/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 257 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), 64 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:

   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   <=(0, y) -> true
   <=(s(x), 0) -> false
   <=(s(x), s(y)) -> <=(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   perfectp(0) -> false
   perfectp(s(x)) -> f(x, s(0), s(x), s(x))
   f(0, y, 0, u) -> true
   f(0, y, s(z), u) -> false
   f(s(x), 0, z, u) -> f(x, u, -(z, s(x)), u)
   f(s(x), s(y), z, u) -> if(<=(x, y), f(s(x), -(y, x), z, u), f(x, u, z, u))

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.
----------------------------------------

(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:

   -(x, 0') -> x
   -(s(x), s(y)) -> -(x, y)
   <=(0', y) -> true
   <=(s(x), 0') -> false
   <=(s(x), s(y)) -> <=(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   perfectp(0') -> false
   perfectp(s(x)) -> f(x, s(0'), s(x), s(x))
   f(0', y, 0', u) -> true
   f(0', y, s(z), u) -> false
   f(s(x), 0', z, u) -> f(x, u, -(z, s(x)), u)
   f(s(x), s(y), z, u) -> if(<=(x, y), f(s(x), -(y, x), z, u), f(x, u, z, u))

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:
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
<=(0', y) -> true
<=(s(x), 0') -> false
<=(s(x), s(y)) -> <=(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
perfectp(0') -> false
perfectp(s(x)) -> f(x, s(0'), s(x), s(x))
f(0', y, 0', u) -> true
f(0', y, s(z), u) -> false
f(s(x), 0', z, u) -> f(x, u, -(z, s(x)), u)
f(s(x), s(y), z, u) -> if(<=(x, y), f(s(x), -(y, x), z, u), f(x, u, z, u))

Types:
- :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
<= :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
if :: true:false -> true:false -> true:false -> true:false
perfectp :: 0':s -> true:false
f :: 0':s -> 0':s -> 0':s -> 0':s -> true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
-, <=, f

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

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(6)
Obligation:
TRS:
Rules:
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
<=(0', y) -> true
<=(s(x), 0') -> false
<=(s(x), s(y)) -> <=(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
perfectp(0') -> false
perfectp(s(x)) -> f(x, s(0'), s(x), s(x))
f(0', y, 0', u) -> true
f(0', y, s(z), u) -> false
f(s(x), 0', z, u) -> f(x, u, -(z, s(x)), u)
f(s(x), s(y), z, u) -> if(<=(x, y), f(s(x), -(y, x), z, u), f(x, u, z, u))

Types:
- :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
<= :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
if :: true:false -> true:false -> true:false -> true:false
perfectp :: 0':s -> true:false
f :: 0':s -> 0':s -> 0':s -> 0':s -> true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
-, <=, f

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

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)

Induction Base:
-(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
gen_0':s3_0(0)

Induction Step:
-(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
gen_0':s3_0(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:
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
<=(0', y) -> true
<=(s(x), 0') -> false
<=(s(x), s(y)) -> <=(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
perfectp(0') -> false
perfectp(s(x)) -> f(x, s(0'), s(x), s(x))
f(0', y, 0', u) -> true
f(0', y, s(z), u) -> false
f(s(x), 0', z, u) -> f(x, u, -(z, s(x)), u)
f(s(x), s(y), z, u) -> if(<=(x, y), f(s(x), -(y, x), z, u), f(x, u, z, u))

Types:
- :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
<= :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
if :: true:false -> true:false -> true:false -> true:false
perfectp :: 0':s -> true:false
f :: 0':s -> 0':s -> 0':s -> 0':s -> true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
-, <=, f

They will be analysed ascendingly in the following order:
- < f
<= < 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:
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
<=(0', y) -> true
<=(s(x), 0') -> false
<=(s(x), s(y)) -> <=(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
perfectp(0') -> false
perfectp(s(x)) -> f(x, s(0'), s(x), s(x))
f(0', y, 0', u) -> true
f(0', y, s(z), u) -> false
f(s(x), 0', z, u) -> f(x, u, -(z, s(x)), u)
f(s(x), s(y), z, u) -> if(<=(x, y), f(s(x), -(y, x), z, u), f(x, u, z, u))

Types:
- :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
<= :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
if :: true:false -> true:false -> true:false -> true:false
perfectp :: 0':s -> true:false
f :: 0':s -> 0':s -> 0':s -> 0':s -> true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
<=, f

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

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
<=(gen_0':s3_0(n273_0), gen_0':s3_0(n273_0)) -> true, rt in Omega(1 + n273_0)

Induction Base:
<=(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
true

Induction Step:
<=(gen_0':s3_0(+(n273_0, 1)), gen_0':s3_0(+(n273_0, 1))) ->_R^Omega(1)
<=(gen_0':s3_0(n273_0), gen_0':s3_0(n273_0)) ->_IH
true

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:
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
<=(0', y) -> true
<=(s(x), 0') -> false
<=(s(x), s(y)) -> <=(x, y)
if(true, x, y) -> x
if(false, x, y) -> y
perfectp(0') -> false
perfectp(s(x)) -> f(x, s(0'), s(x), s(x))
f(0', y, 0', u) -> true
f(0', y, s(z), u) -> false
f(s(x), 0', z, u) -> f(x, u, -(z, s(x)), u)
f(s(x), s(y), z, u) -> if(<=(x, y), f(s(x), -(y, x), z, u), f(x, u, z, u))

Types:
- :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
<= :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
if :: true:false -> true:false -> true:false -> true:false
perfectp :: 0':s -> true:false
f :: 0':s -> 0':s -> 0':s -> 0':s -> true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
-(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> gen_0':s3_0(0), rt in Omega(1 + n5_0)
<=(gen_0':s3_0(n273_0), gen_0':s3_0(n273_0)) -> true, rt in Omega(1 + n273_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
f