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import Mathlib.Topology.MetricSpace.Basic

section

variable {α : Type _} [PartialOrder α]

variable (x y z : α)

#check x ≤ y

#check (le_refl x : x ≤ x)

#check (le_trans : x ≤ y → y ≤ z → x ≤ z)

#check x < y

#check (lt_irrefl x : ¬x < x)

#check (lt_trans : x < y → y < z → x < z)

#check (lt_of_le_of_lt : x ≤ y → y < z → x < z)

#check (lt_of_lt_of_le : x < y → y ≤ z → x < z)

example : x < y ↔ x ≤ y ∧ x ≠ y :=
  lt_iff_le_and_ne

end

section

variable {α : Type _} [Lattice α]

variable (x y z : α)

#check x ⊓ y

#check (inf_le_left : x ⊓ y ≤ x)

#check (inf_le_right : x ⊓ y ≤ y)

#check (le_inf : z ≤ x → z ≤ y → z ≤ x ⊓ y)

#check x ⊔ y

#check (le_sup_left : x ≤ x ⊔ y)

#check (le_sup_right : y ≤ x ⊔ y)

#check (sup_le : x ≤ z → y ≤ z → x ⊔ y ≤ z)

example : x ⊓ y = y ⊓ x := by
  sorry

example : x ⊓ y ⊓ z = x ⊓ (y ⊓ z) := by
  sorry

example : x ⊔ y = y ⊔ x := by
  sorry

example : x ⊔ y ⊔ z = x ⊔ (y ⊔ z) := by
  sorry

theorem absorb1 : x ⊓ (x ⊔ y) = x := by
  sorry

theorem absorb2 : x ⊔ x ⊓ y = x := by
  sorry

end

section

variable {α : Type _} [DistribLattice α]

variable (x y z : α)

#check (inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z)

#check (inf_sup_right : (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z)

#check (sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z))

#check (sup_inf_right : x ⊓ y ⊔ z = (x ⊔ z) ⊓ (y ⊔ z))

end

section

variable {α : Type _} [Lattice α]

variable (a b c : α)

example (h : ∀ x y z : α, x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) : a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c) := by
  sorry

example (h : ∀ x y z : α, x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) : a ⊓ (b ⊔ c) = a ⊓ b ⊔ a ⊓ c := by
  sorry

end

section

variable {R : Type _} [StrictOrderedRing R]

variable (a b c : R)

#check (add_le_add_left : a ≤ b → ∀ c, c + a ≤ c + b)

#check (mul_pos : 0 < a → 0 < b → 0 < a * b)

#check (mul_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a * b)

example : a ≤ b → 0 ≤ b - a := by
  sorry

example : 0 ≤ b - a → a ≤ b := by
  sorry

example (h : a ≤ b) (h' : 0 ≤ c) : a * c ≤ b * c := by
  sorry

end

section

variable {X : Type _} [MetricSpace X]

variable (x y z : X)

#check (dist_self x : dist x x = 0)

#check (dist_comm x y : dist x y = dist y x)

#check (dist_triangle x y z : dist x z ≤ dist x y + dist y z)

example (x y : X) : 0 ≤ dist x y := by
  sorry

end