### (0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

eq(0, 0) → true
eq(0, s(x)) → 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)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
min(add(n, nil)) → n
rm(n, nil) → nil
rm(n, add(m, x)) → if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) → rm(n, x)
if_rm(false, n, add(m, x)) → add(m, rm(n, x))
minsort(nil, nil) → nil
minsort(add(n, x), y) → if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) → add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) → minsort(x, add(n, y))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
eq(s(x), s(y)) →+ eq(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:

eq(0', 0') → true
eq(0', s(x)) → 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)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
min(add(n, nil)) → n
rm(n, nil) → nil
rm(n, add(m, x)) → if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) → rm(n, x)
if_rm(false, n, add(m, x)) → add(m, rm(n, x))
minsort(nil, nil) → nil
minsort(add(n, x), y) → if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) → add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) → minsort(x, add(n, y))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(x)) → 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)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
min(add(n, nil)) → n
rm(n, nil) → nil
rm(n, add(m, x)) → if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) → rm(n, x)
if_rm(false, n, add(m, x)) → add(m, rm(n, x))
minsort(nil, nil) → nil
minsort(add(n, x), y) → if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) → add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) → minsort(x, add(n, y))

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:add → 0':s
if_min :: true:false → nil:add → 0':s
rm :: 0':s → nil:add → nil:add
if_rm :: true:false → 0':s → nil:add → nil:add
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

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

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

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

### (8) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(x)) → 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)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
min(add(n, nil)) → n
rm(n, nil) → nil
rm(n, add(m, x)) → if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) → rm(n, x)
if_rm(false, n, add(m, x)) → add(m, rm(n, x))
minsort(nil, nil) → nil
minsort(add(n, x), y) → if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) → add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) → minsort(x, add(n, y))

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:add → 0':s
if_min :: true:false → nil:add → 0':s
rm :: 0':s → nil:add → nil:add
if_rm :: true:false → 0':s → nil:add → nil:add
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

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

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

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

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

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

Induction Step:
eq(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) →RΩ(1)
eq(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:
eq(0', 0') → true
eq(0', s(x)) → 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)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
min(add(n, nil)) → n
rm(n, nil) → nil
rm(n, add(m, x)) → if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) → rm(n, x)
if_rm(false, n, add(m, x)) → add(m, rm(n, x))
minsort(nil, nil) → nil
minsort(add(n, x), y) → if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) → add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) → minsort(x, add(n, y))

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:add → 0':s
if_min :: true:false → nil:add → 0':s
rm :: 0':s → nil:add → nil:add
if_rm :: true:false → 0':s → nil:add → nil:add
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
eq(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))

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

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

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

Proved the following rewrite lemma:
le(gen_0':s4_0(n582_0), gen_0':s4_0(n582_0)) → true, rt ∈ Ω(1 + n5820)

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

Induction Step:
le(gen_0':s4_0(+(n582_0, 1)), gen_0':s4_0(+(n582_0, 1))) →RΩ(1)
le(gen_0':s4_0(n582_0), gen_0':s4_0(n582_0)) →IH
true

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

### (14) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(x)) → 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)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
min(add(n, nil)) → n
rm(n, nil) → nil
rm(n, add(m, x)) → if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) → rm(n, x)
if_rm(false, n, add(m, x)) → add(m, rm(n, x))
minsort(nil, nil) → nil
minsort(add(n, x), y) → if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) → add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) → minsort(x, add(n, y))

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:add → 0':s
if_min :: true:false → nil:add → 0':s
rm :: 0':s → nil:add → nil:add
if_rm :: true:false → 0':s → nil:add → nil:add
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)
le(gen_0':s4_0(n582_0), gen_0':s4_0(n582_0)) → true, rt ∈ Ω(1 + n5820)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

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

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

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

Proved the following rewrite lemma:
app(gen_nil:add5_0(n929_0), gen_nil:add5_0(b)) → gen_nil:add5_0(+(n929_0, b)), rt ∈ Ω(1 + n9290)

Induction Base:

Induction Step:

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

### (17) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(x)) → 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)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
min(add(n, nil)) → n
rm(n, nil) → nil
rm(n, add(m, x)) → if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) → rm(n, x)
if_rm(false, n, add(m, x)) → add(m, rm(n, x))
minsort(nil, nil) → nil
minsort(add(n, x), y) → if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) → add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) → minsort(x, add(n, y))

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:add → 0':s
if_min :: true:false → nil:add → 0':s
rm :: 0':s → nil:add → nil:add
if_rm :: true:false → 0':s → nil:add → nil:add
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)
le(gen_0':s4_0(n582_0), gen_0':s4_0(n582_0)) → true, rt ∈ Ω(1 + n5820)
app(gen_nil:add5_0(n929_0), gen_nil:add5_0(b)) → gen_nil:add5_0(+(n929_0, b)), rt ∈ Ω(1 + n9290)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

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

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

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

Proved the following rewrite lemma:
min(gen_nil:add5_0(+(1, n1890_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n18900)

Induction Base:
0'

Induction Step:
min(gen_nil:add5_0(+(1, +(n1890_0, 1)))) →RΩ(1)
gen_0':s4_0(0)

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

### (20) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(x)) → 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)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
min(add(n, nil)) → n
rm(n, nil) → nil
rm(n, add(m, x)) → if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) → rm(n, x)
if_rm(false, n, add(m, x)) → add(m, rm(n, x))
minsort(nil, nil) → nil
minsort(add(n, x), y) → if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) → add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) → minsort(x, add(n, y))

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:add → 0':s
if_min :: true:false → nil:add → 0':s
rm :: 0':s → nil:add → nil:add
if_rm :: true:false → 0':s → nil:add → nil:add
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)
le(gen_0':s4_0(n582_0), gen_0':s4_0(n582_0)) → true, rt ∈ Ω(1 + n5820)
app(gen_nil:add5_0(n929_0), gen_nil:add5_0(b)) → gen_nil:add5_0(+(n929_0, b)), rt ∈ Ω(1 + n9290)
min(gen_nil:add5_0(+(1, n1890_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n18900)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

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

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

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

Proved the following rewrite lemma:
rm(gen_0':s4_0(0), gen_nil:add5_0(n2303_0)) → gen_nil:add5_0(0), rt ∈ Ω(1 + n23030)

Induction Base:
nil

Induction Step:
rm(gen_0':s4_0(0), gen_nil:add5_0(+(n2303_0, 1))) →RΩ(1)

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

### (23) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(x)) → 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)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
min(add(n, nil)) → n
rm(n, nil) → nil
rm(n, add(m, x)) → if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) → rm(n, x)
if_rm(false, n, add(m, x)) → add(m, rm(n, x))
minsort(nil, nil) → nil
minsort(add(n, x), y) → if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) → add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) → minsort(x, add(n, y))

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:add → 0':s
if_min :: true:false → nil:add → 0':s
rm :: 0':s → nil:add → nil:add
if_rm :: true:false → 0':s → nil:add → nil:add
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)
le(gen_0':s4_0(n582_0), gen_0':s4_0(n582_0)) → true, rt ∈ Ω(1 + n5820)
app(gen_nil:add5_0(n929_0), gen_nil:add5_0(b)) → gen_nil:add5_0(+(n929_0, b)), rt ∈ Ω(1 + n9290)
min(gen_nil:add5_0(+(1, n1890_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n18900)
rm(gen_0':s4_0(0), gen_nil:add5_0(n2303_0)) → gen_nil:add5_0(0), rt ∈ Ω(1 + n23030)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
minsort

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

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

### (25) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(x)) → 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)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
min(add(n, nil)) → n
rm(n, nil) → nil
rm(n, add(m, x)) → if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) → rm(n, x)
if_rm(false, n, add(m, x)) → add(m, rm(n, x))
minsort(nil, nil) → nil
minsort(add(n, x), y) → if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) → add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) → minsort(x, add(n, y))

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:add → 0':s
if_min :: true:false → nil:add → 0':s
rm :: 0':s → nil:add → nil:add
if_rm :: true:false → 0':s → nil:add → nil:add
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)
le(gen_0':s4_0(n582_0), gen_0':s4_0(n582_0)) → true, rt ∈ Ω(1 + n5820)
app(gen_nil:add5_0(n929_0), gen_nil:add5_0(b)) → gen_nil:add5_0(+(n929_0, b)), rt ∈ Ω(1 + n9290)
min(gen_nil:add5_0(+(1, n1890_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n18900)
rm(gen_0':s4_0(0), gen_nil:add5_0(n2303_0)) → gen_nil:add5_0(0), rt ∈ Ω(1 + n23030)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (26) LowerBoundsProof (EQUIVALENT transformation)

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

### (28) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(x)) → 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)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
min(add(n, nil)) → n
rm(n, nil) → nil
rm(n, add(m, x)) → if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) → rm(n, x)
if_rm(false, n, add(m, x)) → add(m, rm(n, x))
minsort(nil, nil) → nil
minsort(add(n, x), y) → if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) → add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) → minsort(x, add(n, y))

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:add → 0':s
if_min :: true:false → nil:add → 0':s
rm :: 0':s → nil:add → nil:add
if_rm :: true:false → 0':s → nil:add → nil:add
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)
le(gen_0':s4_0(n582_0), gen_0':s4_0(n582_0)) → true, rt ∈ Ω(1 + n5820)
app(gen_nil:add5_0(n929_0), gen_nil:add5_0(b)) → gen_nil:add5_0(+(n929_0, b)), rt ∈ Ω(1 + n9290)
min(gen_nil:add5_0(+(1, n1890_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n18900)
rm(gen_0':s4_0(0), gen_nil:add5_0(n2303_0)) → gen_nil:add5_0(0), rt ∈ Ω(1 + n23030)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (29) LowerBoundsProof (EQUIVALENT transformation)

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

### (31) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(x)) → 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)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
min(add(n, nil)) → n
rm(n, nil) → nil
rm(n, add(m, x)) → if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) → rm(n, x)
if_rm(false, n, add(m, x)) → add(m, rm(n, x))
minsort(nil, nil) → nil
minsort(add(n, x), y) → if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) → add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) → minsort(x, add(n, y))

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:add → 0':s
if_min :: true:false → nil:add → 0':s
rm :: 0':s → nil:add → nil:add
if_rm :: true:false → 0':s → nil:add → nil:add
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)
le(gen_0':s4_0(n582_0), gen_0':s4_0(n582_0)) → true, rt ∈ Ω(1 + n5820)
app(gen_nil:add5_0(n929_0), gen_nil:add5_0(b)) → gen_nil:add5_0(+(n929_0, b)), rt ∈ Ω(1 + n9290)
min(gen_nil:add5_0(+(1, n1890_0))) → gen_0':s4_0(0), rt ∈ Ω(1 + n18900)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (32) LowerBoundsProof (EQUIVALENT transformation)

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

### (34) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(x)) → 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)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
min(add(n, nil)) → n
rm(n, nil) → nil
rm(n, add(m, x)) → if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) → rm(n, x)
if_rm(false, n, add(m, x)) → add(m, rm(n, x))
minsort(nil, nil) → nil
minsort(add(n, x), y) → if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) → add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) → minsort(x, add(n, y))

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:add → 0':s
if_min :: true:false → nil:add → 0':s
rm :: 0':s → nil:add → nil:add
if_rm :: true:false → 0':s → nil:add → nil:add
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)
le(gen_0':s4_0(n582_0), gen_0':s4_0(n582_0)) → true, rt ∈ Ω(1 + n5820)
app(gen_nil:add5_0(n929_0), gen_nil:add5_0(b)) → gen_nil:add5_0(+(n929_0, b)), rt ∈ Ω(1 + n9290)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (35) LowerBoundsProof (EQUIVALENT transformation)

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

### (37) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(x)) → 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)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
min(add(n, nil)) → n
rm(n, nil) → nil
rm(n, add(m, x)) → if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) → rm(n, x)
if_rm(false, n, add(m, x)) → add(m, rm(n, x))
minsort(nil, nil) → nil
minsort(add(n, x), y) → if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) → add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) → minsort(x, add(n, y))

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:add → 0':s
if_min :: true:false → nil:add → 0':s
rm :: 0':s → nil:add → nil:add
if_rm :: true:false → 0':s → nil:add → nil:add
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)
le(gen_0':s4_0(n582_0), gen_0':s4_0(n582_0)) → true, rt ∈ Ω(1 + n5820)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (38) LowerBoundsProof (EQUIVALENT transformation)

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

### (40) Obligation:

TRS:
Rules:
eq(0', 0') → true
eq(0', s(x)) → 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)
app(nil, y) → y
app(add(n, x), y) → add(n, app(x, y))
min(add(n, nil)) → n
rm(n, nil) → nil
rm(n, add(m, x)) → if_rm(eq(n, m), n, add(m, x))
if_rm(true, n, add(m, x)) → rm(n, x)
if_rm(false, n, add(m, x)) → add(m, rm(n, x))
minsort(nil, nil) → nil
minsort(add(n, x), y) → if_minsort(eq(n, min(add(n, x))), add(n, x), y)
if_minsort(true, add(n, x), y) → add(n, minsort(app(rm(n, x), y), nil))
if_minsort(false, add(n, x), y) → minsort(x, add(n, y))

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:add → 0':s
if_min :: true:false → nil:add → 0':s
rm :: 0':s → nil:add → nil:add
if_rm :: true:false → 0':s → nil:add → nil:add
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
eq(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))