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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 262 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), 73 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 82 ms]
        (16) 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:

   f(true, x, y) -> f(and(gt(x, y), gt(y, s(s(0)))), plus(s(0), x), double(y))
   gt(0, v) -> false
   gt(s(u), 0) -> true
   gt(s(u), s(v)) -> gt(u, v)
   and(x, true) -> x
   and(x, false) -> false
   plus(n, 0) -> n
   plus(n, s(m)) -> s(plus(n, m))
   double(0) -> 0
   double(s(x)) -> s(s(double(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:

   f(true, x, y) -> f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y))
   gt(0', v) -> false
   gt(s(u), 0') -> true
   gt(s(u), s(v)) -> gt(u, v)
   and(x, true) -> x
   and(x, false) -> false
   plus(n, 0') -> n
   plus(n, s(m)) -> s(plus(n, m))
   double(0') -> 0'
   double(s(x)) -> s(s(double(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:
f(true, x, y) -> f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)
and(x, true) -> x
and(x, false) -> false
plus(n, 0') -> n
plus(n, s(m)) -> s(plus(n, m))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))

Types:
f :: true:false -> 0':s -> 0':s -> f
true :: true:false
and :: true:false -> true:false -> true:false
gt :: 0':s -> 0':s -> true:false
s :: 0':s -> 0':s
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
f, gt, plus, double

They will be analysed ascendingly in the following order:
gt < f
plus < f
double < f

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(6)
Obligation:
TRS:
Rules:
f(true, x, y) -> f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)
and(x, true) -> x
and(x, false) -> false
plus(n, 0') -> n
plus(n, s(m)) -> s(plus(n, m))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))

Types:
f :: true:false -> 0':s -> 0':s -> f
true :: true:false
and :: true:false -> true:false -> true:false
gt :: 0':s -> 0':s -> true:false
s :: 0':s -> 0':s
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
gt, f, plus, double

They will be analysed ascendingly in the following order:
gt < f
plus < f
double < f

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)

Induction Base:
gt(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1)
false

Induction Step:
gt(gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(+(n6_0, 1))) ->_R^Omega(1)
gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_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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(8)
Complex Obligation (BEST)

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

TRS:
Rules:
f(true, x, y) -> f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)
and(x, true) -> x
and(x, false) -> false
plus(n, 0') -> n
plus(n, s(m)) -> s(plus(n, m))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))

Types:
f :: true:false -> 0':s -> 0':s -> f
true :: true:false
and :: true:false -> true:false -> true:false
gt :: 0':s -> 0':s -> true:false
s :: 0':s -> 0':s
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
gt, f, plus, double

They will be analysed ascendingly in the following order:
gt < f
plus < f
double < f

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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:
f(true, x, y) -> f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)
and(x, true) -> x
and(x, false) -> false
plus(n, 0') -> n
plus(n, s(m)) -> s(plus(n, m))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))

Types:
f :: true:false -> 0':s -> 0':s -> f
true :: true:false
and :: true:false -> true:false -> true:false
gt :: 0':s -> 0':s -> true:false
s :: 0':s -> 0':s
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
plus, f, double

They will be analysed ascendingly in the following order:
plus < f
double < f

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s4_0(a), gen_0':s4_0(n253_0)) -> gen_0':s4_0(+(n253_0, a)), rt in Omega(1 + n253_0)

Induction Base:
plus(gen_0':s4_0(a), gen_0':s4_0(0)) ->_R^Omega(1)
gen_0':s4_0(a)

Induction Step:
plus(gen_0':s4_0(a), gen_0':s4_0(+(n253_0, 1))) ->_R^Omega(1)
s(plus(gen_0':s4_0(a), gen_0':s4_0(n253_0))) ->_IH
s(gen_0':s4_0(+(a, c254_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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(14)
Obligation:
TRS:
Rules:
f(true, x, y) -> f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)
and(x, true) -> x
and(x, false) -> false
plus(n, 0') -> n
plus(n, s(m)) -> s(plus(n, m))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))

Types:
f :: true:false -> 0':s -> 0':s -> f
true :: true:false
and :: true:false -> true:false -> true:false
gt :: 0':s -> 0':s -> true:false
s :: 0':s -> 0':s
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
plus(gen_0':s4_0(a), gen_0':s4_0(n253_0)) -> gen_0':s4_0(+(n253_0, a)), rt in Omega(1 + n253_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
double, f

They will be analysed ascendingly in the following order:
double < f

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(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
double(gen_0':s4_0(n778_0)) -> gen_0':s4_0(*(2, n778_0)), rt in Omega(1 + n778_0)

Induction Base:
double(gen_0':s4_0(0)) ->_R^Omega(1)
0'

Induction Step:
double(gen_0':s4_0(+(n778_0, 1))) ->_R^Omega(1)
s(s(double(gen_0':s4_0(n778_0)))) ->_IH
s(s(gen_0':s4_0(*(2, c779_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:
f(true, x, y) -> f(and(gt(x, y), gt(y, s(s(0')))), plus(s(0'), x), double(y))
gt(0', v) -> false
gt(s(u), 0') -> true
gt(s(u), s(v)) -> gt(u, v)
and(x, true) -> x
and(x, false) -> false
plus(n, 0') -> n
plus(n, s(m)) -> s(plus(n, m))
double(0') -> 0'
double(s(x)) -> s(s(double(x)))

Types:
f :: true:false -> 0':s -> 0':s -> f
true :: true:false
and :: true:false -> true:false -> true:false
gt :: 0':s -> 0':s -> true:false
s :: 0':s -> 0':s
0' :: 0':s
plus :: 0':s -> 0':s -> 0':s
double :: 0':s -> 0':s
false :: true:false
hole_f1_0 :: f
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
gt(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
plus(gen_0':s4_0(a), gen_0':s4_0(n253_0)) -> gen_0':s4_0(+(n253_0, a)), rt in Omega(1 + n253_0)
double(gen_0':s4_0(n778_0)) -> gen_0':s4_0(*(2, n778_0)), rt in Omega(1 + n778_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
f