/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^2), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2, 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), 223 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), 144 ms]
        (14) proven lower bound
        (15) LowerBoundPropagationProof [FINISHED, 0 ms]
        (16) BOUNDS(n^2, INF)


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
   sum(cons(0, x), y) -> sum(x, y)
   sum(nil, y) -> y
   empty(nil) -> true
   empty(cons(n, x)) -> false
   tail(nil) -> nil
   tail(cons(n, x)) -> x
   head(cons(n, x)) -> n
   weight(x) -> if(empty(x), empty(tail(x)), x)
   if(true, b, x) -> weight_undefined_error
   if(false, b, x) -> if2(b, x)
   if2(true, x) -> head(x)
   if2(false, x) -> weight(sum(x, cons(0, tail(tail(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.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
   sum(cons(0', x), y) -> sum(x, y)
   sum(nil, y) -> y
   empty(nil) -> true
   empty(cons(n, x)) -> false
   tail(nil) -> nil
   tail(cons(n, x)) -> x
   head(cons(n, x)) -> n
   weight(x) -> if(empty(x), empty(tail(x)), x)
   if(true, b, x) -> weight_undefined_error
   if(false, b, x) -> if2(b, x)
   if2(true, x) -> head(x)
   if2(false, x) -> weight(sum(x, cons(0', tail(tail(x)))))

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

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

(4)
Obligation:
TRS:
Rules:
sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
sum(cons(0', x), y) -> sum(x, y)
sum(nil, y) -> y
empty(nil) -> true
empty(cons(n, x)) -> false
tail(nil) -> nil
tail(cons(n, x)) -> x
head(cons(n, x)) -> n
weight(x) -> if(empty(x), empty(tail(x)), x)
if(true, b, x) -> weight_undefined_error
if(false, b, x) -> if2(b, x)
if2(true, x) -> head(x)
if2(false, x) -> weight(sum(x, cons(0', tail(tail(x)))))

Types:
sum :: cons:nil -> cons:nil -> cons:nil
cons :: s:0':weight_undefined_error -> cons:nil -> cons:nil
s :: s:0':weight_undefined_error -> s:0':weight_undefined_error
0' :: s:0':weight_undefined_error
nil :: cons:nil
empty :: cons:nil -> true:false
true :: true:false
false :: true:false
tail :: cons:nil -> cons:nil
head :: cons:nil -> s:0':weight_undefined_error
weight :: cons:nil -> s:0':weight_undefined_error
if :: true:false -> true:false -> cons:nil -> s:0':weight_undefined_error
weight_undefined_error :: s:0':weight_undefined_error
if2 :: true:false -> cons:nil -> s:0':weight_undefined_error
hole_cons:nil1_0 :: cons:nil
hole_s:0':weight_undefined_error2_0 :: s:0':weight_undefined_error
hole_true:false3_0 :: true:false
gen_cons:nil4_0 :: Nat -> cons:nil
gen_s:0':weight_undefined_error5_0 :: Nat -> s:0':weight_undefined_error

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
sum, weight

They will be analysed ascendingly in the following order:
sum < weight

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

(6)
Obligation:
TRS:
Rules:
sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
sum(cons(0', x), y) -> sum(x, y)
sum(nil, y) -> y
empty(nil) -> true
empty(cons(n, x)) -> false
tail(nil) -> nil
tail(cons(n, x)) -> x
head(cons(n, x)) -> n
weight(x) -> if(empty(x), empty(tail(x)), x)
if(true, b, x) -> weight_undefined_error
if(false, b, x) -> if2(b, x)
if2(true, x) -> head(x)
if2(false, x) -> weight(sum(x, cons(0', tail(tail(x)))))

Types:
sum :: cons:nil -> cons:nil -> cons:nil
cons :: s:0':weight_undefined_error -> cons:nil -> cons:nil
s :: s:0':weight_undefined_error -> s:0':weight_undefined_error
0' :: s:0':weight_undefined_error
nil :: cons:nil
empty :: cons:nil -> true:false
true :: true:false
false :: true:false
tail :: cons:nil -> cons:nil
head :: cons:nil -> s:0':weight_undefined_error
weight :: cons:nil -> s:0':weight_undefined_error
if :: true:false -> true:false -> cons:nil -> s:0':weight_undefined_error
weight_undefined_error :: s:0':weight_undefined_error
if2 :: true:false -> cons:nil -> s:0':weight_undefined_error
hole_cons:nil1_0 :: cons:nil
hole_s:0':weight_undefined_error2_0 :: s:0':weight_undefined_error
hole_true:false3_0 :: true:false
gen_cons:nil4_0 :: Nat -> cons:nil
gen_s:0':weight_undefined_error5_0 :: Nat -> s:0':weight_undefined_error


Generator Equations:
gen_cons:nil4_0(0) <=> nil
gen_cons:nil4_0(+(x, 1)) <=> cons(0', gen_cons:nil4_0(x))
gen_s:0':weight_undefined_error5_0(0) <=> 0'
gen_s:0':weight_undefined_error5_0(+(x, 1)) <=> s(gen_s:0':weight_undefined_error5_0(x))


The following defined symbols remain to be analysed:
sum, weight

They will be analysed ascendingly in the following order:
sum < weight

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sum(gen_cons:nil4_0(n7_0), gen_cons:nil4_0(b)) -> gen_cons:nil4_0(b), rt in Omega(1 + n7_0)

Induction Base:
sum(gen_cons:nil4_0(0), gen_cons:nil4_0(b)) ->_R^Omega(1)
gen_cons:nil4_0(b)

Induction Step:
sum(gen_cons:nil4_0(+(n7_0, 1)), gen_cons:nil4_0(b)) ->_R^Omega(1)
sum(gen_cons:nil4_0(n7_0), gen_cons:nil4_0(b)) ->_IH
gen_cons:nil4_0(b)

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:
sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
sum(cons(0', x), y) -> sum(x, y)
sum(nil, y) -> y
empty(nil) -> true
empty(cons(n, x)) -> false
tail(nil) -> nil
tail(cons(n, x)) -> x
head(cons(n, x)) -> n
weight(x) -> if(empty(x), empty(tail(x)), x)
if(true, b, x) -> weight_undefined_error
if(false, b, x) -> if2(b, x)
if2(true, x) -> head(x)
if2(false, x) -> weight(sum(x, cons(0', tail(tail(x)))))

Types:
sum :: cons:nil -> cons:nil -> cons:nil
cons :: s:0':weight_undefined_error -> cons:nil -> cons:nil
s :: s:0':weight_undefined_error -> s:0':weight_undefined_error
0' :: s:0':weight_undefined_error
nil :: cons:nil
empty :: cons:nil -> true:false
true :: true:false
false :: true:false
tail :: cons:nil -> cons:nil
head :: cons:nil -> s:0':weight_undefined_error
weight :: cons:nil -> s:0':weight_undefined_error
if :: true:false -> true:false -> cons:nil -> s:0':weight_undefined_error
weight_undefined_error :: s:0':weight_undefined_error
if2 :: true:false -> cons:nil -> s:0':weight_undefined_error
hole_cons:nil1_0 :: cons:nil
hole_s:0':weight_undefined_error2_0 :: s:0':weight_undefined_error
hole_true:false3_0 :: true:false
gen_cons:nil4_0 :: Nat -> cons:nil
gen_s:0':weight_undefined_error5_0 :: Nat -> s:0':weight_undefined_error


Generator Equations:
gen_cons:nil4_0(0) <=> nil
gen_cons:nil4_0(+(x, 1)) <=> cons(0', gen_cons:nil4_0(x))
gen_s:0':weight_undefined_error5_0(0) <=> 0'
gen_s:0':weight_undefined_error5_0(+(x, 1)) <=> s(gen_s:0':weight_undefined_error5_0(x))


The following defined symbols remain to be analysed:
sum, weight

They will be analysed ascendingly in the following order:
sum < weight

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(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
sum(cons(0', x), y) -> sum(x, y)
sum(nil, y) -> y
empty(nil) -> true
empty(cons(n, x)) -> false
tail(nil) -> nil
tail(cons(n, x)) -> x
head(cons(n, x)) -> n
weight(x) -> if(empty(x), empty(tail(x)), x)
if(true, b, x) -> weight_undefined_error
if(false, b, x) -> if2(b, x)
if2(true, x) -> head(x)
if2(false, x) -> weight(sum(x, cons(0', tail(tail(x)))))

Types:
sum :: cons:nil -> cons:nil -> cons:nil
cons :: s:0':weight_undefined_error -> cons:nil -> cons:nil
s :: s:0':weight_undefined_error -> s:0':weight_undefined_error
0' :: s:0':weight_undefined_error
nil :: cons:nil
empty :: cons:nil -> true:false
true :: true:false
false :: true:false
tail :: cons:nil -> cons:nil
head :: cons:nil -> s:0':weight_undefined_error
weight :: cons:nil -> s:0':weight_undefined_error
if :: true:false -> true:false -> cons:nil -> s:0':weight_undefined_error
weight_undefined_error :: s:0':weight_undefined_error
if2 :: true:false -> cons:nil -> s:0':weight_undefined_error
hole_cons:nil1_0 :: cons:nil
hole_s:0':weight_undefined_error2_0 :: s:0':weight_undefined_error
hole_true:false3_0 :: true:false
gen_cons:nil4_0 :: Nat -> cons:nil
gen_s:0':weight_undefined_error5_0 :: Nat -> s:0':weight_undefined_error


Lemmas:
sum(gen_cons:nil4_0(n7_0), gen_cons:nil4_0(b)) -> gen_cons:nil4_0(b), rt in Omega(1 + n7_0)


Generator Equations:
gen_cons:nil4_0(0) <=> nil
gen_cons:nil4_0(+(x, 1)) <=> cons(0', gen_cons:nil4_0(x))
gen_s:0':weight_undefined_error5_0(0) <=> 0'
gen_s:0':weight_undefined_error5_0(+(x, 1)) <=> s(gen_s:0':weight_undefined_error5_0(x))


The following defined symbols remain to be analysed:
weight
----------------------------------------

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
weight(gen_cons:nil4_0(+(1, n640_0))) -> gen_s:0':weight_undefined_error5_0(0), rt in Omega(1 + n640_0 + n640_0^2)

Induction Base:
weight(gen_cons:nil4_0(+(1, 0))) ->_R^Omega(1)
if(empty(gen_cons:nil4_0(+(1, 0))), empty(tail(gen_cons:nil4_0(+(1, 0)))), gen_cons:nil4_0(+(1, 0))) ->_R^Omega(1)
if(false, empty(tail(gen_cons:nil4_0(1))), gen_cons:nil4_0(1)) ->_R^Omega(1)
if(false, empty(gen_cons:nil4_0(0)), gen_cons:nil4_0(1)) ->_R^Omega(1)
if(false, true, gen_cons:nil4_0(1)) ->_R^Omega(1)
if2(true, gen_cons:nil4_0(1)) ->_R^Omega(1)
head(gen_cons:nil4_0(1)) ->_R^Omega(1)
0'

Induction Step:
weight(gen_cons:nil4_0(+(1, +(n640_0, 1)))) ->_R^Omega(1)
if(empty(gen_cons:nil4_0(+(1, +(n640_0, 1)))), empty(tail(gen_cons:nil4_0(+(1, +(n640_0, 1))))), gen_cons:nil4_0(+(1, +(n640_0, 1)))) ->_R^Omega(1)
if(false, empty(tail(gen_cons:nil4_0(+(2, n640_0)))), gen_cons:nil4_0(+(2, n640_0))) ->_R^Omega(1)
if(false, empty(gen_cons:nil4_0(+(1, n640_0))), gen_cons:nil4_0(+(2, n640_0))) ->_R^Omega(1)
if(false, false, gen_cons:nil4_0(+(2, n640_0))) ->_R^Omega(1)
if2(false, gen_cons:nil4_0(+(2, n640_0))) ->_R^Omega(1)
weight(sum(gen_cons:nil4_0(+(2, n640_0)), cons(0', tail(tail(gen_cons:nil4_0(+(2, n640_0))))))) ->_R^Omega(1)
weight(sum(gen_cons:nil4_0(+(2, n640_0)), cons(0', tail(gen_cons:nil4_0(+(1, n640_0)))))) ->_R^Omega(1)
weight(sum(gen_cons:nil4_0(+(2, n640_0)), cons(0', gen_cons:nil4_0(n640_0)))) ->_L^Omega(3 + n640_0)
weight(gen_cons:nil4_0(+(n640_0, 1))) ->_IH
gen_s:0':weight_undefined_error5_0(0)

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(14)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
sum(cons(s(n), x), cons(m, y)) -> sum(cons(n, x), cons(s(m), y))
sum(cons(0', x), y) -> sum(x, y)
sum(nil, y) -> y
empty(nil) -> true
empty(cons(n, x)) -> false
tail(nil) -> nil
tail(cons(n, x)) -> x
head(cons(n, x)) -> n
weight(x) -> if(empty(x), empty(tail(x)), x)
if(true, b, x) -> weight_undefined_error
if(false, b, x) -> if2(b, x)
if2(true, x) -> head(x)
if2(false, x) -> weight(sum(x, cons(0', tail(tail(x)))))

Types:
sum :: cons:nil -> cons:nil -> cons:nil
cons :: s:0':weight_undefined_error -> cons:nil -> cons:nil
s :: s:0':weight_undefined_error -> s:0':weight_undefined_error
0' :: s:0':weight_undefined_error
nil :: cons:nil
empty :: cons:nil -> true:false
true :: true:false
false :: true:false
tail :: cons:nil -> cons:nil
head :: cons:nil -> s:0':weight_undefined_error
weight :: cons:nil -> s:0':weight_undefined_error
if :: true:false -> true:false -> cons:nil -> s:0':weight_undefined_error
weight_undefined_error :: s:0':weight_undefined_error
if2 :: true:false -> cons:nil -> s:0':weight_undefined_error
hole_cons:nil1_0 :: cons:nil
hole_s:0':weight_undefined_error2_0 :: s:0':weight_undefined_error
hole_true:false3_0 :: true:false
gen_cons:nil4_0 :: Nat -> cons:nil
gen_s:0':weight_undefined_error5_0 :: Nat -> s:0':weight_undefined_error


Lemmas:
sum(gen_cons:nil4_0(n7_0), gen_cons:nil4_0(b)) -> gen_cons:nil4_0(b), rt in Omega(1 + n7_0)


Generator Equations:
gen_cons:nil4_0(0) <=> nil
gen_cons:nil4_0(+(x, 1)) <=> cons(0', gen_cons:nil4_0(x))
gen_s:0':weight_undefined_error5_0(0) <=> 0'
gen_s:0':weight_undefined_error5_0(+(x, 1)) <=> s(gen_s:0':weight_undefined_error5_0(x))


The following defined symbols remain to be analysed:
weight
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(15) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(16)
BOUNDS(n^2, INF)