/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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 (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) SlicingProof [LOWER BOUND(ID), 0 ms]
(4) CpxTRS
(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) typed CpxTrs
(7) OrderProof [LOWER BOUND(ID), 0 ms]
(8) typed CpxTrs
(9) RewriteLemmaProof [LOWER BOUND(ID), 286 ms]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 64 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 51 ms]
        (18) 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:

   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   rev(x) -> if(x, eq(0, length(x)), nil, 0, length(x))
   if(x, true, z, c, l) -> z
   if(x, false, z, c, l) -> help(s(c), l, x, z)
   help(c, l, cons(x, y), z) -> if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l)
   append(nil, y) -> y
   append(cons(x, y), z) -> cons(x, append(y, z))
   length(nil) -> 0
   length(cons(x, y)) -> s(length(y))

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:

   ge(x, 0') -> true
   ge(0', s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   rev(x) -> if(x, eq(0', length(x)), nil, 0', length(x))
   if(x, true, z, c, l) -> z
   if(x, false, z, c, l) -> help(s(c), l, x, z)
   help(c, l, cons(x, y), z) -> if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l)
   append(nil, y) -> y
   append(cons(x, y), z) -> cons(x, append(y, z))
   length(nil) -> 0'
   length(cons(x, y)) -> s(length(y))

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

(3) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
eq/0
cons/0

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

(4)
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:

   ge(x, 0') -> true
   ge(0', s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   rev(x) -> if(x, eq(length(x)), nil, 0', length(x))
   if(x, true, z, c, l) -> z
   if(x, false, z, c, l) -> help(s(c), l, x, z)
   help(c, l, cons(y), z) -> if(append(y, cons(nil)), ge(c, l), cons(z), c, l)
   append(nil, y) -> y
   append(cons(y), z) -> cons(append(y, z))
   length(nil) -> 0'
   length(cons(y)) -> s(length(y))

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

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

(6)
Obligation:
TRS:
Rules:
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
rev(x) -> if(x, eq(length(x)), nil, 0', length(x))
if(x, true, z, c, l) -> z
if(x, false, z, c, l) -> help(s(c), l, x, z)
help(c, l, cons(y), z) -> if(append(y, cons(nil)), ge(c, l), cons(z), c, l)
append(nil, y) -> y
append(cons(y), z) -> cons(append(y, z))
length(nil) -> 0'
length(cons(y)) -> s(length(y))

Types:
ge :: 0':s -> 0':s -> true:false:eq
0' :: 0':s
true :: true:false:eq
s :: 0':s -> 0':s
false :: true:false:eq
rev :: nil:cons -> nil:cons
if :: nil:cons -> true:false:eq -> nil:cons -> 0':s -> 0':s -> nil:cons
eq :: 0':s -> true:false:eq
length :: nil:cons -> 0':s
nil :: nil:cons
help :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
cons :: nil:cons -> nil:cons
append :: nil:cons -> nil:cons -> nil:cons
hole_true:false:eq1_0 :: true:false:eq
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons

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

(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
ge, length, help, append

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

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

(8)
Obligation:
TRS:
Rules:
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
rev(x) -> if(x, eq(length(x)), nil, 0', length(x))
if(x, true, z, c, l) -> z
if(x, false, z, c, l) -> help(s(c), l, x, z)
help(c, l, cons(y), z) -> if(append(y, cons(nil)), ge(c, l), cons(z), c, l)
append(nil, y) -> y
append(cons(y), z) -> cons(append(y, z))
length(nil) -> 0'
length(cons(y)) -> s(length(y))

Types:
ge :: 0':s -> 0':s -> true:false:eq
0' :: 0':s
true :: true:false:eq
s :: 0':s -> 0':s
false :: true:false:eq
rev :: nil:cons -> nil:cons
if :: nil:cons -> true:false:eq -> nil:cons -> 0':s -> 0':s -> nil:cons
eq :: 0':s -> true:false:eq
length :: nil:cons -> 0':s
nil :: nil:cons
help :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
cons :: nil:cons -> nil:cons
append :: nil:cons -> nil:cons -> nil:cons
hole_true:false:eq1_0 :: true:false:eq
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
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(gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
ge, length, help, append

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

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

(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_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(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) ->_R^Omega(1)
ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) ->_IH
true

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

(10)
Complex Obligation (BEST)

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

(11)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
rev(x) -> if(x, eq(length(x)), nil, 0', length(x))
if(x, true, z, c, l) -> z
if(x, false, z, c, l) -> help(s(c), l, x, z)
help(c, l, cons(y), z) -> if(append(y, cons(nil)), ge(c, l), cons(z), c, l)
append(nil, y) -> y
append(cons(y), z) -> cons(append(y, z))
length(nil) -> 0'
length(cons(y)) -> s(length(y))

Types:
ge :: 0':s -> 0':s -> true:false:eq
0' :: 0':s
true :: true:false:eq
s :: 0':s -> 0':s
false :: true:false:eq
rev :: nil:cons -> nil:cons
if :: nil:cons -> true:false:eq -> nil:cons -> 0':s -> 0':s -> nil:cons
eq :: 0':s -> true:false:eq
length :: nil:cons -> 0':s
nil :: nil:cons
help :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
cons :: nil:cons -> nil:cons
append :: nil:cons -> nil:cons -> nil:cons
hole_true:false:eq1_0 :: true:false:eq
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
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(gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
ge, length, help, append

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

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

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

(13)
BOUNDS(n^1, INF)

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

(14)
Obligation:
TRS:
Rules:
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
rev(x) -> if(x, eq(length(x)), nil, 0', length(x))
if(x, true, z, c, l) -> z
if(x, false, z, c, l) -> help(s(c), l, x, z)
help(c, l, cons(y), z) -> if(append(y, cons(nil)), ge(c, l), cons(z), c, l)
append(nil, y) -> y
append(cons(y), z) -> cons(append(y, z))
length(nil) -> 0'
length(cons(y)) -> s(length(y))

Types:
ge :: 0':s -> 0':s -> true:false:eq
0' :: 0':s
true :: true:false:eq
s :: 0':s -> 0':s
false :: true:false:eq
rev :: nil:cons -> nil:cons
if :: nil:cons -> true:false:eq -> nil:cons -> 0':s -> 0':s -> nil:cons
eq :: 0':s -> true:false:eq
length :: nil:cons -> 0':s
nil :: nil:cons
help :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
cons :: nil:cons -> nil:cons
append :: nil:cons -> nil:cons -> nil:cons
hole_true:false:eq1_0 :: true:false:eq
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons


Lemmas:
ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, 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(gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
length, help, append

They will be analysed ascendingly in the following order:
append < help

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length(gen_nil:cons5_0(n263_0)) -> gen_0':s4_0(n263_0), rt in Omega(1 + n263_0)

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

Induction Step:
length(gen_nil:cons5_0(+(n263_0, 1))) ->_R^Omega(1)
s(length(gen_nil:cons5_0(n263_0))) ->_IH
s(gen_0':s4_0(c264_0))

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

(16)
Obligation:
TRS:
Rules:
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
rev(x) -> if(x, eq(length(x)), nil, 0', length(x))
if(x, true, z, c, l) -> z
if(x, false, z, c, l) -> help(s(c), l, x, z)
help(c, l, cons(y), z) -> if(append(y, cons(nil)), ge(c, l), cons(z), c, l)
append(nil, y) -> y
append(cons(y), z) -> cons(append(y, z))
length(nil) -> 0'
length(cons(y)) -> s(length(y))

Types:
ge :: 0':s -> 0':s -> true:false:eq
0' :: 0':s
true :: true:false:eq
s :: 0':s -> 0':s
false :: true:false:eq
rev :: nil:cons -> nil:cons
if :: nil:cons -> true:false:eq -> nil:cons -> 0':s -> 0':s -> nil:cons
eq :: 0':s -> true:false:eq
length :: nil:cons -> 0':s
nil :: nil:cons
help :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
cons :: nil:cons -> nil:cons
append :: nil:cons -> nil:cons -> nil:cons
hole_true:false:eq1_0 :: true:false:eq
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons


Lemmas:
ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)
length(gen_nil:cons5_0(n263_0)) -> gen_0':s4_0(n263_0), rt in Omega(1 + n263_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(gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
append, help

They will be analysed ascendingly in the following order:
append < help

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
append(gen_nil:cons5_0(n465_0), gen_nil:cons5_0(b)) -> gen_nil:cons5_0(+(n465_0, b)), rt in Omega(1 + n465_0)

Induction Base:
append(gen_nil:cons5_0(0), gen_nil:cons5_0(b)) ->_R^Omega(1)
gen_nil:cons5_0(b)

Induction Step:
append(gen_nil:cons5_0(+(n465_0, 1)), gen_nil:cons5_0(b)) ->_R^Omega(1)
cons(append(gen_nil:cons5_0(n465_0), gen_nil:cons5_0(b))) ->_IH
cons(gen_nil:cons5_0(+(b, c466_0)))

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

(18)
Obligation:
TRS:
Rules:
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
rev(x) -> if(x, eq(length(x)), nil, 0', length(x))
if(x, true, z, c, l) -> z
if(x, false, z, c, l) -> help(s(c), l, x, z)
help(c, l, cons(y), z) -> if(append(y, cons(nil)), ge(c, l), cons(z), c, l)
append(nil, y) -> y
append(cons(y), z) -> cons(append(y, z))
length(nil) -> 0'
length(cons(y)) -> s(length(y))

Types:
ge :: 0':s -> 0':s -> true:false:eq
0' :: 0':s
true :: true:false:eq
s :: 0':s -> 0':s
false :: true:false:eq
rev :: nil:cons -> nil:cons
if :: nil:cons -> true:false:eq -> nil:cons -> 0':s -> 0':s -> nil:cons
eq :: 0':s -> true:false:eq
length :: nil:cons -> 0':s
nil :: nil:cons
help :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
cons :: nil:cons -> nil:cons
append :: nil:cons -> nil:cons -> nil:cons
hole_true:false:eq1_0 :: true:false:eq
hole_0':s2_0 :: 0':s
hole_nil:cons3_0 :: nil:cons
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons5_0 :: Nat -> nil:cons


Lemmas:
ge(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)
length(gen_nil:cons5_0(n263_0)) -> gen_0':s4_0(n263_0), rt in Omega(1 + n263_0)
append(gen_nil:cons5_0(n465_0), gen_nil:cons5_0(b)) -> gen_nil:cons5_0(+(n465_0, b)), rt in Omega(1 + n465_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(gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
help