Delete more LLM-generated comments from the migration

lean/Spa/Lattice/AboveBelow.lean

@@ -1,25 +1,7 @@
-/-
-Port of `Lattice/AboveBelow.agda`: the flat lattice obtained by adjoining a
-top and bottom element to an (unordered, decidable-equality) type.
-
-With propositional equality the `_≈_` data type and its equivalence/decidability
-proofs disappear (`deriving DecidableEq`). The lattice itself cannot be lifted:
-mathlib has no "flat lattice on a discrete type". The `Lattice` instance is
-built with `Lattice.mk'`, which — exactly like the Agda module — consumes the
-two semilattices (comm/assoc, idempotence derived) plus the absorption laws,
-and defines `a ≤ b ↔ a ⊔ b = b` (Agda's `_≼_`).
-
-The Agda module's `Plain x` submodule (the witness `x` seeds the longest chain
-`⊥ ≺ [x] ≺ ⊤`) becomes `plainFixedHeight x`; the boundedness proof `isLongest`
-is restated through a rank function since chains are mathlib `LTSeries` rather
-than a pattern-matchable inductive (the `¬-Chain-⊤`-style case analysis lives
-in `rank_strictMono`).
--/
 import Spa.Lattice
 
 namespace Spa
 
-/-- Agda: `AboveBelow` with constructors `⊥`, `⊤`, `[_]`. -/
 inductive AboveBelow (α : Type*) where
   | bot
   | top
@@ -28,7 +10,6 @@ inductive AboveBelow (α : Type*) where
 
 namespace AboveBelow
 
-/-- Agda: the `Showable` instance. -/
 instance {α : Type*} [ToString α] : ToString (AboveBelow α) where
   toString
     | bot => "⊥"
@@ -53,9 +34,6 @@ instance : Min (AboveBelow α) where
     | mk _, bot => bot
     | mk x, top => mk x
 
-/-! Agda: `⊥⊔x≡x`, `⊤⊔x≡⊤`, `x⊔⊥≡x`, `x⊔⊤≡⊤`, and the `[x]⊔[y]` reductions
-(`x≈y⇒[x]⊔[y]≡[x]` / `x̷≈y⇒[x]⊔[y]≡⊤` are the two branches of `mk_sup_mk`). -/
-
 @[simp] theorem bot_sup (x : AboveBelow α) : bot ⊔ x = x := rfl
 @[simp] theorem top_sup (x : AboveBelow α) : top ⊔ x = top := rfl
 @[simp] theorem sup_bot (x : AboveBelow α) : x ⊔ bot = x := by cases x <;> rfl
@@ -70,46 +48,38 @@ instance : Min (AboveBelow α) where
 @[simp] theorem mk_inf_mk (x y : α) :
     (mk x ⊓ mk y : AboveBelow α) = if x = y then mk x else bot := rfl
 
-/-- Agda: `⊔-comm`. -/
 protected theorem sup_comm (a b : AboveBelow α) : a ⊔ b = b ⊔ a := by
   rcases a with _ | _ | x <;> rcases b with _ | _ | y <;> simp only
     [bot_sup, sup_bot, top_sup, sup_top, mk_sup_mk]
   split_ifs with h₁ h₂ h₂ <;> simp_all
 
-/-- Agda: `⊔-assoc`. -/
 protected theorem sup_assoc (a b c : AboveBelow α) : a ⊔ b ⊔ c = a ⊔ (b ⊔ c) := by
   rcases a with _ | _ | x <;> rcases b with _ | _ | y <;> rcases c with _ | _ | z <;>
     simp only [bot_sup, sup_bot, top_sup, sup_top, mk_sup_mk]
   split_ifs <;> simp_all
 
-/-- Agda: `⊓-comm`. -/
 protected theorem inf_comm (a b : AboveBelow α) : a ⊓ b = b ⊓ a := by
   rcases a with _ | _ | x <;> rcases b with _ | _ | y <;> simp only
     [bot_inf, inf_bot, top_inf, inf_top, mk_inf_mk]
   split_ifs with h₁ h₂ h₂ <;> simp_all
 
-/-- Agda: `⊓-assoc`. -/
 protected theorem inf_assoc (a b c : AboveBelow α) : a ⊓ b ⊓ c = a ⊓ (b ⊓ c) := by
   rcases a with _ | _ | x <;> rcases b with _ | _ | y <;> rcases c with _ | _ | z <;>
     simp only [bot_inf, inf_bot, top_inf, inf_top, mk_inf_mk]
   split_ifs <;> simp_all
 
-/-- Agda: `absorb-⊔-⊓`. -/
 protected theorem sup_inf_self (a b : AboveBelow α) : a ⊔ a ⊓ b = a := by
   rcases a with _ | _ | x <;> rcases b with _ | _ | y <;>
     simp only [bot_sup, sup_bot, top_sup, sup_top, mk_sup_mk,
                bot_inf, inf_bot, top_inf, inf_top, mk_inf_mk] <;>
     try (split_ifs <;> simp_all)
 
-/-- Agda: `absorb-⊓-⊔`. -/
 protected theorem inf_sup_self (a b : AboveBelow α) : a ⊓ (a ⊔ b) = a := by
   rcases a with _ | _ | x <;> rcases b with _ | _ | y <;>
     simp only [bot_sup, sup_bot, top_sup, sup_top, mk_sup_mk,
                bot_inf, inf_bot, top_inf, inf_top, mk_inf_mk] <;>
     try (split_ifs <;> simp_all)
 
-/-- Agda: `isLattice` (via the two semilattices + absorption, like the Agda
-record; `Lattice.mk'` derives idempotence and sets `a ≤ b ↔ a ⊔ b = b`). -/
 instance : Lattice (AboveBelow α) :=
   Lattice.mk' AboveBelow.sup_comm AboveBelow.sup_assoc
     AboveBelow.inf_comm AboveBelow.inf_assoc
@@ -117,11 +87,9 @@ instance : Lattice (AboveBelow α) :=
 
 theorem le_iff {a b : AboveBelow α} : a ≤ b ↔ a ⊔ b = b := sup_eq_right.symm
 
-/-- Agda: `⊥≺[x]` (the `≤` part; `⊥` is least). -/
 theorem bot_le' (a : AboveBelow α) : (bot : AboveBelow α) ≤ a :=
   le_iff.mpr (bot_sup a)
 
-/-- Agda: `[x]≺⊤` (the `≤` part; `⊤` is greatest). -/
 theorem le_top' (a : AboveBelow α) : a ≤ (top : AboveBelow α) :=
   le_iff.mpr (sup_top a)
 
@@ -134,9 +102,6 @@ theorem mk_lt_top (x : α) : (mk x : AboveBelow α) < top :=
 theorem bot_lt_top : (bot : AboveBelow α) < top :=
   lt_of_le_of_ne (bot_le' _) (by simp)
 
-/-- The order of the flat lattice, by cases (used to discharge the
-monotonicity obligations that were `postulate`d in `Analysis/Sign.agda` and
-`Analysis/Constant.agda`). -/
 theorem le_cases {a b : AboveBelow α} (h : a ≤ b) :
     a = bot ∨ b = top ∨ a = b := by
   have hsup := le_iff.mp h
@@ -183,18 +148,12 @@ theorem monotone₂_of_strict {β γ : Type*} [DecidableEq β] [DecidableEq γ]
       · rw [htopr x hx]; exact le_top' _
     · exact le_rfl
 
-/-! ### Interpretations of flat lattices
-
-The `⟦⟧-⊔-∨` / `⟦⟧-⊓-∧` proofs of `Analysis/Sign.agda` and
-`Analysis/Constant.agda` are the same case analysis; only the meaning of the
-plain elements differs. Factored here, they need just `P ⊥ ↦ False`,
-`P ⊤ ↦ True`, and (for `⊓`) disjointness of distinct plain elements. -/
+/-! ### Interpretations of flat lattices -/
 
 section Interp
 
 variable {V : Type*} {P : AboveBelow α → V → Prop}
 
-/-- Agda: `⟦⟧ᵍ-⊔ᵍ-∨` / `⟦⟧ᶜ-⊔ᶜ-∨`, generalized. -/
 theorem interp_sup_of (hbot : ∀ v, ¬P bot v) (htop : ∀ v, P top v)
     {s₁ s₂ : AboveBelow α} (v : V) (h : P s₁ v ∨ P s₂ v) : P (s₁ ⊔ s₂) v := by
   rcases s₁ with _ | _ | x
@@ -208,7 +167,6 @@ theorem interp_sup_of (hbot : ∀ v, ¬P bot v) (htop : ∀ v, P top v)
       · next heq => subst heq; exact h.elim id id
       · exact htop v
 
-/-- Agda: `⟦⟧ᵍ-⊓ᵍ-∧` / `⟦⟧ᶜ-⊓ᶜ-∧`, generalized. -/
 theorem interp_inf_of
     (hdisj : ∀ {x y : α}, x ≠ y → ∀ v, ¬(P (mk x) v ∧ P (mk y) v))
     {s₁ s₂ : AboveBelow α} (v : V) (h : P s₁ v ∧ P s₂ v) : P (s₁ ⊓ s₂) v := by
@@ -258,17 +216,12 @@ theorem rank_strictMono : StrictMono (rank : AboveBelow α → ℕ) := by
   · simp [rank]
   · exact absurd hab (not_mk_lt_mk x y)
 
-/-- Agda: `isLongest` — no chain is longer than 2. -/
 theorem boundedChains : BoundedChains (AboveBelow α) 2 := fun c => by
   have h := LTSeries.head_add_length_le_nat (c.map rank rank_strictMono)
   rw [LTSeries.head_map, LTSeries.last_map, LTSeries.map_length] at h
   have h2 : rank c.last ≤ 2 := by cases c.last <;> simp [rank]
   omega
 
-/-- Agda: `Plain.longestChain`/`Plain.fixedHeight` and
-`Plain.isFiniteHeightLattice`/`Plain.finiteHeightLattice` — the witness
-(`default`, playing the role of the Agda module parameter `x`) seeds the chain
-`⊥ ≺ [x] ≺ ⊤` of length 2. -/
 instance [Inhabited α] : FiniteHeightLattice (AboveBelow α) where
   bot := bot
   top := top