/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^2), ?)
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 304 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), 43 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 26 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 53 ms]
        (18) proven lower bound
        (19) LowerBoundPropagationProof [FINISHED, 0 ms]
        (20) 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:

   eq(0, 0) -> true
   eq(0, s(Y)) -> false
   eq(s(X), 0) -> false
   eq(s(X), s(Y)) -> eq(X, Y)
   le(0, Y) -> true
   le(s(X), 0) -> false
   le(s(X), s(Y)) -> le(X, Y)
   min(cons(0, nil)) -> 0
   min(cons(s(N), nil)) -> s(N)
   min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L)))
   ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L))
   ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L))
   replace(N, M, nil) -> nil
   replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L))
   ifrepl(true, N, M, cons(K, L)) -> cons(M, L)
   ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L))
   selsort(nil) -> nil
   selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L))
   ifselsort(true, cons(N, L)) -> cons(N, selsort(L))
   ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

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^2, INF).


The TRS R consists of the following rules:

   eq(0', 0') -> true
   eq(0', s(Y)) -> false
   eq(s(X), 0') -> false
   eq(s(X), s(Y)) -> eq(X, Y)
   le(0', Y) -> true
   le(s(X), 0') -> false
   le(s(X), s(Y)) -> le(X, Y)
   min(cons(0', nil)) -> 0'
   min(cons(s(N), nil)) -> s(N)
   min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L)))
   ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L))
   ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L))
   replace(N, M, nil) -> nil
   replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L))
   ifrepl(true, N, M, cons(K, L)) -> cons(M, L)
   ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L))
   selsort(nil) -> nil
   selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L))
   ifselsort(true, cons(N, L)) -> cons(N, selsort(L))
   ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

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

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

(4)
Obligation:
TRS:
Rules:
eq(0', 0') -> true
eq(0', s(Y)) -> false
eq(s(X), 0') -> false
eq(s(X), s(Y)) -> eq(X, Y)
le(0', Y) -> true
le(s(X), 0') -> false
le(s(X), s(Y)) -> le(X, Y)
min(cons(0', nil)) -> 0'
min(cons(s(N), nil)) -> s(N)
min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L)))
ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L))
ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L))
replace(N, M, nil) -> nil
replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L))
ifrepl(true, N, M, cons(K, L)) -> cons(M, L)
ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L))
selsort(nil) -> nil
selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L))
ifselsort(true, cons(N, L)) -> cons(N, selsort(L))
ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

Types:
eq :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
le :: 0':s -> 0':s -> true:false
min :: nil:cons -> 0':s
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
ifmin :: true:false -> nil:cons -> 0':s
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
selsort :: nil:cons -> nil:cons
ifselsort :: true:false -> 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:
eq, le, min, replace, selsort

They will be analysed ascendingly in the following order:
eq < replace
eq < selsort
le < min
min < selsort
replace < selsort

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

(6)
Obligation:
TRS:
Rules:
eq(0', 0') -> true
eq(0', s(Y)) -> false
eq(s(X), 0') -> false
eq(s(X), s(Y)) -> eq(X, Y)
le(0', Y) -> true
le(s(X), 0') -> false
le(s(X), s(Y)) -> le(X, Y)
min(cons(0', nil)) -> 0'
min(cons(s(N), nil)) -> s(N)
min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L)))
ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L))
ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L))
replace(N, M, nil) -> nil
replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L))
ifrepl(true, N, M, cons(K, L)) -> cons(M, L)
ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L))
selsort(nil) -> nil
selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L))
ifselsort(true, cons(N, L)) -> cons(N, selsort(L))
ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

Types:
eq :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
le :: 0':s -> 0':s -> true:false
min :: nil:cons -> 0':s
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
ifmin :: true:false -> nil:cons -> 0':s
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
selsort :: nil:cons -> nil:cons
ifselsort :: true:false -> 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:
eq, le, min, replace, selsort

They will be analysed ascendingly in the following order:
eq < replace
eq < selsort
le < min
min < selsort
replace < selsort

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_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(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) ->_R^Omega(1)
eq(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:
eq(0', 0') -> true
eq(0', s(Y)) -> false
eq(s(X), 0') -> false
eq(s(X), s(Y)) -> eq(X, Y)
le(0', Y) -> true
le(s(X), 0') -> false
le(s(X), s(Y)) -> le(X, Y)
min(cons(0', nil)) -> 0'
min(cons(s(N), nil)) -> s(N)
min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L)))
ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L))
ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L))
replace(N, M, nil) -> nil
replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L))
ifrepl(true, N, M, cons(K, L)) -> cons(M, L)
ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L))
selsort(nil) -> nil
selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L))
ifselsort(true, cons(N, L)) -> cons(N, selsort(L))
ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

Types:
eq :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
le :: 0':s -> 0':s -> true:false
min :: nil:cons -> 0':s
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
ifmin :: true:false -> nil:cons -> 0':s
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
selsort :: nil:cons -> nil:cons
ifselsort :: true:false -> 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:
eq, le, min, replace, selsort

They will be analysed ascendingly in the following order:
eq < replace
eq < selsort
le < min
min < selsort
replace < selsort

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
eq(0', 0') -> true
eq(0', s(Y)) -> false
eq(s(X), 0') -> false
eq(s(X), s(Y)) -> eq(X, Y)
le(0', Y) -> true
le(s(X), 0') -> false
le(s(X), s(Y)) -> le(X, Y)
min(cons(0', nil)) -> 0'
min(cons(s(N), nil)) -> s(N)
min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L)))
ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L))
ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L))
replace(N, M, nil) -> nil
replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L))
ifrepl(true, N, M, cons(K, L)) -> cons(M, L)
ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L))
selsort(nil) -> nil
selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L))
ifselsort(true, cons(N, L)) -> cons(N, selsort(L))
ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

Types:
eq :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
le :: 0':s -> 0':s -> true:false
min :: nil:cons -> 0':s
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
ifmin :: true:false -> nil:cons -> 0':s
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
selsort :: nil:cons -> nil:cons
ifselsort :: true:false -> 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:
eq(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:
le, min, replace, selsort

They will be analysed ascendingly in the following order:
le < min
min < selsort
replace < selsort

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s4_0(n530_0), gen_0':s4_0(n530_0)) -> true, rt in Omega(1 + n530_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(+(n530_0, 1)), gen_0':s4_0(+(n530_0, 1))) ->_R^Omega(1)
le(gen_0':s4_0(n530_0), gen_0':s4_0(n530_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:
eq(0', 0') -> true
eq(0', s(Y)) -> false
eq(s(X), 0') -> false
eq(s(X), s(Y)) -> eq(X, Y)
le(0', Y) -> true
le(s(X), 0') -> false
le(s(X), s(Y)) -> le(X, Y)
min(cons(0', nil)) -> 0'
min(cons(s(N), nil)) -> s(N)
min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L)))
ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L))
ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L))
replace(N, M, nil) -> nil
replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L))
ifrepl(true, N, M, cons(K, L)) -> cons(M, L)
ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L))
selsort(nil) -> nil
selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L))
ifselsort(true, cons(N, L)) -> cons(N, selsort(L))
ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

Types:
eq :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
le :: 0':s -> 0':s -> true:false
min :: nil:cons -> 0':s
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
ifmin :: true:false -> nil:cons -> 0':s
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
selsort :: nil:cons -> nil:cons
ifselsort :: true:false -> 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:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)
le(gen_0':s4_0(n530_0), gen_0':s4_0(n530_0)) -> true, rt in Omega(1 + n530_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, replace, selsort

They will be analysed ascendingly in the following order:
min < selsort
replace < selsort

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
min(gen_nil:cons5_0(+(1, n847_0))) -> gen_0':s4_0(0), rt in Omega(1 + n847_0)

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

Induction Step:
min(gen_nil:cons5_0(+(1, +(n847_0, 1)))) ->_R^Omega(1)
ifmin(le(0', 0'), cons(0', cons(0', gen_nil:cons5_0(n847_0)))) ->_L^Omega(1)
ifmin(true, cons(0', cons(0', gen_nil:cons5_0(n847_0)))) ->_R^Omega(1)
min(cons(0', gen_nil:cons5_0(n847_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:
eq(0', 0') -> true
eq(0', s(Y)) -> false
eq(s(X), 0') -> false
eq(s(X), s(Y)) -> eq(X, Y)
le(0', Y) -> true
le(s(X), 0') -> false
le(s(X), s(Y)) -> le(X, Y)
min(cons(0', nil)) -> 0'
min(cons(s(N), nil)) -> s(N)
min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L)))
ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L))
ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L))
replace(N, M, nil) -> nil
replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L))
ifrepl(true, N, M, cons(K, L)) -> cons(M, L)
ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L))
selsort(nil) -> nil
selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L))
ifselsort(true, cons(N, L)) -> cons(N, selsort(L))
ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

Types:
eq :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
le :: 0':s -> 0':s -> true:false
min :: nil:cons -> 0':s
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
ifmin :: true:false -> nil:cons -> 0':s
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
selsort :: nil:cons -> nil:cons
ifselsort :: true:false -> 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:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)
le(gen_0':s4_0(n530_0), gen_0':s4_0(n530_0)) -> true, rt in Omega(1 + n530_0)
min(gen_nil:cons5_0(+(1, n847_0))) -> gen_0':s4_0(0), rt in Omega(1 + n847_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:
replace, selsort

They will be analysed ascendingly in the following order:
replace < selsort

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
selsort(gen_nil:cons5_0(n1534_0)) -> gen_nil:cons5_0(n1534_0), rt in Omega(1 + n1534_0 + n1534_0^2)

Induction Base:
selsort(gen_nil:cons5_0(0)) ->_R^Omega(1)
nil

Induction Step:
selsort(gen_nil:cons5_0(+(n1534_0, 1))) ->_R^Omega(1)
ifselsort(eq(0', min(cons(0', gen_nil:cons5_0(n1534_0)))), cons(0', gen_nil:cons5_0(n1534_0))) ->_L^Omega(1 + n1534_0)
ifselsort(eq(0', gen_0':s4_0(0)), cons(0', gen_nil:cons5_0(n1534_0))) ->_L^Omega(1)
ifselsort(true, cons(0', gen_nil:cons5_0(n1534_0))) ->_R^Omega(1)
cons(0', selsort(gen_nil:cons5_0(n1534_0))) ->_IH
cons(0', gen_nil:cons5_0(c1535_0))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
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(18)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
eq(0', 0') -> true
eq(0', s(Y)) -> false
eq(s(X), 0') -> false
eq(s(X), s(Y)) -> eq(X, Y)
le(0', Y) -> true
le(s(X), 0') -> false
le(s(X), s(Y)) -> le(X, Y)
min(cons(0', nil)) -> 0'
min(cons(s(N), nil)) -> s(N)
min(cons(N, cons(M, L))) -> ifmin(le(N, M), cons(N, cons(M, L)))
ifmin(true, cons(N, cons(M, L))) -> min(cons(N, L))
ifmin(false, cons(N, cons(M, L))) -> min(cons(M, L))
replace(N, M, nil) -> nil
replace(N, M, cons(K, L)) -> ifrepl(eq(N, K), N, M, cons(K, L))
ifrepl(true, N, M, cons(K, L)) -> cons(M, L)
ifrepl(false, N, M, cons(K, L)) -> cons(K, replace(N, M, L))
selsort(nil) -> nil
selsort(cons(N, L)) -> ifselsort(eq(N, min(cons(N, L))), cons(N, L))
ifselsort(true, cons(N, L)) -> cons(N, selsort(L))
ifselsort(false, cons(N, L)) -> cons(min(cons(N, L)), selsort(replace(min(cons(N, L)), N, L)))

Types:
eq :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
le :: 0':s -> 0':s -> true:false
min :: nil:cons -> 0':s
cons :: 0':s -> nil:cons -> nil:cons
nil :: nil:cons
ifmin :: true:false -> nil:cons -> 0':s
replace :: 0':s -> 0':s -> nil:cons -> nil:cons
ifrepl :: true:false -> 0':s -> 0':s -> nil:cons -> nil:cons
selsort :: nil:cons -> nil:cons
ifselsort :: true:false -> 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:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)
le(gen_0':s4_0(n530_0), gen_0':s4_0(n530_0)) -> true, rt in Omega(1 + n530_0)
min(gen_nil:cons5_0(+(1, n847_0))) -> gen_0':s4_0(0), rt in Omega(1 + n847_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:
selsort
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(19) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(20)
BOUNDS(n^2, INF)