-- Mathlib/Topology/Instances/EReal/Lemmas.lean
theorem continuousAt_add {p : EReal × EReal} (h : p.1 ≠ ⊤ ∨ p.2 ≠ ⊥) (h' : p.1 ≠ ⊥ ∨ p.2 ≠ ⊤) :
    ContinuousAt (fun p : EReal × EReal => p.1 + p.2) p

Proof state:
case mk.h_real.h_real
a✝¹ a✝ : ℝ
h : (↑a✝¹, ↑a✝).1 ≠ ⊤ ∨ (↑a✝¹, ↑a✝).2 ≠ ⊥
h' : (↑a✝¹, ↑a✝).1 ≠ ⊥ ∨ (↑a✝¹, ↑a✝).2 ≠ ⊤
⊢ ContinuousAt (fun p => p.1 + p.2) (↑a✝¹, ↑a✝)

Tactic:
exact continuousAt_add_coe_coe _ _

Result:
no goals

Proof state:
case mk.h_top.h_bot
h : (⊤, ⊥).1 ≠ ⊤ ∨ (⊤, ⊥).2 ≠ ⊥
h' : (⊤, ⊥).1 ≠ ⊥ ∨ (⊤, ⊥).2 ≠ ⊤
⊢ ContinuousAt (fun p => p.1 + p.2) (⊤, ⊥)

Tactic:
simp at h

Result:
no goals

Proof state:
case mk.h_bot.h_top
h : (⊥, ⊤).1 ≠ ⊤ ∨ (⊥, ⊤).2 ≠ ⊥
h' : (⊥, ⊤).1 ≠ ⊥ ∨ (⊥, ⊤).2 ≠ ⊤
⊢ ContinuousAt (fun p => p.1 + p.2) (⊥, ⊤)

Tactic:
simp at h'

Result:
no goals

Proof state:
case mk
x y : EReal
h : (x, y).1 ≠ ⊤ ∨ (x, y).2 ≠ ⊥
h' : (x, y).1 ≠ ⊥ ∨ (x, y).2 ≠ ⊤
⊢ ContinuousAt (fun p => p.1 + p.2) (x, y)

Tactic:
induction x <;> induction y

Result:
case mk.h_bot.h_bot
h : (⊥, ⊥).1 ≠ ⊤ ∨ (⊥, ⊥).2 ≠ ⊥
h' : (⊥, ⊥).1 ≠ ⊥ ∨ (⊥, ⊥).2 ≠ ⊤
⊢ ContinuousAt (fun p => p.1 + p.2) (⊥, ⊥)

case mk.h_bot.h_real
a✝ : ℝ
h : (⊥, ↑a✝).1 ≠ ⊤ ∨ (⊥, ↑a✝).2 ≠ ⊥
h' : (⊥, ↑a✝).1 ≠ ⊥ ∨ (⊥, ↑a✝).2 ≠ ⊤
⊢ ContinuousAt (fun p => p.1 + p.2) (⊥, ↑a✝)

case mk.h_bot.h_top
h : (⊥, ⊤).1 ≠ ⊤ ∨ (⊥, ⊤).2 ≠ ⊥
h' : (⊥, ⊤).1 ≠ ⊥ ∨ (⊥, ⊤).2 ≠ ⊤
⊢ ContinuousAt (fun p => p.1 + p.2) (⊥, ⊤)

case mk.h_real.h_bot
a✝ : ℝ
h : (↑a✝, ⊥).1 ≠ ⊤ ∨ (↑a✝, ⊥).2 ≠ ⊥
h' : (↑a✝, ⊥).1 ≠ ⊥ ∨ (↑a✝, ⊥).2 ≠ ⊤
⊢ ContinuousAt (fun p => p.1 + p.2) (↑a✝, ⊥)

case mk.h_real.h_real
a✝¹ a✝ : ℝ
h : (↑a✝¹, ↑a✝).1 ≠ ⊤ ∨ (↑a✝¹, ↑a✝).2 ≠ ⊥
h' : (↑a✝¹, ↑a✝).1 ≠ ⊥ ∨ (↑a✝¹, ↑a✝).2 ≠ ⊤
⊢ ContinuousAt (fun p => p.1 + p.2) (↑a✝¹, ↑a✝)

case mk.h_real.h_top
a✝ : ℝ
h : (↑a✝, ⊤).1 ≠ ⊤ ∨ (↑a✝, ⊤).2 ≠ ⊥
h' : (↑a✝, ⊤).1 ≠ ⊥ ∨ (↑a✝, ⊤).2 ≠ ⊤
⊢ ContinuousAt (fun p => p.1 + p.2) (↑a✝, ⊤)

case mk.h_top.h_bot
h : (⊤, ⊥).1 ≠ ⊤ ∨ (⊤, ⊥).2 ≠ ⊥
h' : (⊤, ⊥).1 ≠ ⊥ ∨ (⊤, ⊥).2 ≠ ⊤
⊢ ContinuousAt (fun p => p.1 + p.2) (⊤, ⊥)

case mk.h_top.h_real
a✝ : ℝ
h : (⊤, ↑a✝).1 ≠ ⊤ ∨ (⊤, ↑a✝).2 ≠ ⊥
h' : (⊤, ↑a✝).1 ≠ ⊥ ∨ (⊤, ↑a✝).2 ≠ ⊤
⊢ ContinuousAt (fun p => p.1 + p.2) (⊤, ↑a✝)

case mk.h_top.h_top
h : (⊤, ⊤).1 ≠ ⊤ ∨ (⊤, ⊤).2 ≠ ⊥
h' : (⊤, ⊤).1 ≠ ⊥ ∨ (⊤, ⊤).2 ≠ ⊤
⊢ ContinuousAt (fun p => p.1 + p.2) (⊤, ⊤)

Proof state:
case mk.h_real.h_bot
a✝ : ℝ
h : (↑a✝, ⊥).1 ≠ ⊤ ∨ (↑a✝, ⊥).2 ≠ ⊥
h' : (↑a✝, ⊥).1 ≠ ⊥ ∨ (↑a✝, ⊥).2 ≠ ⊤
⊢ ContinuousAt (fun p => p.1 + p.2) (↑a✝, ⊥)

Tactic:
exact continuousAt_add_coe_bot _

Result:
no goals

Proof state:
p : EReal × EReal
h : p.1 ≠ ⊤ ∨ p.2 ≠ ⊥
h' : p.1 ≠ ⊥ ∨ p.2 ≠ ⊤
⊢ ContinuousAt (fun p => p.1 + p.2) p

Tactic:
rcases p with ⟨x, y⟩

Result:
case mk
x y : EReal
h : (x, y).1 ≠ ⊤ ∨ (x, y).2 ≠ ⊥
h' : (x, y).1 ≠ ⊥ ∨ (x, y).2 ≠ ⊤
⊢ ContinuousAt (fun p => p.1 + p.2) (x, y)

Proof state:
case mk.h_real.h_top
a✝ : ℝ
h : (↑a✝, ⊤).1 ≠ ⊤ ∨ (↑a✝, ⊤).2 ≠ ⊥
h' : (↑a✝, ⊤).1 ≠ ⊥ ∨ (↑a✝, ⊤).2 ≠ ⊤
⊢ ContinuousAt (fun p => p.1 + p.2) (↑a✝, ⊤)

Tactic:
exact continuousAt_add_coe_top _

Result:
no goals

Proof state:
case mk.h_top.h_real
a✝ : ℝ
h : (⊤, ↑a✝).1 ≠ ⊤ ∨ (⊤, ↑a✝).2 ≠ ⊥
h' : (⊤, ↑a✝).1 ≠ ⊥ ∨ (⊤, ↑a✝).2 ≠ ⊤
⊢ ContinuousAt (fun p => p.1 + p.2) (⊤, ↑a✝)

Tactic:
exact continuousAt_add_top_coe _

Result:
no goals

Proof state:
case mk.h_bot.h_bot
h : (⊥, ⊥).1 ≠ ⊤ ∨ (⊥, ⊥).2 ≠ ⊥
h' : (⊥, ⊥).1 ≠ ⊥ ∨ (⊥, ⊥).2 ≠ ⊤
⊢ ContinuousAt (fun p => p.1 + p.2) (⊥, ⊥)

Tactic:
exact continuousAt_add_bot_bot

Result:
no goals

Proof state:
case mk.h_top.h_top
h : (⊤, ⊤).1 ≠ ⊤ ∨ (⊤, ⊤).2 ≠ ⊥
h' : (⊤, ⊤).1 ≠ ⊥ ∨ (⊤, ⊤).2 ≠ ⊤
⊢ ContinuousAt (fun p => p.1 + p.2) (⊤, ⊤)

Tactic:
exact continuousAt_add_top_top

Result:
no goals

Proof state:
case mk.h_bot.h_real
a✝ : ℝ
h : (⊥, ↑a✝).1 ≠ ⊤ ∨ (⊥, ↑a✝).2 ≠ ⊥
h' : (⊥, ↑a✝).1 ≠ ⊥ ∨ (⊥, ↑a✝).2 ≠ ⊤
⊢ ContinuousAt (fun p => p.1 + p.2) (⊥, ↑a✝)

Tactic:
exact continuousAt_add_bot_coe _

Result:
no goals