p q r : ℕ+
hpq : p ≤ q
hqr : q ≤ r
H : 1 < sumInv {p, q, r}
hp3 : p < 3
⊢ Admissible {p, q, r}
p q r : ℕ+
hpq : p ≤ q
hqr : q ≤ r
H : 1 < sumInv {p, q, r}
⊢ Admissible {p, q, r}
have hp3 : p < 3 := lt_three hpq hqr H
p q r : ℕ+
hpq : p ≤ q
hqr : q ≤ r
H : 1 < sumInv {p, q, r}
hp3 : p ∈ Finset.Iio 3
⊢ Admissible {p, q, r}
p q r : ℕ+
hpq : p ≤ q
hqr : q ≤ r
H : 1 < sumInv {p, q, r}
hp3 : p < 3
⊢ Admissible {p, q, r}
replace hp3 := Finset.mem_Iio.mpr hp3
p q r : ℕ+
hpq : p ≤ q
hqr : q ≤ r
H : 1 < sumInv {p, q, r}
hp3 : p ∈ {1, 2}
⊢ Admissible {p, q, r}
p q r : ℕ+
hpq : p ≤ q
hqr : q ≤ r
H : 1 < sumInv {p, q, r}
hp3 : p ∈ Finset.Iio 3
⊢ Admissible {p, q, r}
conv at hp3 => change p ∈ ({1, 2} : Multiset ℕ+)
case head
q r : ℕ+
hqr : q ≤ r
hpq : 1 ≤ q
H : 1 < sumInv {1, q, r}
⊢ Admissible {1, q, r}

case tail.head
q r : ℕ+
hqr : q ≤ r
hpq : 2 ≤ q
H : 1 < sumInv {2, q, r}
⊢ Admissible {2, q, r}
p q r : ℕ+
hpq : p ≤ q
hqr : q ≤ r
H : 1 < sumInv {p, q, r}
hp3 : p ∈ {1, 2}
⊢ Admissible {p, q, r}
fin_cases hp3
case tail.head
q r : ℕ+
hqr : q ≤ r
hpq : 2 ≤ q
H : 1 < sumInv {2, q, r}
hq4 : q < 4
⊢ Admissible {2, q, r}
case tail.head
q r : ℕ+
hqr : q ≤ r
hpq : 2 ≤ q
H : 1 < sumInv {2, q, r}
⊢ Admissible {2, q, r}
have hq4 : q < 4 := lt_four hqr H
case tail.head
q r : ℕ+
hqr : q ≤ r
hpq : 2 ≤ q
H : 1 < sumInv {2, q, r}
hq4 : q ∈ Finset.Ico 2 4
⊢ Admissible {2, q, r}
case tail.head
q r : ℕ+
hqr : q ≤ r
hpq : 2 ≤ q
H : 1 < sumInv {2, q, r}
hq4 : q < 4
⊢ Admissible {2, q, r}
replace hq4 := Finset.mem_Ico.mpr ⟨hpq, hq4⟩
case tail.head
q r : ℕ+
hqr : q ≤ r
H : 1 < sumInv {2, q, r}
hq4 : q ∈ Finset.Ico 2 4
⊢ Admissible {2, q, r}
case tail.head
q r : ℕ+
hqr : q ≤ r
hpq : 2 ≤ q
H : 1 < sumInv {2, q, r}
hq4 : q ∈ Finset.Ico 2 4
⊢ Admissible {2, q, r}
clear hpq
case tail.head
q r : ℕ+
hqr : q ≤ r
H : 1 < sumInv {2, q, r}
hq4 : q ∈ {2, 3}
⊢ Admissible {2, q, r}
case tail.head
q r : ℕ+
hqr : q ≤ r
H : 1 < sumInv {2, q, r}
hq4 : q ∈ Finset.Ico 2 4
⊢ Admissible {2, q, r}
conv at hq4 => change q ∈ ({2, 3} : Multiset ℕ+)
case tail.head.head
r : ℕ+
hqr : 2 ≤ r
H : 1 < sumInv {2, 2, r}
⊢ Admissible {2, 2, r}

case tail.head.tail.head
r : ℕ+
hqr : 3 ≤ r
H : 1 < sumInv {2, 3, r}
⊢ Admissible {2, 3, r}
case tail.head
q r : ℕ+
hqr : q ≤ r
H : 1 < sumInv {2, q, r}
hq4 : q ∈ {2, 3}
⊢ Admissible {2, q, r}
fin_cases hq4
case tail.head.tail.head
r : ℕ+
hqr : 3 ≤ r
H : 1 < sumInv {2, 3, r}
hr6 : r < 6
⊢ Admissible {2, 3, r}
case tail.head.tail.head
r : ℕ+
hqr : 3 ≤ r
H : 1 < sumInv {2, 3, r}
⊢ Admissible {2, 3, r}
have hr6 : r < 6 := lt_six H
case tail.head.tail.head
r : ℕ+
hqr : 3 ≤ r
H : 1 < sumInv {2, 3, r}
hr6 : r ∈ Finset.Ico 3 6
⊢ Admissible {2, 3, r}
case tail.head.tail.head
r : ℕ+
hqr : 3 ≤ r
H : 1 < sumInv {2, 3, r}
hr6 : r < 6
⊢ Admissible {2, 3, r}
replace hr6 := Finset.mem_Ico.mpr ⟨hqr, hr6⟩
case tail.head.tail.head
r : ℕ+
H : 1 < sumInv {2, 3, r}
hr6 : r ∈ Finset.Ico 3 6
⊢ Admissible {2, 3, r}
case tail.head.tail.head
r : ℕ+
hqr : 3 ≤ r
H : 1 < sumInv {2, 3, r}
hr6 : r ∈ Finset.Ico 3 6
⊢ Admissible {2, 3, r}
clear hqr
case tail.head.tail.head
r : ℕ+
H : 1 < sumInv {2, 3, r}
hr6 : r ∈ {3, 4, 5}
⊢ Admissible {2, 3, r}
case tail.head.tail.head
r : ℕ+
H : 1 < sumInv {2, 3, r}
hr6 : r ∈ Finset.Ico 3 6
⊢ Admissible {2, 3, r}
conv at hr6 => change r ∈ ({3, 4, 5} : Multiset ℕ+)
case tail.head.tail.head.head
H : 1 < sumInv {2, 3, 3}
⊢ Admissible {2, 3, 3}

case tail.head.tail.head.tail.head
H : 1 < sumInv {2, 3, 4}
⊢ Admissible {2, 3, 4}

case tail.head.tail.head.tail.tail.head
H : 1 < sumInv {2, 3, 5}
⊢ Admissible {2, 3, 5}
case tail.head.tail.head
r : ℕ+
H : 1 < sumInv {2, 3, r}
hr6 : r ∈ {3, 4, 5}
⊢ Admissible {2, 3, r}
fin_cases hr6
no goals
case head
q r : ℕ+
hqr : q ≤ r
hpq : 1 ≤ q
H : 1 < sumInv {1, q, r}
⊢ Admissible {1, q, r}
exact admissible_A' q r
no goals
case tail.head.head
r : ℕ+
hqr : 2 ≤ r
H : 1 < sumInv {2, 2, r}
⊢ Admissible {2, 2, r}
exact admissible_D' r
no goals
case tail.head.tail.head.head
H : 1 < sumInv {2, 3, 3}
⊢ Admissible {2, 3, 3}
exact admissible_E6
no goals
case tail.head.tail.head.tail.head
H : 1 < sumInv {2, 3, 4}
⊢ Admissible {2, 3, 4}
exact admissible_E7
no goals
case tail.head.tail.head.tail.tail.head
H : 1 < sumInv {2, 3, 5}
⊢ Admissible {2, 3, 5}
exact admissible_E8