case inl
α : Type u_1
β : Type ?u.21410
inst✝³ : CanonicallyOrderedCommSemiring α
inst✝² : Sub α
inst✝¹ : OrderedSub α
inst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
m n : ℕ
h : m ≤ n
⊢ ↑(m - n) = ↑m - ↑n

case inr
α : Type u_1
β : Type ?u.21410
inst✝³ : CanonicallyOrderedCommSemiring α
inst✝² : Sub α
inst✝¹ : OrderedSub α
inst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
m n : ℕ
h : n ≤ m
⊢ ↑(m - n) = ↑m - ↑n
α : Type u_1
β : Type ?u.21410
inst✝³ : CanonicallyOrderedCommSemiring α
inst✝² : Sub α
inst✝¹ : OrderedSub α
inst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
m n : ℕ
⊢ ↑(m - n) = ↑m - ↑n
cases' le_total m n with h h
case inl
α : Type u_1
β : Type ?u.21410
inst✝³ : CanonicallyOrderedCommSemiring α
inst✝² : Sub α
inst✝¹ : OrderedSub α
inst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
m n : ℕ
h : m ≤ n
⊢ ↑m ≤ ↑n
case inl
α : Type u_1
β : Type ?u.21410
inst✝³ : CanonicallyOrderedCommSemiring α
inst✝² : Sub α
inst✝¹ : OrderedSub α
inst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
m n : ℕ
h : m ≤ n
⊢ ↑(m - n) = ↑m - ↑n
rw [tsub_eq_zero_of_le h, cast_zero, tsub_eq_zero_of_le]
no goals
case inl
α : Type u_1
β : Type ?u.21410
inst✝³ : CanonicallyOrderedCommSemiring α
inst✝² : Sub α
inst✝¹ : OrderedSub α
inst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
m n : ℕ
h : m ≤ n
⊢ ↑m ≤ ↑n
exact mono_cast h
case inr.intro
α : Type u_1
β : Type ?u.21410
inst✝³ : CanonicallyOrderedCommSemiring α
inst✝² : Sub α
inst✝¹ : OrderedSub α
inst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
n m : ℕ
h : n ≤ m + n
⊢ ↑(m + n - n) = ↑(m + n) - ↑n
case inr
α : Type u_1
β : Type ?u.21410
inst✝³ : CanonicallyOrderedCommSemiring α
inst✝² : Sub α
inst✝¹ : OrderedSub α
inst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
m n : ℕ
h : n ≤ m
⊢ ↑(m - n) = ↑m - ↑n
rcases le_iff_exists_add'.mp h with ⟨m, rfl⟩
no goals
case inr.intro
α : Type u_1
β : Type ?u.21410
inst✝³ : CanonicallyOrderedCommSemiring α
inst✝² : Sub α
inst✝¹ : OrderedSub α
inst✝ : ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
n m : ℕ
h : n ≤ m + n
⊢ ↑(m + n - n) = ↑(m + n) - ↑n
rw [add_tsub_cancel_right, cast_add, add_tsub_cancel_right]