/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The Runtime Complexity (innermost) 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), 244 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), 84 ms]
        (14) typed CpxTrs


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(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   times(x, y) -> sum(generate(x, y))
   generate(x, y) -> gen(x, y, 0)
   gen(x, y, z) -> if(ge(z, x), x, y, z)
   if(true, x, y, z) -> nil
   if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
   sum(nil) -> 0
   sum(cons(0, xs)) -> sum(xs)
   sum(cons(s(x), xs)) -> s(sum(cons(x, xs)))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)

S is empty.
Rewrite Strategy: INNERMOST
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(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   times(x, y) -> sum(generate(x, y))
   generate(x, y) -> gen(x, y, 0')
   gen(x, y, z) -> if(ge(z, x), x, y, z)
   if(true, x, y, z) -> nil
   if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
   sum(nil) -> 0'
   sum(cons(0', xs)) -> sum(xs)
   sum(cons(s(x), xs)) -> s(sum(cons(x, xs)))
   ge(x, 0') -> true
   ge(0', s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)

S is empty.
Rewrite Strategy: INNERMOST
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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
Innermost TRS:
Rules:
times(x, y) -> sum(generate(x, y))
generate(x, y) -> gen(x, y, 0')
gen(x, y, z) -> if(ge(z, x), x, y, z)
if(true, x, y, z) -> nil
if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
sum(nil) -> 0'
sum(cons(0', xs)) -> sum(xs)
sum(cons(s(x), xs)) -> s(sum(cons(x, xs)))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)

Types:
times :: 0':s -> 0':s -> 0':s
sum :: nil:cons -> 0':s
generate :: 0':s -> 0':s -> nil:cons
gen :: 0':s -> 0':s -> 0':s -> nil:cons
0' :: 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> nil:cons
ge :: 0':s -> 0':s -> true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0':s -> nil:cons -> nil:cons
s :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons

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

They will be analysed ascendingly in the following order:
ge < gen

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(6)
Obligation:
Innermost TRS:
Rules:
times(x, y) -> sum(generate(x, y))
generate(x, y) -> gen(x, y, 0')
gen(x, y, z) -> if(ge(z, x), x, y, z)
if(true, x, y, z) -> nil
if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
sum(nil) -> 0'
sum(cons(0', xs)) -> sum(xs)
sum(cons(s(x), xs)) -> s(sum(cons(x, xs)))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)

Types:
times :: 0':s -> 0':s -> 0':s
sum :: nil:cons -> 0':s
generate :: 0':s -> 0':s -> nil:cons
gen :: 0':s -> 0':s -> 0':s -> nil:cons
0' :: 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> nil:cons
ge :: 0':s -> 0':s -> true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0':s -> nil:cons -> nil:cons
s :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:cons5_0(0) <=> nil
gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
sum, gen, ge

They will be analysed ascendingly in the following order:
ge < gen

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sum(gen_nil:cons5_0(n7_0)) -> gen_0':s4_0(0), rt in Omega(1 + n7_0)

Induction Base:
sum(gen_nil:cons5_0(0)) ->_R^Omega(1)
0'

Induction Step:
sum(gen_nil:cons5_0(+(n7_0, 1))) ->_R^Omega(1)
sum(gen_nil:cons5_0(n7_0)) ->_IH
gen_0':s4_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:

Innermost TRS:
Rules:
times(x, y) -> sum(generate(x, y))
generate(x, y) -> gen(x, y, 0')
gen(x, y, z) -> if(ge(z, x), x, y, z)
if(true, x, y, z) -> nil
if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
sum(nil) -> 0'
sum(cons(0', xs)) -> sum(xs)
sum(cons(s(x), xs)) -> s(sum(cons(x, xs)))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)

Types:
times :: 0':s -> 0':s -> 0':s
sum :: nil:cons -> 0':s
generate :: 0':s -> 0':s -> nil:cons
gen :: 0':s -> 0':s -> 0':s -> nil:cons
0' :: 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> nil:cons
ge :: 0':s -> 0':s -> true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0':s -> nil:cons -> nil:cons
s :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:cons5_0(0) <=> nil
gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
sum, gen, ge

They will be analysed ascendingly in the following order:
ge < gen

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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:
Innermost TRS:
Rules:
times(x, y) -> sum(generate(x, y))
generate(x, y) -> gen(x, y, 0')
gen(x, y, z) -> if(ge(z, x), x, y, z)
if(true, x, y, z) -> nil
if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
sum(nil) -> 0'
sum(cons(0', xs)) -> sum(xs)
sum(cons(s(x), xs)) -> s(sum(cons(x, xs)))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)

Types:
times :: 0':s -> 0':s -> 0':s
sum :: nil:cons -> 0':s
generate :: 0':s -> 0':s -> nil:cons
gen :: 0':s -> 0':s -> 0':s -> nil:cons
0' :: 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> nil:cons
ge :: 0':s -> 0':s -> true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0':s -> nil:cons -> nil:cons
s :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons


Lemmas:
sum(gen_nil:cons5_0(n7_0)) -> gen_0':s4_0(0), rt in Omega(1 + n7_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:cons5_0(0) <=> nil
gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
ge, gen

They will be analysed ascendingly in the following order:
ge < gen

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(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_0':s4_0(n300_0), gen_0':s4_0(n300_0)) -> true, rt in Omega(1 + n300_0)

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

Induction Step:
ge(gen_0':s4_0(+(n300_0, 1)), gen_0':s4_0(+(n300_0, 1))) ->_R^Omega(1)
ge(gen_0':s4_0(n300_0), gen_0':s4_0(n300_0)) ->_IH
true

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(14)
Obligation:
Innermost TRS:
Rules:
times(x, y) -> sum(generate(x, y))
generate(x, y) -> gen(x, y, 0')
gen(x, y, z) -> if(ge(z, x), x, y, z)
if(true, x, y, z) -> nil
if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
sum(nil) -> 0'
sum(cons(0', xs)) -> sum(xs)
sum(cons(s(x), xs)) -> s(sum(cons(x, xs)))
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)

Types:
times :: 0':s -> 0':s -> 0':s
sum :: nil:cons -> 0':s
generate :: 0':s -> 0':s -> nil:cons
gen :: 0':s -> 0':s -> 0':s -> nil:cons
0' :: 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> nil:cons
ge :: 0':s -> 0':s -> true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0':s -> nil:cons -> nil:cons
s :: 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_nil:cons2_0 :: nil:cons
hole_true:false3_0 :: true:false
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons


Lemmas:
sum(gen_nil:cons5_0(n7_0)) -> gen_0':s4_0(0), rt in Omega(1 + n7_0)
ge(gen_0':s4_0(n300_0), gen_0':s4_0(n300_0)) -> true, rt in Omega(1 + n300_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:cons5_0(0) <=> nil
gen_nil:cons5_0(+(x, 1)) <=> cons(0', gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
gen