/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) 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), 260 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), 50 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 35 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:

   eq(0, 0) -> true
   eq(0, s(m)) -> false
   eq(s(n), 0) -> false
   eq(s(n), s(m)) -> eq(n, m)
   le(0, m) -> true
   le(s(n), 0) -> false
   le(s(n), s(m)) -> le(n, m)
   min(cons(x, nil)) -> x
   min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x)))
   if_min(true, cons(n, cons(m, x))) -> min(cons(n, x))
   if_min(false, cons(n, cons(m, x))) -> min(cons(m, x))
   replace(n, m, nil) -> nil
   replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x))
   if_replace(true, n, m, cons(k, x)) -> cons(m, x)
   if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x))
   empty(nil) -> true
   empty(cons(n, x)) -> false
   head(cons(n, x)) -> n
   tail(nil) -> nil
   tail(cons(n, x)) -> x
   sort(x) -> sortIter(x, nil)
   sortIter(x, y) -> if(empty(x), x, y, append(y, cons(min(x), nil)))
   if(true, x, y, z) -> y
   if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z)

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:

   eq(0', 0') -> true
   eq(0', s(m)) -> false
   eq(s(n), 0') -> false
   eq(s(n), s(m)) -> eq(n, m)
   le(0', m) -> true
   le(s(n), 0') -> false
   le(s(n), s(m)) -> le(n, m)
   min(cons(x, nil)) -> x
   min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x)))
   if_min(true, cons(n, cons(m, x))) -> min(cons(n, x))
   if_min(false, cons(n, cons(m, x))) -> min(cons(m, x))
   replace(n, m, nil) -> nil
   replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x))
   if_replace(true, n, m, cons(k, x)) -> cons(m, x)
   if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x))
   empty(nil) -> true
   empty(cons(n, x)) -> false
   head(cons(n, x)) -> n
   tail(nil) -> nil
   tail(cons(n, x)) -> x
   sort(x) -> sortIter(x, nil)
   sortIter(x, y) -> if(empty(x), x, y, append(y, cons(min(x), nil)))
   if(true, x, y, z) -> y
   if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z)

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

(3) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
append/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:

   eq(0', 0') -> true
   eq(0', s(m)) -> false
   eq(s(n), 0') -> false
   eq(s(n), s(m)) -> eq(n, m)
   le(0', m) -> true
   le(s(n), 0') -> false
   le(s(n), s(m)) -> le(n, m)
   min(cons(x, nil)) -> x
   min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x)))
   if_min(true, cons(n, cons(m, x))) -> min(cons(n, x))
   if_min(false, cons(n, cons(m, x))) -> min(cons(m, x))
   replace(n, m, nil) -> nil
   replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x))
   if_replace(true, n, m, cons(k, x)) -> cons(m, x)
   if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x))
   empty(nil) -> true
   empty(cons(n, x)) -> false
   head(cons(n, x)) -> n
   tail(nil) -> nil
   tail(cons(n, x)) -> x
   sort(x) -> sortIter(x, nil)
   sortIter(x, y) -> if(empty(x), x, y, append(cons(min(x), nil)))
   if(true, x, y, z) -> y
   if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z)

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

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

(6)
Obligation:
TRS:
Rules:
eq(0', 0') -> true
eq(0', s(m)) -> false
eq(s(n), 0') -> false
eq(s(n), s(m)) -> eq(n, m)
le(0', m) -> true
le(s(n), 0') -> false
le(s(n), s(m)) -> le(n, m)
min(cons(x, nil)) -> x
min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x)))
if_min(true, cons(n, cons(m, x))) -> min(cons(n, x))
if_min(false, cons(n, cons(m, x))) -> min(cons(m, x))
replace(n, m, nil) -> nil
replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x))
if_replace(true, n, m, cons(k, x)) -> cons(m, x)
if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x))
empty(nil) -> true
empty(cons(n, x)) -> false
head(cons(n, x)) -> n
tail(nil) -> nil
tail(cons(n, x)) -> x
sort(x) -> sortIter(x, nil)
sortIter(x, y) -> if(empty(x), x, y, append(cons(min(x), nil)))
if(true, x, y, z) -> y
if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z)

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:append -> 0':s
cons :: 0':s -> nil:cons:append -> nil:cons:append
nil :: nil:cons:append
if_min :: true:false -> nil:cons:append -> 0':s
replace :: 0':s -> 0':s -> nil:cons:append -> nil:cons:append
if_replace :: true:false -> 0':s -> 0':s -> nil:cons:append -> nil:cons:append
empty :: nil:cons:append -> true:false
head :: nil:cons:append -> 0':s
tail :: nil:cons:append -> nil:cons:append
sort :: nil:cons:append -> nil:cons:append
sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append
if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append
append :: nil:cons:append -> nil:cons:append
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons:append3_0 :: nil:cons:append
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons:append5_0 :: Nat -> nil:cons:append

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

(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
eq, le, min, replace, sortIter

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

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

(8)
Obligation:
TRS:
Rules:
eq(0', 0') -> true
eq(0', s(m)) -> false
eq(s(n), 0') -> false
eq(s(n), s(m)) -> eq(n, m)
le(0', m) -> true
le(s(n), 0') -> false
le(s(n), s(m)) -> le(n, m)
min(cons(x, nil)) -> x
min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x)))
if_min(true, cons(n, cons(m, x))) -> min(cons(n, x))
if_min(false, cons(n, cons(m, x))) -> min(cons(m, x))
replace(n, m, nil) -> nil
replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x))
if_replace(true, n, m, cons(k, x)) -> cons(m, x)
if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x))
empty(nil) -> true
empty(cons(n, x)) -> false
head(cons(n, x)) -> n
tail(nil) -> nil
tail(cons(n, x)) -> x
sort(x) -> sortIter(x, nil)
sortIter(x, y) -> if(empty(x), x, y, append(cons(min(x), nil)))
if(true, x, y, z) -> y
if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z)

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:append -> 0':s
cons :: 0':s -> nil:cons:append -> nil:cons:append
nil :: nil:cons:append
if_min :: true:false -> nil:cons:append -> 0':s
replace :: 0':s -> 0':s -> nil:cons:append -> nil:cons:append
if_replace :: true:false -> 0':s -> 0':s -> nil:cons:append -> nil:cons:append
empty :: nil:cons:append -> true:false
head :: nil:cons:append -> 0':s
tail :: nil:cons:append -> nil:cons:append
sort :: nil:cons:append -> nil:cons:append
sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append
if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append
append :: nil:cons:append -> nil:cons:append
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons:append3_0 :: nil:cons:append
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons:append5_0 :: Nat -> nil:cons:append


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:cons:append5_0(0) <=> nil
gen_nil:cons:append5_0(+(x, 1)) <=> cons(0', gen_nil:cons:append5_0(x))


The following defined symbols remain to be analysed:
eq, le, min, replace, sortIter

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

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

(9) 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).
----------------------------------------

(10)
Complex Obligation (BEST)

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

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

TRS:
Rules:
eq(0', 0') -> true
eq(0', s(m)) -> false
eq(s(n), 0') -> false
eq(s(n), s(m)) -> eq(n, m)
le(0', m) -> true
le(s(n), 0') -> false
le(s(n), s(m)) -> le(n, m)
min(cons(x, nil)) -> x
min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x)))
if_min(true, cons(n, cons(m, x))) -> min(cons(n, x))
if_min(false, cons(n, cons(m, x))) -> min(cons(m, x))
replace(n, m, nil) -> nil
replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x))
if_replace(true, n, m, cons(k, x)) -> cons(m, x)
if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x))
empty(nil) -> true
empty(cons(n, x)) -> false
head(cons(n, x)) -> n
tail(nil) -> nil
tail(cons(n, x)) -> x
sort(x) -> sortIter(x, nil)
sortIter(x, y) -> if(empty(x), x, y, append(cons(min(x), nil)))
if(true, x, y, z) -> y
if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z)

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:append -> 0':s
cons :: 0':s -> nil:cons:append -> nil:cons:append
nil :: nil:cons:append
if_min :: true:false -> nil:cons:append -> 0':s
replace :: 0':s -> 0':s -> nil:cons:append -> nil:cons:append
if_replace :: true:false -> 0':s -> 0':s -> nil:cons:append -> nil:cons:append
empty :: nil:cons:append -> true:false
head :: nil:cons:append -> 0':s
tail :: nil:cons:append -> nil:cons:append
sort :: nil:cons:append -> nil:cons:append
sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append
if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append
append :: nil:cons:append -> nil:cons:append
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons:append3_0 :: nil:cons:append
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons:append5_0 :: Nat -> nil:cons:append


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:cons:append5_0(0) <=> nil
gen_nil:cons:append5_0(+(x, 1)) <=> cons(0', gen_nil:cons:append5_0(x))


The following defined symbols remain to be analysed:
eq, le, min, replace, sortIter

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

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

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

(13)
BOUNDS(n^1, INF)

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

(14)
Obligation:
TRS:
Rules:
eq(0', 0') -> true
eq(0', s(m)) -> false
eq(s(n), 0') -> false
eq(s(n), s(m)) -> eq(n, m)
le(0', m) -> true
le(s(n), 0') -> false
le(s(n), s(m)) -> le(n, m)
min(cons(x, nil)) -> x
min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x)))
if_min(true, cons(n, cons(m, x))) -> min(cons(n, x))
if_min(false, cons(n, cons(m, x))) -> min(cons(m, x))
replace(n, m, nil) -> nil
replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x))
if_replace(true, n, m, cons(k, x)) -> cons(m, x)
if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x))
empty(nil) -> true
empty(cons(n, x)) -> false
head(cons(n, x)) -> n
tail(nil) -> nil
tail(cons(n, x)) -> x
sort(x) -> sortIter(x, nil)
sortIter(x, y) -> if(empty(x), x, y, append(cons(min(x), nil)))
if(true, x, y, z) -> y
if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z)

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:append -> 0':s
cons :: 0':s -> nil:cons:append -> nil:cons:append
nil :: nil:cons:append
if_min :: true:false -> nil:cons:append -> 0':s
replace :: 0':s -> 0':s -> nil:cons:append -> nil:cons:append
if_replace :: true:false -> 0':s -> 0':s -> nil:cons:append -> nil:cons:append
empty :: nil:cons:append -> true:false
head :: nil:cons:append -> 0':s
tail :: nil:cons:append -> nil:cons:append
sort :: nil:cons:append -> nil:cons:append
sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append
if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append
append :: nil:cons:append -> nil:cons:append
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons:append3_0 :: nil:cons:append
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons:append5_0 :: Nat -> nil:cons:append


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:cons:append5_0(0) <=> nil
gen_nil:cons:append5_0(+(x, 1)) <=> cons(0', gen_nil:cons:append5_0(x))


The following defined symbols remain to be analysed:
le, min, replace, sortIter

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

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s4_0(n554_0), gen_0':s4_0(n554_0)) -> true, rt in Omega(1 + n554_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(+(n554_0, 1)), gen_0':s4_0(+(n554_0, 1))) ->_R^Omega(1)
le(gen_0':s4_0(n554_0), gen_0':s4_0(n554_0)) ->_IH
true

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(m)) -> false
eq(s(n), 0') -> false
eq(s(n), s(m)) -> eq(n, m)
le(0', m) -> true
le(s(n), 0') -> false
le(s(n), s(m)) -> le(n, m)
min(cons(x, nil)) -> x
min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x)))
if_min(true, cons(n, cons(m, x))) -> min(cons(n, x))
if_min(false, cons(n, cons(m, x))) -> min(cons(m, x))
replace(n, m, nil) -> nil
replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x))
if_replace(true, n, m, cons(k, x)) -> cons(m, x)
if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x))
empty(nil) -> true
empty(cons(n, x)) -> false
head(cons(n, x)) -> n
tail(nil) -> nil
tail(cons(n, x)) -> x
sort(x) -> sortIter(x, nil)
sortIter(x, y) -> if(empty(x), x, y, append(cons(min(x), nil)))
if(true, x, y, z) -> y
if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z)

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:append -> 0':s
cons :: 0':s -> nil:cons:append -> nil:cons:append
nil :: nil:cons:append
if_min :: true:false -> nil:cons:append -> 0':s
replace :: 0':s -> 0':s -> nil:cons:append -> nil:cons:append
if_replace :: true:false -> 0':s -> 0':s -> nil:cons:append -> nil:cons:append
empty :: nil:cons:append -> true:false
head :: nil:cons:append -> 0':s
tail :: nil:cons:append -> nil:cons:append
sort :: nil:cons:append -> nil:cons:append
sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append
if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append
append :: nil:cons:append -> nil:cons:append
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons:append3_0 :: nil:cons:append
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons:append5_0 :: Nat -> nil:cons:append


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(n554_0), gen_0':s4_0(n554_0)) -> true, rt in Omega(1 + n554_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:cons:append5_0(0) <=> nil
gen_nil:cons:append5_0(+(x, 1)) <=> cons(0', gen_nil:cons:append5_0(x))


The following defined symbols remain to be analysed:
min, replace, sortIter

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

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
min(gen_nil:cons:append5_0(+(1, n895_0))) -> gen_0':s4_0(0), rt in Omega(1 + n895_0)

Induction Base:
min(gen_nil:cons:append5_0(+(1, 0))) ->_R^Omega(1)
0'

Induction Step:
min(gen_nil:cons:append5_0(+(1, +(n895_0, 1)))) ->_R^Omega(1)
if_min(le(0', 0'), cons(0', cons(0', gen_nil:cons:append5_0(n895_0)))) ->_L^Omega(1)
if_min(true, cons(0', cons(0', gen_nil:cons:append5_0(n895_0)))) ->_R^Omega(1)
min(cons(0', gen_nil:cons:append5_0(n895_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).
----------------------------------------

(18)
Obligation:
TRS:
Rules:
eq(0', 0') -> true
eq(0', s(m)) -> false
eq(s(n), 0') -> false
eq(s(n), s(m)) -> eq(n, m)
le(0', m) -> true
le(s(n), 0') -> false
le(s(n), s(m)) -> le(n, m)
min(cons(x, nil)) -> x
min(cons(n, cons(m, x))) -> if_min(le(n, m), cons(n, cons(m, x)))
if_min(true, cons(n, cons(m, x))) -> min(cons(n, x))
if_min(false, cons(n, cons(m, x))) -> min(cons(m, x))
replace(n, m, nil) -> nil
replace(n, m, cons(k, x)) -> if_replace(eq(n, k), n, m, cons(k, x))
if_replace(true, n, m, cons(k, x)) -> cons(m, x)
if_replace(false, n, m, cons(k, x)) -> cons(k, replace(n, m, x))
empty(nil) -> true
empty(cons(n, x)) -> false
head(cons(n, x)) -> n
tail(nil) -> nil
tail(cons(n, x)) -> x
sort(x) -> sortIter(x, nil)
sortIter(x, y) -> if(empty(x), x, y, append(cons(min(x), nil)))
if(true, x, y, z) -> y
if(false, x, y, z) -> sortIter(replace(min(x), head(x), tail(x)), z)

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:append -> 0':s
cons :: 0':s -> nil:cons:append -> nil:cons:append
nil :: nil:cons:append
if_min :: true:false -> nil:cons:append -> 0':s
replace :: 0':s -> 0':s -> nil:cons:append -> nil:cons:append
if_replace :: true:false -> 0':s -> 0':s -> nil:cons:append -> nil:cons:append
empty :: nil:cons:append -> true:false
head :: nil:cons:append -> 0':s
tail :: nil:cons:append -> nil:cons:append
sort :: nil:cons:append -> nil:cons:append
sortIter :: nil:cons:append -> nil:cons:append -> nil:cons:append
if :: true:false -> nil:cons:append -> nil:cons:append -> nil:cons:append -> nil:cons:append
append :: nil:cons:append -> nil:cons:append
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_nil:cons:append3_0 :: nil:cons:append
gen_0':s4_0 :: Nat -> 0':s
gen_nil:cons:append5_0 :: Nat -> nil:cons:append


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(n554_0), gen_0':s4_0(n554_0)) -> true, rt in Omega(1 + n554_0)
min(gen_nil:cons:append5_0(+(1, n895_0))) -> gen_0':s4_0(0), rt in Omega(1 + n895_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_nil:cons:append5_0(0) <=> nil
gen_nil:cons:append5_0(+(x, 1)) <=> cons(0', gen_nil:cons:append5_0(x))


The following defined symbols remain to be analysed:
replace, sortIter

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