case mp
m n k : ℕ
p q : ℕ → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
h : ∃ n, p n
⊢ Nat.find h = m → p m ∧ ∀ (n : ℕ), n < m → ¬p n

case mpr
m n k : ℕ
p q : ℕ → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
h : ∃ n, p n
⊢ (p m ∧ ∀ (n : ℕ), n < m → ¬p n) → Nat.find h = m
m n k : ℕ
p q : ℕ → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
h : ∃ n, p n
⊢ Nat.find h = m ↔ p m ∧ ∀ (n : ℕ), n < m → ¬p n
constructor
case mp
n k : ℕ
p q : ℕ → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
h : ∃ n, p n
⊢ p (Nat.find h) ∧ ∀ (n : ℕ), n < Nat.find h → ¬p n
case mp
m n k : ℕ
p q : ℕ → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
h : ∃ n, p n
⊢ Nat.find h = m → p m ∧ ∀ (n : ℕ), n < m → ¬p n
rintro rfl
no goals
case mp
n k : ℕ
p q : ℕ → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
h : ∃ n, p n
⊢ p (Nat.find h) ∧ ∀ (n : ℕ), n < Nat.find h → ¬p n
exact ⟨Nat.find_spec h, fun _ => Nat.find_min h⟩
case mpr.intro
m n k : ℕ
p q : ℕ → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
h : ∃ n, p n
hm : p m
hlt : ∀ (n : ℕ), n < m → ¬p n
⊢ Nat.find h = m
case mpr
m n k : ℕ
p q : ℕ → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
h : ∃ n, p n
⊢ (p m ∧ ∀ (n : ℕ), n < m → ¬p n) → Nat.find h = m
rintro ⟨hm, hlt⟩
no goals
case mpr.intro
m n k : ℕ
p q : ℕ → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
h : ∃ n, p n
hm : p m
hlt : ∀ (n : ℕ), n < m → ¬p n
⊢ Nat.find h = m
exact le_antisymm (Nat.find_min' h hm) (not_lt.1 <| imp_not_comm.1 (hlt _) <| Nat.find_spec h)