Lean migration: Phase 3 (Unit, Prod, AboveBelow lattices)
- Spa.Lattice.Unit: PUnit fixed height 0 (lattice lifted from mathlib) - Spa.Lattice.Prod: chain unzip + FixedHeight (h1+h2) on products (componentwise lattice lifted from mathlib's Prod.instLattice) - Spa.Lattice.AboveBelow: flat lattice via Lattice.mk' (mirrors the Agda semilattices+absorption construction), boundedness via rank into Nat, Plain x ↦ plainFixedHeight x, height 2 Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
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import Spa.Lattice
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import Spa.Lattice
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import Spa.Fixedpoint
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import Spa.Fixedpoint
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import Spa.Isomorphism
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import Spa.Isomorphism
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import Spa.Lattice.Unit
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import Spa.Lattice.Prod
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import Spa.Lattice.AboveBelow
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199
lean/Spa/Lattice/AboveBelow.lean
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199
lean/Spa/Lattice/AboveBelow.lean
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/-
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Port of `Lattice/AboveBelow.agda`: the flat lattice obtained by adjoining a
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top and bottom element to an (unordered, decidable-equality) type.
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With propositional equality the `_≈_` data type and its equivalence/decidability
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proofs disappear (`deriving DecidableEq`). The lattice itself cannot be lifted:
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mathlib has no "flat lattice on a discrete type". The `Lattice` instance is
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built with `Lattice.mk'`, which — exactly like the Agda module — consumes the
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two semilattices (comm/assoc, idempotence derived) plus the absorption laws,
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and defines `a ≤ b ↔ a ⊔ b = b` (Agda's `_≼_`).
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The Agda module's `Plain x` submodule (the witness `x` seeds the longest chain
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`⊥ ≺ [x] ≺ ⊤`) becomes `plainFixedHeight x`; the boundedness proof `isLongest`
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is restated through a rank function since chains are mathlib `LTSeries` rather
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than a pattern-matchable inductive (the `¬-Chain-⊤`-style case analysis lives
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in `rank_strictMono`).
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-/
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import Spa.Lattice
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namespace Spa
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/-- Agda: `AboveBelow` with constructors `⊥`, `⊤`, `[_]`. -/
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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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/-- Agda: the `Showable` instance. -/
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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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/-! Agda: `⊥⊔x≡x`, `⊤⊔x≡⊤`, `x⊔⊥≡x`, `x⊔⊤≡⊤`, and the `[x]⊔[y]` reductions
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(`x≈y⇒[x]⊔[y]≡[x]` / `x̷≈y⇒[x]⊔[y]≡⊤` are the two branches of `mk_sup_mk`). -/
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@[simp] theorem bot_sup (x : AboveBelow α) : bot ⊔ x = x := rfl
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@[simp] theorem top_sup (x : AboveBelow α) : top ⊔ x = top := rfl
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@[simp] theorem sup_bot (x : AboveBelow α) : x ⊔ bot = x := by cases x <;> rfl
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@[simp] theorem sup_top (x : AboveBelow α) : x ⊔ top = top := by cases x <;> rfl
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@[simp] theorem 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] theorem bot_inf (x : AboveBelow α) : bot ⊓ x = bot := rfl
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@[simp] theorem top_inf (x : AboveBelow α) : top ⊓ x = x := rfl
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@[simp] theorem inf_bot (x : AboveBelow α) : x ⊓ bot = bot := by cases x <;> rfl
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@[simp] theorem inf_top (x : AboveBelow α) : x ⊓ top = x := by cases x <;> rfl
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@[simp] theorem 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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/-- Agda: `⊔-comm`. -/
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protected theorem 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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/-- Agda: `⊔-assoc`. -/
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protected theorem 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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/-- Agda: `⊓-comm`. -/
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protected theorem 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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/-- Agda: `⊓-assoc`. -/
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protected theorem 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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/-- Agda: `absorb-⊔-⊓`. -/
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protected theorem 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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/-- Agda: `absorb-⊓-⊔`. -/
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protected theorem 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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/-- Agda: `isLattice` (via the two semilattices + absorption, like the Agda
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record; `Lattice.mk'` derives idempotence and sets `a ≤ b ↔ a ⊔ b = b`). -/
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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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theorem le_iff {a b : AboveBelow α} : a ≤ b ↔ a ⊔ b = b := sup_eq_right.symm
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/-- Agda: `⊥≺[x]` (the `≤` part; `⊥` is least). -/
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theorem bot_le' (a : AboveBelow α) : (bot : AboveBelow α) ≤ a :=
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le_iff.mpr (bot_sup a)
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/-- Agda: `[x]≺⊤` (the `≤` part; `⊤` is greatest). -/
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theorem le_top' (a : AboveBelow α) : a ≤ (top : AboveBelow α) :=
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le_iff.mpr (sup_top a)
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theorem 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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theorem 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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theorem bot_lt_top : (bot : AboveBelow α) < top :=
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lt_of_le_of_ne (bot_le' _) (by simp)
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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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theorem 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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theorem 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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/-- Agda: `isLongest` — no chain is longer than 2. -/
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theorem 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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/-- Agda: `Plain.longestChain` and `Plain.fixedHeight` — the witness `x`
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seeds the chain `⊥ ≺ [x] ≺ ⊤` of length 2. -/
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def plainFixedHeight (x : α) : FixedHeight (AboveBelow α) 2 where
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bot := bot
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top := top
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longestChain :=
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((RelSeries.singleton _ bot).snoc (mk x)
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(by rw [RelSeries.last_singleton]; exact bot_lt_mk x)).snoc top
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(by rw [RelSeries.last_snoc]; exact mk_lt_top x)
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head_longestChain := by simp
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last_longestChain := by simp
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length_longestChain := by simp [RelSeries.snoc, RelSeries.append]
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bounded := boundedChains
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/-- Agda: `Plain.isFiniteHeightLattice` / `Plain.finiteHeightLattice`
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(`default` plays the role of the Agda module parameter `x`). -/
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instance [Inhabited α] : FiniteHeightLattice (AboveBelow α) where
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height := 2
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fixedHeight := plainFixedHeight default
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end AboveBelow
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end Spa
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113
lean/Spa/Lattice/Prod.lean
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lean/Spa/Lattice/Prod.lean
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/-
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Port of `Lattice/Prod.agda`.
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The component-wise lattice structure on `α × β` is lifted into mathlib
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(`Prod.instLattice`), as is decidability of equality. What remains custom is
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the fixed-height content:
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unzip ↦ LTSeries.exists_unzip
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a,∙-Monotonic/∙,b-Monotonic ↦ Prod.mk_lt_mk_iff_right/left (strict mono of
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the two injections, used to map the chains)
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fixedHeight (h₁ + h₂) ↦ FixedHeight.prod
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isFiniteHeightLattice ↦ instance FiniteHeightLattice (α × β)
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-/
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import Spa.Lattice
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namespace Spa
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section Unzip
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variable {α β : Type*} [PartialOrder α] [PartialOrder β]
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/-- Agda: `unzip` — a chain in the product splits into chains of the
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components whose lengths sum to at least the original length. -/
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theorem LTSeries.exists_unzip (c : LTSeries (α × β)) :
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∃ (c₁ : LTSeries α) (c₂ : LTSeries β),
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c₁.head = c.head.1 ∧ c₁.last = c.last.1 ∧
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c₂.head = c.head.2 ∧ c₂.last = c.last.2 ∧
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c.length ≤ c₁.length + c₂.length := by
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suffices H : ∀ (n : ℕ) (c : LTSeries (α × β)), c.length = n →
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∃ (c₁ : LTSeries α) (c₂ : LTSeries β),
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c₁.head = c.head.1 ∧ c₁.last = c.last.1 ∧
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c₂.head = c.head.2 ∧ c₂.last = c.last.2 ∧
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c.length ≤ c₁.length + c₂.length from H c.length c rfl
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intro n
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induction n with
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| zero =>
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intro c hn
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refine ⟨RelSeries.singleton _ c.head.1, RelSeries.singleton _ c.head.2,
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rfl, ?_, rfl, ?_, by simp [hn]⟩ <;>
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· have hlast : Fin.last c.length = 0 := by ext; simp [hn]
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simp [RelSeries.last, RelSeries.head, hlast]
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| succ n ih =>
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intro c hn
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have h0 : c.length ≠ 0 := by omega
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obtain ⟨c₁, c₂, hh₁, hl₁, hh₂, hl₂, hlen⟩ :=
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ih (c.tail h0) (by simp [RelSeries.tail_length, hn])
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rw [RelSeries.last_tail] at hl₁ hl₂
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rw [RelSeries.head_tail] at hh₁ hh₂
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rw [RelSeries.tail_length] at hlen
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have hstep : c.head < c 1 := by
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have h := c.step ⟨0, by omega⟩
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have h1 : (⟨0, by omega⟩ : Fin c.length).succ = 1 := by
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ext; simp [Fin.val_one, Nat.mod_eq_of_lt (by omega : 1 < c.length + 1)]
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rwa [h1] at h
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obtain ⟨hle1, hle2⟩ := Prod.le_def.mp hstep.le
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rcases eq_or_lt_of_le hle1 with heq1 | hlt1 <;>
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rcases eq_or_lt_of_le hle2 with heq2 | hlt2
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· exact absurd (Prod.ext heq1 heq2) hstep.ne
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· refine ⟨c₁, c₂.cons c.head.2 (hh₂ ▸ hlt2),
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hh₁.trans heq1.symm, hl₁, RelSeries.head_cons .., by
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rw [RelSeries.last_cons]; exact hl₂, by
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simp only [RelSeries.cons_length]; omega⟩
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· refine ⟨c₁.cons c.head.1 (hh₁ ▸ hlt1), c₂,
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RelSeries.head_cons .., by
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rw [RelSeries.last_cons]; exact hl₁,
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hh₂.trans heq2.symm, hl₂, by
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simp only [RelSeries.cons_length]; omega⟩
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· refine ⟨c₁.cons c.head.1 (hh₁ ▸ hlt1), c₂.cons c.head.2 (hh₂ ▸ hlt2),
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RelSeries.head_cons .., by
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rw [RelSeries.last_cons]; exact hl₁,
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RelSeries.head_cons .., by
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rw [RelSeries.last_cons]; exact hl₂, by
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simp only [RelSeries.cons_length]; omega⟩
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end Unzip
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section FixedHeight
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variable {α β : Type*} [Lattice α] [Lattice β]
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/-- Agda: `Lattice/Prod.agda`'s `fixedHeight` — the product of lattices of
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|
heights `h₁` and `h₂` has height `h₁ + h₂`. The longest chain climbs the first
|
||||||
|
component (at `⊥₂`), then the second component (at `⊤₁`). -/
|
||||||
|
def FixedHeight.prod {h₁ h₂ : ℕ} (fhA : FixedHeight α h₁) (fhB : FixedHeight β h₂) :
|
||||||
|
FixedHeight (α × β) (h₁ + h₂) where
|
||||||
|
bot := (fhA.bot, fhB.bot)
|
||||||
|
top := (fhA.top, fhB.top)
|
||||||
|
longestChain :=
|
||||||
|
RelSeries.smash
|
||||||
|
(fhA.longestChain.map (fun a => (a, fhB.bot))
|
||||||
|
(fun _ _ h => Prod.mk_lt_mk_iff_left.mpr h))
|
||||||
|
(fhB.longestChain.map (fun b => (fhA.top, b))
|
||||||
|
(fun _ _ h => Prod.mk_lt_mk_iff_right.mpr h))
|
||||||
|
(by simp [fhA.last_longestChain, fhB.head_longestChain])
|
||||||
|
head_longestChain := by simp [fhA.head_longestChain]
|
||||||
|
last_longestChain := by simp [fhB.last_longestChain]
|
||||||
|
length_longestChain := by
|
||||||
|
simp [fhA.length_longestChain, fhB.length_longestChain]
|
||||||
|
bounded := fun c => by
|
||||||
|
obtain ⟨c₁, c₂, -, -, -, -, hlen⟩ := LTSeries.exists_unzip c
|
||||||
|
have h₁ := fhA.bounded c₁
|
||||||
|
have h₂ := fhB.bounded c₂
|
||||||
|
omega
|
||||||
|
|
||||||
|
/-- Agda: `Lattice/Prod.agda`'s `isFiniteHeightLattice`/`finiteHeightLattice`. -/
|
||||||
|
instance [IA : FiniteHeightLattice α] [IB : FiniteHeightLattice β] :
|
||||||
|
FiniteHeightLattice (α × β) where
|
||||||
|
height := IA.height + IB.height
|
||||||
|
fixedHeight := IA.fixedHeight.prod IB.fixedHeight
|
||||||
|
|
||||||
|
end FixedHeight
|
||||||
|
|
||||||
|
end Spa
|
||||||
32
lean/Spa/Lattice/Unit.lean
Normal file
32
lean/Spa/Lattice/Unit.lean
Normal file
@@ -0,0 +1,32 @@
|
|||||||
|
/-
|
||||||
|
Port of `Lattice/Unit.agda`.
|
||||||
|
|
||||||
|
The lattice structure itself (`_⊔_`, `_⊓_`, all semilattice/lattice laws) is
|
||||||
|
lifted into mathlib: `PUnit.instLinearOrder` provides `Lattice PUnit`.
|
||||||
|
What remains is the fixed-height structure: the unit lattice has height 0.
|
||||||
|
-/
|
||||||
|
import Spa.Lattice
|
||||||
|
|
||||||
|
namespace Spa
|
||||||
|
|
||||||
|
/-- Chains in a subsingleton order are bounded by any `n` (Agda: the `bounded`
|
||||||
|
field of `Lattice/Unit.agda`'s `fixedHeight`, generalized). -/
|
||||||
|
theorem boundedChains_of_subsingleton (α : Type*) [Preorder α] [Subsingleton α]
|
||||||
|
(n : ℕ) : BoundedChains α n := fun c => by
|
||||||
|
by_contra hc
|
||||||
|
push_neg at hc
|
||||||
|
exact (c.step ⟨0, by omega⟩).ne (Subsingleton.elim _ _)
|
||||||
|
|
||||||
|
/-- Agda: `Lattice/Unit.agda`'s `fixedHeight`/`isFiniteHeightLattice`. -/
|
||||||
|
instance : FiniteHeightLattice PUnit where
|
||||||
|
height := 0
|
||||||
|
fixedHeight :=
|
||||||
|
{ bot := PUnit.unit
|
||||||
|
top := PUnit.unit
|
||||||
|
longestChain := RelSeries.singleton _ PUnit.unit
|
||||||
|
head_longestChain := rfl
|
||||||
|
last_longestChain := rfl
|
||||||
|
length_longestChain := rfl
|
||||||
|
bounded := boundedChains_of_subsingleton PUnit 0 }
|
||||||
|
|
||||||
|
end Spa
|
||||||
Reference in New Issue
Block a user