Theorem: ADEInequality.Admissible.one_lt_sumInv
File: Mathlib/NumberTheory/ADEInequality.lean

tactic: rw [Admissible]
state_before: pqr : Multiset ℕ+
⊢ Admissible pqr → 1 < sumInv pqr
state_after: pqr : Multiset ℕ+
⊢ (∃ q r, A' q r = pqr) ∨ (∃ r, D' r = pqr) ∨ E' 3 = pqr ∨ E' 4 = pqr ∨ E' 5 = pqr → 1 < sumInv pqr

tactic: rintro (⟨p', q', H⟩ | ⟨n, H⟩ | H | H | H)
state_before: pqr : Multiset ℕ+
⊢ (∃ q r, A' q r = pqr) ∨ (∃ r, D' r = pqr) ∨ E' 3 = pqr ∨ E' 4 = pqr ∨ E' 5 = pqr → 1 < sumInv pqr
state_after: case inl.intro.intro
pqr : Multiset ℕ+
p' q' : ℕ+
H : A' p' q' = pqr
⊢ 1 < sumInv pqr

case inr.inl.intro
pqr : Multiset ℕ+
n : ℕ+
H : D' n = pqr
⊢ 1 < sumInv pqr

case inr.inr.inl
pqr : Multiset ℕ+
H : E' 3 = pqr
⊢ 1 < sumInv pqr

case inr.inr.inr.inl
pqr : Multiset ℕ+
H : E' 4 = pqr
⊢ 1 < sumInv pqr

case inr.inr.inr.inr
pqr : Multiset ℕ+
H : E' 5 = pqr
⊢ 1 < sumInv pqr

tactic: all_goals
  rw [← H, E', sumInv_pqr]
  conv_rhs => simp only [OfNat.ofNat, PNat.mk_coe]
  rfl
state_before: case inr.inr.inl
pqr : Multiset ℕ+
H : E' 3 = pqr
⊢ 1 < sumInv pqr

case inr.inr.inr.inl
pqr : Multiset ℕ+
H : E' 4 = pqr
⊢ 1 < sumInv pqr

case inr.inr.inr.inr
pqr : Multiset ℕ+
H : E' 5 = pqr
⊢ 1 < sumInv pqr
state_after: no goals

tactic: rw [← H, A', sumInv_pqr, add_assoc]
state_before: case inl.intro.intro
pqr : Multiset ℕ+
p' q' : ℕ+
H : A' p' q' = pqr
⊢ 1 < sumInv pqr
state_after: case inl.intro.intro
pqr : Multiset ℕ+
p' q' : ℕ+
H : A' p' q' = pqr
⊢ 1 < (↑↑1)⁻¹ + ((↑↑p')⁻¹ + (↑↑q')⁻¹)

tactic: simp only [lt_add_iff_pos_right, PNat.one_coe, inv_one, Nat.cast_one]
state_before: case inl.intro.intro
pqr : Multiset ℕ+
p' q' : ℕ+
H : A' p' q' = pqr
⊢ 1 < (↑↑1)⁻¹ + ((↑↑p')⁻¹ + (↑↑q')⁻¹)
state_after: case inl.intro.intro
pqr : Multiset ℕ+
p' q' : ℕ+
H : A' p' q' = pqr
⊢ 0 < (↑↑p')⁻¹ + (↑↑q')⁻¹

tactic: apply add_pos <;> simp only [PNat.pos, Nat.cast_pos, inv_pos]
state_before: case inl.intro.intro
pqr : Multiset ℕ+
p' q' : ℕ+
H : A' p' q' = pqr
⊢ 0 < (↑↑p')⁻¹ + (↑↑q')⁻¹
state_after: no goals

tactic: rw [← H, D', sumInv_pqr]
state_before: case inr.inl.intro
pqr : Multiset ℕ+
n : ℕ+
H : D' n = pqr
⊢ 1 < sumInv pqr
state_after: case inr.inl.intro
pqr : Multiset ℕ+
n : ℕ+
H : D' n = pqr
⊢ 1 < (↑↑2)⁻¹ + (↑↑2)⁻¹ + (↑↑n)⁻¹

tactic: conv_rhs => simp only [OfNat.ofNat, PNat.mk_coe]
state_before: case inr.inl.intro
pqr : Multiset ℕ+
n : ℕ+
H : D' n = pqr
⊢ 1 < (↑↑2)⁻¹ + (↑↑2)⁻¹ + (↑↑n)⁻¹
state_after: case inr.inl.intro
pqr : Multiset ℕ+
n : ℕ+
H : D' n = pqr
⊢ 1 < (↑2)⁻¹ + (↑2)⁻¹ + (↑↑n)⁻¹

tactic: norm_num
state_before: case inr.inl.intro
pqr : Multiset ℕ+
n : ℕ+
H : D' n = pqr
⊢ 1 < (↑2)⁻¹ + (↑2)⁻¹ + (↑↑n)⁻¹
state_after: no goals

tactic: rw [← H, E', sumInv_pqr]
state_before: case inr.inr.inr.inr
pqr : Multiset ℕ+
H : E' 5 = pqr
⊢ 1 < sumInv pqr
state_after: case inr.inr.inr.inr
pqr : Multiset ℕ+
H : E' 5 = pqr
⊢ 1 < (↑↑2)⁻¹ + (↑↑3)⁻¹ + (↑↑5)⁻¹

tactic: conv_rhs => simp only [OfNat.ofNat, PNat.mk_coe]
state_before: case inr.inr.inr.inr
pqr : Multiset ℕ+
H : E' 5 = pqr
⊢ 1 < (↑↑2)⁻¹ + (↑↑3)⁻¹ + (↑↑5)⁻¹
state_after: case inr.inr.inr.inr
pqr : Multiset ℕ+
H : E' 5 = pqr
⊢ 1 < (↑2)⁻¹ + (↑3)⁻¹ + (↑5)⁻¹

tactic: rfl
state_before: case inr.inr.inr.inr
pqr : Multiset ℕ+
H : E' 5 = pqr
⊢ 1 < (↑2)⁻¹ + (↑3)⁻¹ + (↑5)⁻¹
state_after: no goals