### (0) Obligation:

Runtime Complexity TRS:
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)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
le(s(x), s(y)) →+ le(x, y)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x / s(x), y / s(y)].
The result substitution is [ ].

### (3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

### (4) Obligation:

Runtime Complexity Relative TRS:
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)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

S is empty.
Rewrite Strategy: FULL

Infered types.

### (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)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

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
if1 :: true:false → 0':s → 0':s → cons:nil → 0':s
min :: 0':s → cons:nil → 0':s
if2 :: true:false → 0':s → 0':s → cons:nil → cons:nil
cons :: 0':s → cons:nil → cons:nil
del :: 0':s → cons:nil → cons:nil
minsort :: cons:nil → cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

### (7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
le, eq, min, del, minsort

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

### (8) 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)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

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
if1 :: true:false → 0':s → 0':s → cons:nil → 0':s
min :: 0':s → cons:nil → 0':s
if2 :: true:false → 0':s → 0':s → cons:nil → cons:nil
cons :: 0':s → cons:nil → cons:nil
del :: 0':s → cons:nil → cons:nil
minsort :: cons:nil → cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

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

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

### (9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

Induction Base:
le(gen_0':s4_0(0), gen_0':s4_0(0)) →RΩ(1)
true

Induction Step:
le(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) →RΩ(1)
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) →IH
true

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (11) 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)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

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
if1 :: true:false → 0':s → 0':s → cons:nil → 0':s
min :: 0':s → cons:nil → 0':s
if2 :: true:false → 0':s → 0':s → cons:nil → cons:nil
cons :: 0':s → cons:nil → cons:nil
del :: 0':s → cons:nil → cons:nil
minsort :: cons:nil → cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

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

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

### (12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
eq(gen_0':s4_0(n324_0), gen_0':s4_0(n324_0)) → true, rt ∈ Ω(1 + n3240)

Induction Base:
eq(gen_0':s4_0(0), gen_0':s4_0(0)) →RΩ(1)
true

Induction Step:
eq(gen_0':s4_0(+(n324_0, 1)), gen_0':s4_0(+(n324_0, 1))) →RΩ(1)
eq(gen_0':s4_0(n324_0), gen_0':s4_0(n324_0)) →IH
true

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(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)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

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
if1 :: true:false → 0':s → 0':s → cons:nil → 0':s
min :: 0':s → cons:nil → 0':s
if2 :: true:false → 0':s → 0':s → cons:nil → cons:nil
cons :: 0':s → cons:nil → cons:nil
del :: 0':s → cons:nil → cons:nil
minsort :: cons:nil → cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)
eq(gen_0':s4_0(n324_0), gen_0':s4_0(n324_0)) → true, rt ∈ Ω(1 + n3240)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

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

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

### (15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
min(gen_0':s4_0(0), gen_cons:nil5_0(n881_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n8810)

Induction Base:
min(gen_0':s4_0(0), gen_cons:nil5_0(0)) →RΩ(1)
gen_0':s4_0(0)

Induction Step:
min(gen_0':s4_0(0), gen_cons:nil5_0(+(n881_0, 1))) →RΩ(1)
if1(le(gen_0':s4_0(0), 0'), gen_0':s4_0(0), 0', gen_cons:nil5_0(n881_0)) →LΩ(1)
if1(true, gen_0':s4_0(0), 0', gen_cons:nil5_0(n881_0)) →RΩ(1)
min(gen_0':s4_0(0), gen_cons:nil5_0(n881_0)) →IH
gen_0':s4_0(0)

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (17) 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)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

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
if1 :: true:false → 0':s → 0':s → cons:nil → 0':s
min :: 0':s → cons:nil → 0':s
if2 :: true:false → 0':s → 0':s → cons:nil → cons:nil
cons :: 0':s → cons:nil → cons:nil
del :: 0':s → cons:nil → cons:nil
minsort :: cons:nil → cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)
eq(gen_0':s4_0(n324_0), gen_0':s4_0(n324_0)) → true, rt ∈ Ω(1 + n3240)
min(gen_0':s4_0(0), gen_cons:nil5_0(n881_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n8810)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

The following defined symbols remain to be analysed:
del, minsort

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

### (18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol del.

### (19) 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)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

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
if1 :: true:false → 0':s → 0':s → cons:nil → 0':s
min :: 0':s → cons:nil → 0':s
if2 :: true:false → 0':s → 0':s → cons:nil → cons:nil
cons :: 0':s → cons:nil → cons:nil
del :: 0':s → cons:nil → cons:nil
minsort :: cons:nil → cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)
eq(gen_0':s4_0(n324_0), gen_0':s4_0(n324_0)) → true, rt ∈ Ω(1 + n3240)
min(gen_0':s4_0(0), gen_cons:nil5_0(n881_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n8810)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

The following defined symbols remain to be analysed:
minsort

### (20) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
minsort(gen_cons:nil5_0(n1604_0)) → gen_cons:nil5_0(n1604_0), rt ∈ Ω(1 + n16040 + n160402)

Induction Base:
minsort(gen_cons:nil5_0(0)) →RΩ(1)
nil

Induction Step:
minsort(gen_cons:nil5_0(+(n1604_0, 1))) →RΩ(1)
cons(min(0', gen_cons:nil5_0(n1604_0)), minsort(del(min(0', gen_cons:nil5_0(n1604_0)), cons(0', gen_cons:nil5_0(n1604_0))))) →LΩ(1 + n16040)
cons(gen_0':s4_0(0), minsort(del(min(0', gen_cons:nil5_0(n1604_0)), cons(0', gen_cons:nil5_0(n1604_0))))) →LΩ(1 + n16040)
cons(gen_0':s4_0(0), minsort(del(gen_0':s4_0(0), cons(0', gen_cons:nil5_0(n1604_0))))) →RΩ(1)
cons(gen_0':s4_0(0), minsort(if2(eq(gen_0':s4_0(0), 0'), gen_0':s4_0(0), 0', gen_cons:nil5_0(n1604_0)))) →LΩ(1)
cons(gen_0':s4_0(0), minsort(if2(true, gen_0':s4_0(0), 0', gen_cons:nil5_0(n1604_0)))) →RΩ(1)
cons(gen_0':s4_0(0), minsort(gen_cons:nil5_0(n1604_0))) →IH
cons(gen_0':s4_0(0), gen_cons:nil5_0(c1605_0))

We have rt ∈ Ω(n2) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n2).

### (22) 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)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

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
if1 :: true:false → 0':s → 0':s → cons:nil → 0':s
min :: 0':s → cons:nil → 0':s
if2 :: true:false → 0':s → 0':s → cons:nil → cons:nil
cons :: 0':s → cons:nil → cons:nil
del :: 0':s → cons:nil → cons:nil
minsort :: cons:nil → cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)
eq(gen_0':s4_0(n324_0), gen_0':s4_0(n324_0)) → true, rt ∈ Ω(1 + n3240)
min(gen_0':s4_0(0), gen_cons:nil5_0(n881_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n8810)
minsort(gen_cons:nil5_0(n1604_0)) → gen_cons:nil5_0(n1604_0), rt ∈ Ω(1 + n16040 + n160402)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

No more defined symbols left to analyse.

### (23) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
minsort(gen_cons:nil5_0(n1604_0)) → gen_cons:nil5_0(n1604_0), rt ∈ Ω(1 + n16040 + n160402)

### (25) 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)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

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
if1 :: true:false → 0':s → 0':s → cons:nil → 0':s
min :: 0':s → cons:nil → 0':s
if2 :: true:false → 0':s → 0':s → cons:nil → cons:nil
cons :: 0':s → cons:nil → cons:nil
del :: 0':s → cons:nil → cons:nil
minsort :: cons:nil → cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)
eq(gen_0':s4_0(n324_0), gen_0':s4_0(n324_0)) → true, rt ∈ Ω(1 + n3240)
min(gen_0':s4_0(0), gen_cons:nil5_0(n881_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n8810)
minsort(gen_cons:nil5_0(n1604_0)) → gen_cons:nil5_0(n1604_0), rt ∈ Ω(1 + n16040 + n160402)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

No more defined symbols left to analyse.

### (26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:
minsort(gen_cons:nil5_0(n1604_0)) → gen_cons:nil5_0(n1604_0), rt ∈ Ω(1 + n16040 + n160402)

### (28) 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)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

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
if1 :: true:false → 0':s → 0':s → cons:nil → 0':s
min :: 0':s → cons:nil → 0':s
if2 :: true:false → 0':s → 0':s → cons:nil → cons:nil
cons :: 0':s → cons:nil → cons:nil
del :: 0':s → cons:nil → cons:nil
minsort :: cons:nil → cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)
eq(gen_0':s4_0(n324_0), gen_0':s4_0(n324_0)) → true, rt ∈ Ω(1 + n3240)
min(gen_0':s4_0(0), gen_cons:nil5_0(n881_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n8810)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

No more defined symbols left to analyse.

### (29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

### (31) 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)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

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
if1 :: true:false → 0':s → 0':s → cons:nil → 0':s
min :: 0':s → cons:nil → 0':s
if2 :: true:false → 0':s → 0':s → cons:nil → cons:nil
cons :: 0':s → cons:nil → cons:nil
del :: 0':s → cons:nil → cons:nil
minsort :: cons:nil → cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)
eq(gen_0':s4_0(n324_0), gen_0':s4_0(n324_0)) → true, rt ∈ Ω(1 + n3240)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

No more defined symbols left to analyse.

### (32) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

### (34) 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)
if1(true, x, y, xs) → min(x, xs)
if1(false, x, y, xs) → min(y, xs)
if2(true, x, y, xs) → xs
if2(false, x, y, xs) → cons(y, del(x, xs))
minsort(nil) → nil
minsort(cons(x, y)) → cons(min(x, y), minsort(del(min(x, y), cons(x, y))))
min(x, nil) → x
min(x, cons(y, z)) → if1(le(x, y), x, y, z)
del(x, nil) → nil
del(x, cons(y, z)) → if2(eq(x, y), x, y, z)

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
if1 :: true:false → 0':s → 0':s → cons:nil → 0':s
min :: 0':s → cons:nil → 0':s
if2 :: true:false → 0':s → 0':s → cons:nil → cons:nil
cons :: 0':s → cons:nil → cons:nil
del :: 0':s → cons:nil → cons:nil
minsort :: cons:nil → cons:nil
nil :: cons:nil
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_cons:nil3_0 :: cons:nil
gen_0':s4_0 :: Nat → 0':s
gen_cons:nil5_0 :: Nat → cons:nil

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))
gen_cons:nil5_0(0) ⇔ nil
gen_cons:nil5_0(+(x, 1)) ⇔ cons(0', gen_cons:nil5_0(x))

No more defined symbols left to analyse.

### (35) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)