/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 289 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), 59 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 21 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 28 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:

   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   eq(0, 0) -> true
   eq(0, s(y)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   minsort(nil) -> nil
   minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
   min(nil) -> 0
   min(cons(x, nil)) -> x
   min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs)
   if1(true, x, y, xs) -> min(cons(x, xs))
   if1(false, x, y, xs) -> min(cons(y, xs))
   rm(x, nil) -> nil
   rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
   if2(true, x, y, xs) -> rm(x, xs)
   if2(false, x, y, xs) -> cons(y, rm(x, xs))

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:

   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)
   eq(0', 0') -> true
   eq(0', s(y)) -> false
   eq(s(x), 0') -> false
   eq(s(x), s(y)) -> eq(x, y)
   minsort(nil) -> nil
   minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
   min(nil) -> 0'
   min(cons(x, nil)) -> x
   min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs)
   if1(true, x, y, xs) -> min(cons(x, xs))
   if1(false, x, y, xs) -> min(cons(y, xs))
   rm(x, nil) -> nil
   rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
   if2(true, x, y, xs) -> rm(x, xs)
   if2(false, x, y, xs) -> cons(y, rm(x, xs))

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

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

(4)
Obligation:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
eq(0', 0') -> true
eq(0', s(y)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
minsort(nil) -> nil
minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
min(nil) -> 0'
min(cons(x, nil)) -> x
min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs)
if1(true, x, y, xs) -> min(cons(x, xs))
if1(false, x, y, xs) -> min(cons(y, xs))
rm(x, nil) -> nil
rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) -> rm(x, xs)
if2(false, x, y, xs) -> cons(y, rm(x, xs))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
eq :: 0':s -> 0':s -> true:false
minsort :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
min :: nil:cons -> 0':s
rm :: 0':s -> nil:cons -> nil:cons
if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s
if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
hole_true:false1_0 :: true:false
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

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
le, eq, minsort, min, rm

They will be analysed ascendingly in the following order:
le < min
eq < rm
min < minsort
rm < minsort

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

(6)
Obligation:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
eq(0', 0') -> true
eq(0', s(y)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
minsort(nil) -> nil
minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
min(nil) -> 0'
min(cons(x, nil)) -> x
min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs)
if1(true, x, y, xs) -> min(cons(x, xs))
if1(false, x, y, xs) -> min(cons(y, xs))
rm(x, nil) -> nil
rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) -> rm(x, xs)
if2(false, x, y, xs) -> cons(y, rm(x, xs))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
eq :: 0':s -> 0':s -> true:false
minsort :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
min :: nil:cons -> 0':s
rm :: 0':s -> nil:cons -> nil:cons
if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s
if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
hole_true:false1_0 :: true:false
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(0', gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
le, eq, minsort, min, rm

They will be analysed ascendingly in the following order:
le < min
eq < rm
min < minsort
rm < minsort

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

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

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

Induction Step:
le(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) ->_R^Omega(1)
le(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).
----------------------------------------

(8)
Complex Obligation (BEST)

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

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
eq(0', 0') -> true
eq(0', s(y)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
minsort(nil) -> nil
minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
min(nil) -> 0'
min(cons(x, nil)) -> x
min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs)
if1(true, x, y, xs) -> min(cons(x, xs))
if1(false, x, y, xs) -> min(cons(y, xs))
rm(x, nil) -> nil
rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) -> rm(x, xs)
if2(false, x, y, xs) -> cons(y, rm(x, xs))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
eq :: 0':s -> 0':s -> true:false
minsort :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
min :: nil:cons -> 0':s
rm :: 0':s -> nil:cons -> nil:cons
if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s
if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
hole_true:false1_0 :: true:false
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(0', gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
le, eq, minsort, min, rm

They will be analysed ascendingly in the following order:
le < min
eq < rm
min < minsort
rm < minsort

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
eq(0', 0') -> true
eq(0', s(y)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
minsort(nil) -> nil
minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
min(nil) -> 0'
min(cons(x, nil)) -> x
min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs)
if1(true, x, y, xs) -> min(cons(x, xs))
if1(false, x, y, xs) -> min(cons(y, xs))
rm(x, nil) -> nil
rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) -> rm(x, xs)
if2(false, x, y, xs) -> cons(y, rm(x, xs))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
eq :: 0':s -> 0':s -> true:false
minsort :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
min :: nil:cons -> 0':s
rm :: 0':s -> nil:cons -> nil:cons
if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s
if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
hole_true:false1_0 :: true:false
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:
le(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(0', gen_nil:cons5_0(x))


The following defined symbols remain to be analysed:
eq, minsort, min, rm

They will be analysed ascendingly in the following order:
eq < rm
min < minsort
rm < minsort

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
eq(gen_0':s4_0(n306_0), gen_0':s4_0(n306_0)) -> true, rt in Omega(1 + n306_0)

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

Induction Step:
eq(gen_0':s4_0(+(n306_0, 1)), gen_0':s4_0(+(n306_0, 1))) ->_R^Omega(1)
eq(gen_0':s4_0(n306_0), gen_0':s4_0(n306_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:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
eq(0', 0') -> true
eq(0', s(y)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
minsort(nil) -> nil
minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
min(nil) -> 0'
min(cons(x, nil)) -> x
min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs)
if1(true, x, y, xs) -> min(cons(x, xs))
if1(false, x, y, xs) -> min(cons(y, xs))
rm(x, nil) -> nil
rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) -> rm(x, xs)
if2(false, x, y, xs) -> cons(y, rm(x, xs))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
eq :: 0':s -> 0':s -> true:false
minsort :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
min :: nil:cons -> 0':s
rm :: 0':s -> nil:cons -> nil:cons
if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s
if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
hole_true:false1_0 :: true:false
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:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)
eq(gen_0':s4_0(n306_0), gen_0':s4_0(n306_0)) -> true, rt in Omega(1 + n306_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:
min, minsort, rm

They will be analysed ascendingly in the following order:
min < minsort
rm < minsort

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
min(gen_nil:cons5_0(+(1, n823_0))) -> gen_0':s4_0(0), rt in Omega(1 + n823_0)

Induction Base:
min(gen_nil:cons5_0(+(1, 0))) ->_R^Omega(1)
0'

Induction Step:
min(gen_nil:cons5_0(+(1, +(n823_0, 1)))) ->_R^Omega(1)
if1(le(0', 0'), 0', 0', gen_nil:cons5_0(n823_0)) ->_L^Omega(1)
if1(true, 0', 0', gen_nil:cons5_0(n823_0)) ->_R^Omega(1)
min(cons(0', gen_nil:cons5_0(n823_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).
----------------------------------------

(16)
Obligation:
TRS:
Rules:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
eq(0', 0') -> true
eq(0', s(y)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
minsort(nil) -> nil
minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
min(nil) -> 0'
min(cons(x, nil)) -> x
min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs)
if1(true, x, y, xs) -> min(cons(x, xs))
if1(false, x, y, xs) -> min(cons(y, xs))
rm(x, nil) -> nil
rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) -> rm(x, xs)
if2(false, x, y, xs) -> cons(y, rm(x, xs))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
eq :: 0':s -> 0':s -> true:false
minsort :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
min :: nil:cons -> 0':s
rm :: 0':s -> nil:cons -> nil:cons
if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s
if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
hole_true:false1_0 :: true:false
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:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)
eq(gen_0':s4_0(n306_0), gen_0':s4_0(n306_0)) -> true, rt in Omega(1 + n306_0)
min(gen_nil:cons5_0(+(1, n823_0))) -> gen_0':s4_0(0), rt in Omega(1 + n823_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:
rm, minsort

They will be analysed ascendingly in the following order:
rm < minsort

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
rm(gen_0':s4_0(0), gen_nil:cons5_0(n1254_0)) -> gen_nil:cons5_0(0), rt in Omega(1 + n1254_0)

Induction Base:
rm(gen_0':s4_0(0), gen_nil:cons5_0(0)) ->_R^Omega(1)
nil

Induction Step:
rm(gen_0':s4_0(0), gen_nil:cons5_0(+(n1254_0, 1))) ->_R^Omega(1)
if2(eq(gen_0':s4_0(0), 0'), gen_0':s4_0(0), 0', gen_nil:cons5_0(n1254_0)) ->_L^Omega(1)
if2(true, gen_0':s4_0(0), 0', gen_nil:cons5_0(n1254_0)) ->_R^Omega(1)
rm(gen_0':s4_0(0), gen_nil:cons5_0(n1254_0)) ->_IH
gen_nil:cons5_0(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:
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
eq(0', 0') -> true
eq(0', s(y)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
minsort(nil) -> nil
minsort(cons(x, xs)) -> cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
min(nil) -> 0'
min(cons(x, nil)) -> x
min(cons(x, cons(y, xs))) -> if1(le(x, y), x, y, xs)
if1(true, x, y, xs) -> min(cons(x, xs))
if1(false, x, y, xs) -> min(cons(y, xs))
rm(x, nil) -> nil
rm(x, cons(y, xs)) -> if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) -> rm(x, xs)
if2(false, x, y, xs) -> cons(y, rm(x, xs))

Types:
le :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
eq :: 0':s -> 0':s -> true:false
minsort :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0':s -> nil:cons -> nil:cons
min :: nil:cons -> 0':s
rm :: 0':s -> nil:cons -> nil:cons
if1 :: true:false -> 0':s -> 0':s -> nil:cons -> 0':s
if2 :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
hole_true:false1_0 :: true:false
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:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)
eq(gen_0':s4_0(n306_0), gen_0':s4_0(n306_0)) -> true, rt in Omega(1 + n306_0)
min(gen_nil:cons5_0(+(1, n823_0))) -> gen_0':s4_0(0), rt in Omega(1 + n823_0)
rm(gen_0':s4_0(0), gen_nil:cons5_0(n1254_0)) -> gen_nil:cons5_0(0), rt in Omega(1 + n1254_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:
minsort