236 lines
8.4 KiB
Lean4
236 lines
8.4 KiB
Lean4
import Spa.Lattice
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namespace Spa
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inductive AboveBelow (α : Type*) where
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| bot
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| top
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| mk (x : α)
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deriving DecidableEq
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namespace AboveBelow
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instance {α : Type*} [ToString α] : ToString (AboveBelow α) where
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toString
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| bot => "⊥"
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| top => "⊤"
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| mk x => toString x
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variable {α : Type*} [DecidableEq α]
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instance : Max (AboveBelow α) where
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max
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| bot, x => x
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| top, _ => top
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| mk x, mk y => if x = y then mk x else top
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| mk x, bot => mk x
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| mk _, top => top
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instance : Min (AboveBelow α) where
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min
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| bot, _ => bot
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| top, x => x
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| mk x, mk y => if x = y then mk x else bot
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| mk _, bot => bot
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| mk x, top => mk x
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@[simp] lemma bot_sup (x : AboveBelow α) : bot ⊔ x = x := rfl
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@[simp] lemma top_sup (x : AboveBelow α) : top ⊔ x = top := rfl
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@[simp] lemma sup_bot (x : AboveBelow α) : x ⊔ bot = x := by cases x <;> rfl
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@[simp] lemma sup_top (x : AboveBelow α) : x ⊔ top = top := by cases x <;> rfl
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@[simp] lemma mk_sup_mk (x y : α) :
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(mk x ⊔ mk y : AboveBelow α) = if x = y then mk x else top := rfl
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@[simp] lemma bot_inf (x : AboveBelow α) : bot ⊓ x = bot := rfl
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@[simp] lemma top_inf (x : AboveBelow α) : top ⊓ x = x := rfl
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@[simp] lemma inf_bot (x : AboveBelow α) : x ⊓ bot = bot := by cases x <;> rfl
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@[simp] lemma inf_top (x : AboveBelow α) : x ⊓ top = x := by cases x <;> rfl
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@[simp] lemma mk_inf_mk (x y : α) :
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(mk x ⊓ mk y : AboveBelow α) = if x = y then mk x else bot := rfl
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protected lemma sup_comm (a b : AboveBelow α) : a ⊔ b = b ⊔ a := by
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rcases a with _ | _ | x <;> rcases b with _ | _ | y <;> simp only
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[bot_sup, sup_bot, top_sup, sup_top, mk_sup_mk]
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split_ifs with h₁ h₂ h₂ <;> simp_all
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protected lemma sup_assoc (a b c : AboveBelow α) : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) := by
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rcases a with _ | _ | x <;> rcases b with _ | _ | y <;> rcases c with _ | _ | z <;>
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simp only [bot_sup, sup_bot, top_sup, sup_top, mk_sup_mk]
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split_ifs <;> simp_all
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protected lemma inf_comm (a b : AboveBelow α) : a ⊓ b = b ⊓ a := by
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rcases a with _ | _ | x <;> rcases b with _ | _ | y <;> simp only
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[bot_inf, inf_bot, top_inf, inf_top, mk_inf_mk]
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split_ifs with h₁ h₂ h₂ <;> simp_all
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protected lemma inf_assoc (a b c : AboveBelow α) : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) := by
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rcases a with _ | _ | x <;> rcases b with _ | _ | y <;> rcases c with _ | _ | z <;>
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simp only [bot_inf, inf_bot, top_inf, inf_top, mk_inf_mk]
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split_ifs <;> simp_all
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protected lemma sup_inf_self (a b : AboveBelow α) : a ⊔ a ⊓ b = a := by
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rcases a with _ | _ | x <;> rcases b with _ | _ | y <;>
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simp only [bot_sup, sup_bot, top_sup, sup_top, mk_sup_mk,
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bot_inf, inf_bot, top_inf, inf_top, mk_inf_mk] <;>
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try (split_ifs <;> simp_all)
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protected lemma inf_sup_self (a b : AboveBelow α) : a ⊓ (a ⊔ b) = a := by
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rcases a with _ | _ | x <;> rcases b with _ | _ | y <;>
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simp only [bot_sup, sup_bot, top_sup, sup_top, mk_sup_mk,
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bot_inf, inf_bot, top_inf, inf_top, mk_inf_mk] <;>
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try (split_ifs <;> simp_all)
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instance : Lattice (AboveBelow α) :=
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Lattice.mk' AboveBelow.sup_comm AboveBelow.sup_assoc
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AboveBelow.inf_comm AboveBelow.inf_assoc
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AboveBelow.sup_inf_self AboveBelow.inf_sup_self
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lemma le_iff {a b : AboveBelow α} : a ≤ b ↔ a ⊔ b = b := sup_eq_right.symm
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lemma bot_le' (a : AboveBelow α) : (bot : AboveBelow α) ≤ a :=
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le_iff.mpr (bot_sup a)
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lemma le_top' (a : AboveBelow α) : a ≤ (top : AboveBelow α) :=
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le_iff.mpr (sup_top a)
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lemma bot_lt_mk (x : α) : (bot : AboveBelow α) < mk x :=
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lt_of_le_of_ne (bot_le' _) (by simp)
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lemma mk_lt_top (x : α) : (mk x : AboveBelow α) < top :=
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lt_of_le_of_ne (le_top' _) (by simp)
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lemma bot_lt_top : (bot : AboveBelow α) < top :=
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lt_of_le_of_ne (bot_le' _) (by simp)
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lemma le_cases {a b : AboveBelow α} (h : a ≤ b) :
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a = bot ∨ b = top ∨ a = b := by
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have hsup := le_iff.mp h
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rcases a with _ | _ | x <;> rcases b with _ | _ | y
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· exact Or.inl rfl
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· exact Or.inr (Or.inl rfl)
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· exact Or.inl rfl
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· exact absurd hsup (by simp)
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· exact Or.inr (Or.inl rfl)
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· exact absurd hsup (by simp)
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· exact absurd hsup (by simp)
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· exact Or.inr (Or.inl rfl)
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· rw [mk_sup_mk] at hsup
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by_cases hxy : x = y
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· exact Or.inr (Or.inr (by rw [hxy]))
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· rw [if_neg hxy] at hsup
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exact absurd hsup (by simp)
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/-- Monotonicity for *strict* operations on flat lattices: if `f` sends `⊥` to
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`⊥` (in either argument) and `⊤` to `⊤` (against any non-`⊥` argument), it is
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monotone in both arguments — regardless of its values on plain elements.
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`Analysis/Sign.agda` and `Analysis/Constant.agda` postulated exactly these
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monotonicity facts for their `plus`/`minus`, all of which have this shape. -/
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lemma monotone₂_of_strict {β γ : Type*} [DecidableEq β] [DecidableEq γ]
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(f : AboveBelow α → AboveBelow β → AboveBelow γ)
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(hbotl : ∀ y, f bot y = bot) (hbotr : ∀ x, f x bot = bot)
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(htopl : ∀ y, y ≠ bot → f top y = top)
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(htopr : ∀ x, x ≠ bot → f x top = top) : Monotone₂ f := by
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constructor
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· intro y a b hab
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show f a y ≤ f b y
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rcases le_cases hab with rfl | rfl | rfl
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· rw [hbotl]; exact bot_le' _
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· rcases eq_or_ne y bot with rfl | hy
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· rw [hbotr, hbotr]
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· rw [htopl y hy]; exact le_top' _
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· exact le_rfl
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· intro x a b hab
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show f x a ≤ f x b
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rcases le_cases hab with rfl | rfl | rfl
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· rw [hbotr]; exact bot_le' _
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· rcases eq_or_ne x bot with rfl | hx
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· rw [hbotl, hbotl]
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· rw [htopr x hx]; exact le_top' _
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· exact le_rfl
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/-! ### Interpretations of flat lattices -/
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section Interp
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variable {V : Type*} {P : AboveBelow α → V → Prop}
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lemma interp_sup_of (hbot : ∀ v, ¬P bot v) (htop : ∀ v, P top v)
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{s₁ s₂ : AboveBelow α} (v : V) (h : P s₁ v ∨ P s₂ v) : P (s₁ ⊔ s₂) v := by
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rcases s₁ with _ | _ | x
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· rw [bot_sup]; exact h.resolve_left (hbot v)
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· rw [top_sup]; exact htop v
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· rcases s₂ with _ | _ | y
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· rw [sup_bot]; exact h.resolve_right (hbot v)
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· rw [sup_top]; exact htop v
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· rw [mk_sup_mk]
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split
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· next heq => subst heq; exact h.elim id id
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· exact htop v
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lemma interp_inf_of
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(hdisj : ∀ {x y : α}, x ≠ y → ∀ v, ¬(P (mk x) v ∧ P (mk y) v))
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{s₁ s₂ : AboveBelow α} (v : V) (h : P s₁ v ∧ P s₂ v) : P (s₁ ⊓ s₂) v := by
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rcases s₁ with _ | _ | x
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· rw [bot_inf]; exact h.1
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· rw [top_inf]; exact h.2
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· rcases s₂ with _ | _ | y
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· rw [inf_bot]; exact h.2
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· rw [inf_top]; exact h.1
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· rw [mk_inf_mk]
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split
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· next heq => subst heq; exact h.1
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· next hne => exact absurd h (hdisj hne v)
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end Interp
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/-- Rank of an element: `⊥ ↦ 0`, `[x] ↦ 1`, `⊤ ↦ 2`. Used to bound chains
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(Agda's `isLongest` / `x≺[y]⇒x≡⊥` / `[x]≺y⇒y≡⊤` case analysis lives here). -/
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def rank : AboveBelow α → ℕ
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| bot => 0
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| mk _ => 1
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| top => 2
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/-- Agda: the impossibility of `[x] ≺ [y]` (combines `x≺[y]⇒x≡⊥` and
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`[x]≺y⇒y≡⊤`: the flat middle layer is an antichain). -/
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lemma not_mk_lt_mk (x y : α) : ¬(mk x : AboveBelow α) < mk y := by
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intro h
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obtain ⟨hle, hne⟩ := lt_iff_le_and_ne.mp h
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have hsup := le_iff.mp hle
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rw [mk_sup_mk] at hsup
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by_cases hxy : x = y
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· rw [if_pos hxy] at hsup
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exact hne hsup
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· rw [if_neg hxy] at hsup
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exact absurd hsup (by simp)
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lemma rank_strictMono : StrictMono (rank : AboveBelow α → ℕ) := by
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intro a b hab
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rcases a with _ | _ | x <;> rcases b with _ | _ | y
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· exact absurd hab (lt_irrefl _)
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· simp [rank]
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· simp [rank]
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· exact absurd hab (bot_le' _).not_lt
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· exact absurd hab (lt_irrefl _)
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· exact absurd hab (le_top' _).not_lt
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· exact absurd hab (bot_le' _).not_lt
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· simp [rank]
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· exact absurd hab (not_mk_lt_mk x y)
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lemma boundedChains : BoundedChains (AboveBelow α) 2 := fun c => by
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have h := LTSeries.head_add_length_le_nat (c.map rank rank_strictMono)
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rw [LTSeries.head_map, LTSeries.last_map, LTSeries.map_length] at h
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have h2 : rank c.last ≤ 2 := by cases c.last <;> simp [rank]
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omega
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instance [Inhabited α] : FiniteHeightLattice (AboveBelow α) where
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toLattice := inferInstance
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longestChain :=
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((RelSeries.singleton _ bot).snoc (mk default)
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(by rw [RelSeries.last_singleton]; exact bot_lt_mk default)).snoc top
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(by rw [RelSeries.last_snoc]; exact mk_lt_top default)
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chains_bounded := boundedChains
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end AboveBelow
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end Spa
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