/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^1), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^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), 232 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) 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:

   cond1(true, x, y) -> cond2(gr(x, y), x, y)
   cond2(true, x, y) -> cond1(gr0(x), y, y)
   cond2(false, x, y) -> cond1(gr0(x), p(x), y)
   gr(0, x) -> false
   gr(s(x), 0) -> true
   gr(s(x), s(y)) -> gr(x, y)
   gr0(0) -> false
   gr0(s(x)) -> true
   p(0) -> 0
   p(s(x)) -> 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:

   cond1(true, x, y) -> cond2(gr(x, y), x, y)
   cond2(true, x, y) -> cond1(gr0(x), y, y)
   cond2(false, x, y) -> cond1(gr0(x), p(x), y)
   gr(0', x) -> false
   gr(s(x), 0') -> true
   gr(s(x), s(y)) -> gr(x, y)
   gr0(0') -> false
   gr0(s(x)) -> true
   p(0') -> 0'
   p(s(x)) -> 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:
cond1(true, x, y) -> cond2(gr(x, y), x, y)
cond2(true, x, y) -> cond1(gr0(x), y, y)
cond2(false, x, y) -> cond1(gr0(x), p(x), y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
gr0(0') -> false
gr0(s(x)) -> true
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
gr :: 0':s -> 0':s -> true:false
gr0 :: 0':s -> true:false
false :: true:false
p :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
hole_cond1:cond21_3 :: cond1:cond2
hole_true:false2_3 :: true:false
hole_0':s3_3 :: 0':s
gen_0':s4_3 :: Nat -> 0':s

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
cond1, cond2, gr

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1

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(6)
Obligation:
TRS:
Rules:
cond1(true, x, y) -> cond2(gr(x, y), x, y)
cond2(true, x, y) -> cond1(gr0(x), y, y)
cond2(false, x, y) -> cond1(gr0(x), p(x), y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
gr0(0') -> false
gr0(s(x)) -> true
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
gr :: 0':s -> 0':s -> true:false
gr0 :: 0':s -> true:false
false :: true:false
p :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
hole_cond1:cond21_3 :: cond1:cond2
hole_true:false2_3 :: true:false
hole_0':s3_3 :: 0':s
gen_0':s4_3 :: Nat -> 0':s


Generator Equations:
gen_0':s4_3(0) <=> 0'
gen_0':s4_3(+(x, 1)) <=> s(gen_0':s4_3(x))


The following defined symbols remain to be analysed:
gr, cond1, cond2

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gr(gen_0':s4_3(n6_3), gen_0':s4_3(n6_3)) -> false, rt in Omega(1 + n6_3)

Induction Base:
gr(gen_0':s4_3(0), gen_0':s4_3(0)) ->_R^Omega(1)
false

Induction Step:
gr(gen_0':s4_3(+(n6_3, 1)), gen_0':s4_3(+(n6_3, 1))) ->_R^Omega(1)
gr(gen_0':s4_3(n6_3), gen_0':s4_3(n6_3)) ->_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:
cond1(true, x, y) -> cond2(gr(x, y), x, y)
cond2(true, x, y) -> cond1(gr0(x), y, y)
cond2(false, x, y) -> cond1(gr0(x), p(x), y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
gr0(0') -> false
gr0(s(x)) -> true
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
gr :: 0':s -> 0':s -> true:false
gr0 :: 0':s -> true:false
false :: true:false
p :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
hole_cond1:cond21_3 :: cond1:cond2
hole_true:false2_3 :: true:false
hole_0':s3_3 :: 0':s
gen_0':s4_3 :: Nat -> 0':s


Generator Equations:
gen_0':s4_3(0) <=> 0'
gen_0':s4_3(+(x, 1)) <=> s(gen_0':s4_3(x))


The following defined symbols remain to be analysed:
gr, cond1, cond2

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1

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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:
cond1(true, x, y) -> cond2(gr(x, y), x, y)
cond2(true, x, y) -> cond1(gr0(x), y, y)
cond2(false, x, y) -> cond1(gr0(x), p(x), y)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
gr0(0') -> false
gr0(s(x)) -> true
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2
gr :: 0':s -> 0':s -> true:false
gr0 :: 0':s -> true:false
false :: true:false
p :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
hole_cond1:cond21_3 :: cond1:cond2
hole_true:false2_3 :: true:false
hole_0':s3_3 :: 0':s
gen_0':s4_3 :: Nat -> 0':s


Lemmas:
gr(gen_0':s4_3(n6_3), gen_0':s4_3(n6_3)) -> false, rt in Omega(1 + n6_3)


Generator Equations:
gen_0':s4_3(0) <=> 0'
gen_0':s4_3(+(x, 1)) <=> s(gen_0':s4_3(x))


The following defined symbols remain to be analysed:
cond2, cond1

They will be analysed ascendingly in the following order:
cond1 = cond2