/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 285 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), 41 ms]
        (14) 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:

   cond1(true, x, y, z) -> cond2(gr(x, 0), x, y, z)
   cond2(true, x, y, z) -> cond1(gr(add(x, y), z), p(x), y, z)
   cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z)
   cond3(true, x, y, z) -> cond1(gr(add(x, y), z), x, p(y), z)
   cond3(false, x, y, z) -> cond1(gr(add(x, y), z), x, y, z)
   gr(0, x) -> false
   gr(s(x), 0) -> true
   gr(s(x), s(y)) -> gr(x, y)
   add(0, x) -> x
   add(s(x), y) -> s(add(x, y))
   p(0) -> 0
   p(s(x)) -> x

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:

   cond1(true, x, y, z) -> cond2(gr(x, 0'), x, y, z)
   cond2(true, x, y, z) -> cond1(gr(add(x, y), z), p(x), y, z)
   cond2(false, x, y, z) -> cond3(gr(y, 0'), x, y, z)
   cond3(true, x, y, z) -> cond1(gr(add(x, y), z), x, p(y), z)
   cond3(false, x, y, z) -> cond1(gr(add(x, y), z), x, y, z)
   gr(0', x) -> false
   gr(s(x), 0') -> true
   gr(s(x), s(y)) -> gr(x, y)
   add(0', x) -> x
   add(s(x), y) -> s(add(x, y))
   p(0') -> 0'
   p(s(x)) -> x

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

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

(4)
Obligation:
TRS:
Rules:
cond1(true, x, y, z) -> cond2(gr(x, 0'), x, y, z)
cond2(true, x, y, z) -> cond1(gr(add(x, y), z), p(x), y, z)
cond2(false, x, y, z) -> cond3(gr(y, 0'), x, y, z)
cond3(true, x, y, z) -> cond1(gr(add(x, y), z), x, p(y), z)
cond3(false, x, y, z) -> cond1(gr(add(x, y), z), x, y, z)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
add(0', x) -> x
add(s(x), y) -> s(add(x, y))
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
add :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
false :: true:false
cond3 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
s :: 0':s -> 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
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:
cond1, cond2, gr, add, cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1
cond1 = cond3
gr < cond2
add < cond2
cond2 = cond3
gr < cond3
add < cond3

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

(6)
Obligation:
TRS:
Rules:
cond1(true, x, y, z) -> cond2(gr(x, 0'), x, y, z)
cond2(true, x, y, z) -> cond1(gr(add(x, y), z), p(x), y, z)
cond2(false, x, y, z) -> cond3(gr(y, 0'), x, y, z)
cond3(true, x, y, z) -> cond1(gr(add(x, y), z), x, p(y), z)
cond3(false, x, y, z) -> cond1(gr(add(x, y), z), x, y, z)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
add(0', x) -> x
add(s(x), y) -> s(add(x, y))
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
add :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
false :: true:false
cond3 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
s :: 0':s -> 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
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:
gr, cond1, cond2, add, cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1
cond1 = cond3
gr < cond2
add < cond2
cond2 = cond3
gr < cond3
add < cond3

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

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

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

Induction Step:
gr(gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(+(n6_0, 1))) ->_R^Omega(1)
gr(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:
cond1(true, x, y, z) -> cond2(gr(x, 0'), x, y, z)
cond2(true, x, y, z) -> cond1(gr(add(x, y), z), p(x), y, z)
cond2(false, x, y, z) -> cond3(gr(y, 0'), x, y, z)
cond3(true, x, y, z) -> cond1(gr(add(x, y), z), x, p(y), z)
cond3(false, x, y, z) -> cond1(gr(add(x, y), z), x, y, z)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
add(0', x) -> x
add(s(x), y) -> s(add(x, y))
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
add :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
false :: true:false
cond3 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
s :: 0':s -> 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
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:
gr, cond1, cond2, add, cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1
cond1 = cond3
gr < cond2
add < cond2
cond2 = cond3
gr < cond3
add < cond3

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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, z) -> cond2(gr(x, 0'), x, y, z)
cond2(true, x, y, z) -> cond1(gr(add(x, y), z), p(x), y, z)
cond2(false, x, y, z) -> cond3(gr(y, 0'), x, y, z)
cond3(true, x, y, z) -> cond1(gr(add(x, y), z), x, p(y), z)
cond3(false, x, y, z) -> cond1(gr(add(x, y), z), x, y, z)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
add(0', x) -> x
add(s(x), y) -> s(add(x, y))
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
add :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
false :: true:false
cond3 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
s :: 0':s -> 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
gr(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:
add, cond1, cond2, cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
cond1 = cond3
add < cond2
cond2 = cond3
add < cond3

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
add(gen_0':s4_0(n265_0), gen_0':s4_0(b)) -> gen_0':s4_0(+(n265_0, b)), rt in Omega(1 + n265_0)

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

Induction Step:
add(gen_0':s4_0(+(n265_0, 1)), gen_0':s4_0(b)) ->_R^Omega(1)
s(add(gen_0':s4_0(n265_0), gen_0':s4_0(b))) ->_IH
s(gen_0':s4_0(+(b, c266_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:
cond1(true, x, y, z) -> cond2(gr(x, 0'), x, y, z)
cond2(true, x, y, z) -> cond1(gr(add(x, y), z), p(x), y, z)
cond2(false, x, y, z) -> cond3(gr(y, 0'), x, y, z)
cond3(true, x, y, z) -> cond1(gr(add(x, y), z), x, p(y), z)
cond3(false, x, y, z) -> cond1(gr(add(x, y), z), x, y, z)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
add(0', x) -> x
add(s(x), y) -> s(add(x, y))
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
add :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
false :: true:false
cond3 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
s :: 0':s -> 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)
add(gen_0':s4_0(n265_0), gen_0':s4_0(b)) -> gen_0':s4_0(+(n265_0, b)), rt in Omega(1 + n265_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:
cond2, cond1, cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
cond1 = cond3
cond2 = cond3