:0'
le :: s:0' -> s:0' -> false:true
false :: false:true
true :: false:true
gcd :: s:0' -> s:0' -> s:0'
if :: false:true -> s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


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

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

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_s:0'3_0(+(1, n2357_0)), gen_s:0'3_0(n2357_0)) -> false, rt in Omega(1 + n2357_0)

Induction Base:
le(gen_s:0'3_0(+(1, 0)), gen_s:0'3_0(0)) ->_R^Omega(1)
false

Induction Step:
le(gen_s:0'3_0(+(1, +(n2357_0, 1))), gen_s:0'3_0(+(n2357_0, 1))) ->_R^Omega(1)
le(gen_s:0'3_0(+(1, n2357_0)), gen_s:0'3_0(n2357_0)) ->_IH
false

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(14)
Obligation:
TRS:
Rules:
minus(X, s(Y)) -> pred(minus(X, Y))
minus(X, 0') -> X
pred(s(X)) -> X
le(s(X), s(Y)) -> le(X, Y)
le(s(X), 0') -> false
le(0', Y) -> true
gcd(0', Y) -> 0'
gcd(s(X), 0') -> s(X)
gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
if(false, s(X), s(Y)) -> gcd(minus(Y, X), s(X))

Types:
minus :: s:0' -> s:0' -> s:0'
s :: s:0' -> s:0'
pred :: s:0' -> s:0'
0' :: s:0'
le :: s:0' -> s:0' -> false:true
false :: false:true
true :: false:true
gcd :: s:0' -> s:0' -> s:0'
if :: false:true -> s:0' -> s:0' -> s:0'
hole_s:0'1_0 :: s:0'
hole_false:true2_0 :: false:true
gen_s:0'3_0 :: Nat -> s:0'


Lemmas:
minus(gen_s:0'3_0(a), gen_s:0'3_0(+(1, n5_0))) -> *4_0, rt in Omega(n5_0)
le(gen_s:0'3_0(+(1, n2357_0)), gen_s:0'3_0(n2357_0)) -> false, rt in Omega(1 + n2357_0)


Generator Equations:
gen_s:0'3_0(0) <=> 0'
gen_s:0'3_0(+(x, 1)) <=> s(gen_s:0'3_0(x))


The following defined symbols remain to be analysed:
gcd