949cf69246
This PR changes the "declaration uses 'sorry'" warning to use backticks instead of single quotes, consistent with Lean's conventions for formatting code identifiers in diagnostic messages.
189 lines
4.7 KiB
Text
189 lines
4.7 KiB
Text
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section Mathlib.Order.Defs
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variable {α : Type _}
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section Preorder
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class Preorder (α : Type _) extends LE α, LT α where
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le_refl : ∀ a : α, a ≤ a
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le_trans : ∀ a b c : α, a ≤ b → b ≤ c → a ≤ c
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lt := fun a b => a ≤ b ∧ ¬b ≤ a
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lt_iff_le_not_le : ∀ a b : α, a < b ↔ a ≤ b ∧ ¬b ≤ a := by intros; rfl
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end Preorder
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section PartialOrder
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class PartialOrder (α : Type _) extends Preorder α where
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le_antisymm : ∀ a b : α, a ≤ b → b ≤ a → a = b
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end PartialOrder
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section LinearOrder
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class LinearOrder (α : Type _) extends PartialOrder α, Min α, Max α, Ord α where
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le_total (a b : α) : a ≤ b ∨ b ≤ a
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decidableLE : DecidableRel (· ≤ · : α → α → Prop)
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decidableEq : DecidableEq α
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decidableLT : DecidableRel (· < · : α → α → Prop)
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min := fun a b => if a ≤ b then a else b
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max := fun a b => if a ≤ b then b else a
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min_def : ∀ a b, min a b = if a ≤ b then a else b := by intros; rfl
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max_def : ∀ a b, max a b = if a ≤ b then b else a := by intros; rfl
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compare a b := compareOfLessAndEq a b
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compare_eq_compareOfLessAndEq : ∀ a b, compare a b = compareOfLessAndEq a b
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end LinearOrder
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end Mathlib.Order.Defs
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section Mathlib.Order.Notation
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class Bot (α : Type _) where
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bot : α
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notation "⊥" => Bot.bot
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attribute [match_pattern] Bot.bot
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end Mathlib.Order.Notation
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section Mathlib.Order.Basic
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variable {α : Type _}
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/-- An order is dense if there is an element between any pair of distinct comparable elements. -/
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class DenselyOrdered (α : Type _) [LT α] : Prop where
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/-- An order is dense if there is an element between any pair of distinct elements. -/
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dense : ∀ a₁ a₂ : α, a₁ < a₂ → ∃ a, a₁ < a ∧ a < a₂
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theorem exists_between [LT α] [DenselyOrdered α] : ∀ {a₁ a₂ : α}, a₁ < a₂ → ∃ a, a₁ < a ∧ a < a₂ :=
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DenselyOrdered.dense _ _
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theorem le_of_forall_le_of_dense [LinearOrder α] [DenselyOrdered α] {a₁ a₂ : α}
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(h : ∀ a, a₂ < a → a₁ ≤ a) : a₁ ≤ a₂ := sorry
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end Mathlib.Order.Basic
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section Mathlib.Order.BoundedOrder
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universe u
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variable {α : Type u}
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/-- An order is an `OrderBot` if it has a least element.
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We state this using a data mixin, holding the value of `⊥` and the least element constraint. -/
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class OrderBot (α : Type u) [LE α] extends Bot α where
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/-- `⊥` is the least element -/
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bot_le : ∀ a : α, ⊥ ≤ a
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section OrderBot
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variable [LE α] [OrderBot α] {a : α}
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@[simp]
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theorem bot_le : ⊥ ≤ a :=
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OrderBot.bot_le a
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end OrderBot
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end Mathlib.Order.BoundedOrder
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section Mathlib.Order.TypeTags
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variable {α : Type _}
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def WithBot (α : Type _) := Option α
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namespace WithBot
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@[coe, match_pattern] def some : α → WithBot α :=
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Option.some
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instance coe : Coe α (WithBot α) :=
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⟨some⟩
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instance bot : Bot (WithBot α) :=
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⟨none⟩
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@[elab_as_elim, induction_eliminator, cases_eliminator]
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def recBotCoe {C : WithBot α → Sort _} (bot : C ⊥) (coe : ∀ a : α, C a) : ∀ n : WithBot α, C n
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| ⊥ => bot
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| (a : α) => coe a
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end WithBot
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end Mathlib.Order.TypeTags
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variable {α β γ δ : Type _}
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namespace WithBot
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variable {a b : α}
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@[simp, norm_cast]
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theorem coe_inj : (a : WithBot α) = b ↔ a = b :=
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Option.some_inj
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@[simp]
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theorem bot_ne_coe : ⊥ ≠ (a : WithBot α) :=
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nofun
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section LE
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variable [LE α]
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instance (priority := 10) le : LE (WithBot α) :=
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⟨fun o₁ o₂ => ∀ a : α, o₁ = ↑a → ∃ b : α, o₂ = ↑b ∧ a ≤ b⟩
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@[simp, norm_cast]
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theorem coe_le_coe : (a : WithBot α) ≤ b ↔ a ≤ b := by
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simp [LE.le]
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instance orderBot : OrderBot (WithBot α) where
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bot_le _ := fun _ h => Option.noConfusion rfl (heq_of_eq h)
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theorem le_coe_iff : ∀ {x : WithBot α}, x ≤ b ↔ ∀ a : α, x = ↑a → a ≤ b
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| (b : α) => by simp
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| ⊥ => by simp
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end LE
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section LT
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variable [LT α]
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instance (priority := 10) lt : LT (WithBot α) :=
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⟨fun o₁ o₂ : WithBot α => ∃ b : α, o₂ = ↑b ∧ ∀ a : α, o₁ = ↑a → a < b⟩
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@[simp, norm_cast]
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theorem coe_lt_coe : (a : WithBot α) < b ↔ a < b := by
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simp [LT.lt]
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end LT
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instance preorder [Preorder α] : Preorder (WithBot α) where
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le := (· ≤ ·)
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lt := (· < ·)
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lt_iff_le_not_le := sorry
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le_refl _ a ha := sorry
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le_trans _ _ _ h₁ h₂ a ha := sorry
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end WithBot
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namespace WithBot
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/-- warning: declaration uses `sorry` -/
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#guard_msgs in
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theorem le_of_forall_lt_iff_le [LinearOrder α] [DenselyOrdered α]
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{x y : WithBot α} : (∀ z : α, x < z → y ≤ z) ↔ y ≤ x := by
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refine ⟨fun h ↦ ?_, fun h z x_z ↦ sorry⟩
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induction x with
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| bot => sorry
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| coe x =>
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rw [le_coe_iff]
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rintro y rfl
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exact le_of_forall_le_of_dense (by exact_mod_cast h)
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end WithBot
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