### (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)
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))

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)
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

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)
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

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

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

### (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)
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

### (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)
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 ∈ Ω(1 + n70)

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

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

Proved the following rewrite lemma:
eq(gen_0':s4_0(n330_0), gen_0':s4_0(n330_0)) → true, rt ∈ Ω(1 + n3300)

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

Induction Step:
eq(gen_0':s4_0(+(n330_0, 1)), gen_0':s4_0(+(n330_0, 1))) →RΩ(1)
eq(gen_0':s4_0(n330_0), gen_0':s4_0(n330_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)
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 ∈ Ω(1 + n70)
eq(gen_0':s4_0(n330_0), gen_0':s4_0(n330_0)) → true, rt ∈ Ω(1 + n3300)

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) transformation)

Proved the following rewrite lemma:
min(gen_nil:cons5_0(+(1, n893_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n8930)

Induction Base:
min(gen_nil:cons5_0(+(1, 0))) →RΩ(1)
0'

Induction Step:
min(gen_nil:cons5_0(+(1, +(n893_0, 1)))) →RΩ(1)
if1(le(0', 0'), 0', 0', gen_nil:cons5_0(n893_0)) →LΩ(1)
if1(true, 0', 0', gen_nil:cons5_0(n893_0)) →RΩ(1)
min(cons(0', gen_nil:cons5_0(n893_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)
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 ∈ Ω(1 + n70)
eq(gen_0':s4_0(n330_0), gen_0':s4_0(n330_0)) → true, rt ∈ Ω(1 + n3300)
min(gen_nil:cons5_0(+(1, n893_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n8930)

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

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

Proved the following rewrite lemma:
rm(gen_0':s4_0(0), gen_nil:cons5_0(n1324_0)) → gen_nil:cons5_0(0), rt ∈ Ω(1 + n13240)

Induction Base:
rm(gen_0':s4_0(0), gen_nil:cons5_0(0)) →RΩ(1)
nil

Induction Step:
rm(gen_0':s4_0(0), gen_nil:cons5_0(+(n1324_0, 1))) →RΩ(1)
if2(eq(gen_0':s4_0(0), 0'), gen_0':s4_0(0), 0', gen_nil:cons5_0(n1324_0)) →LΩ(1)
if2(true, gen_0':s4_0(0), 0', gen_nil:cons5_0(n1324_0)) →RΩ(1)
rm(gen_0':s4_0(0), gen_nil:cons5_0(n1324_0)) →IH
gen_nil:cons5_0(0)

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

### (20) 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 ∈ Ω(1 + n70)
eq(gen_0':s4_0(n330_0), gen_0':s4_0(n330_0)) → true, rt ∈ Ω(1 + n3300)
min(gen_nil:cons5_0(+(1, n893_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n8930)
rm(gen_0':s4_0(0), gen_nil:cons5_0(n1324_0)) → gen_nil:cons5_0(0), rt ∈ Ω(1 + n13240)

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

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

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

### (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)
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 ∈ Ω(1 + n70)
eq(gen_0':s4_0(n330_0), gen_0':s4_0(n330_0)) → true, rt ∈ Ω(1 + n3300)
min(gen_nil:cons5_0(+(1, n893_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n8930)
rm(gen_0':s4_0(0), gen_nil:cons5_0(n1324_0)) → gen_nil:cons5_0(0), rt ∈ Ω(1 + n13240)

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))

No more defined symbols left to analyse.

### (23) 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)

### (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)
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 ∈ Ω(1 + n70)
eq(gen_0':s4_0(n330_0), gen_0':s4_0(n330_0)) → true, rt ∈ Ω(1 + n3300)
min(gen_nil:cons5_0(+(1, n893_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n8930)
rm(gen_0':s4_0(0), gen_nil:cons5_0(n1324_0)) → gen_nil:cons5_0(0), rt ∈ Ω(1 + n13240)

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))

No more defined symbols left to analyse.

### (26) 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)

### (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)
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 ∈ Ω(1 + n70)
eq(gen_0':s4_0(n330_0), gen_0':s4_0(n330_0)) → true, rt ∈ Ω(1 + n3300)
min(gen_nil:cons5_0(+(1, n893_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n8930)

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))

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)
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 ∈ Ω(1 + n70)
eq(gen_0':s4_0(n330_0), gen_0':s4_0(n330_0)) → true, rt ∈ Ω(1 + n3300)

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))

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)
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 ∈ Ω(1 + n70)

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))

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)