Remove maximal chain witness from FiniteHeightLattice

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

lean/Spa/Lattice.lean

@@ -77,45 +77,24 @@ lemma boundedChains_of_subsingleton (α : Type*) [Preorder α] [Subsingleton α]
 
 /-- A finite height lattice is a lattice in which all chains $a < \ldots < z$ have a maximum height `height`. -/
 class FiniteHeightLattice (α : Type*) extends Lattice α, OrderBot α, OrderTop α where
-  longestChain : LTSeries α
-  chains_bounded : BoundedChains α longestChain.length
+  height : ℕ
+  chains_bounded : BoundedChains α height
 
 -- a < ... < z
 -- ----------- length <= height
 
 namespace FiniteHeightLattice
 
-def height (α : Type*) [FiniteHeightLattice α] : ℕ :=
-  (longestChain (α := α)).length
-
-variable (α : Type*) [FiniteHeightLattice α]
-
-/-- Any maximum-length chain in a bounded finite-height lattice starts at `⊥`. -/
-lemma longestChain_head : (longestChain (α := α)).head = ⊥ := by
-  by_contra hne
-  have hbound := chains_bounded ((longestChain (α := α)).cons ⊥ (bot_lt_iff_ne_bot.mpr hne))
-  rw [RelSeries.cons_length] at hbound
-  omega
-
-/-- Any maximum-length chain in a bounded finite-height lattice ends at `⊤`. -/
-lemma longestChain_last : (longestChain (α := α)).last = ⊤ := by
-  by_contra hne
-  have hbound := chains_bounded ((longestChain (α := α)).snoc ⊤ (lt_top_iff_ne_top.mpr hne))
-  rw [RelSeries.snoc_length] at hbound
-  omega
-
 /-- This is something like a lemma about isomorphic types having the same height.
     Given a finite-height lattice `α`, lattice `β`, and a `Monotone` bijection
     between the two, we can show that lattice `β` also has a finite height.
 
-    The proof is fairly trivial: the longest chain in `α` can be transported
-    to be a longest chain in `β` (by monotonicity), establishing a height witness.
-    At the same time, any chain in `β` can be transported to a chain in `α`,
+    The proof is fairly trivial: any chain in `β` can be transported to a chain in `α`,
     and must be bounded by the same height by `FiniteHeightLattice.chains_bounded`. -/
 def transport {α β : Type*} [Lattice β]
     [I : FiniteHeightLattice α] (f : α → β) (g : β → α)
     (hf : Monotone f) (hg : Monotone g)
-    (hgf : Function.LeftInverse g f) (hfg : Function.LeftInverse f g) :
+    (hfg : Function.LeftInverse f g) :
     FiniteHeightLattice β where
   toLattice := inferInstance
   toOrderBot := {
@@ -128,8 +107,7 @@ def transport {α β : Type*} [Lattice β]
     le_top := fun b => by
       rw [← hfg b]
       exact hf (_root_.le_top : g b ≤ (⊤ : α)) }
-  longestChain :=
-    I.longestChain.map f (hf.strictMono_of_injective hgf.injective)
+  height := I.height
   chains_bounded := fun c =>
     I.chains_bounded (c.map g (hg.strictMono_of_injective hfg.injective))
 
@@ -144,7 +122,7 @@ def ofUnique (α : Type*) [Lattice α] [Unique α] :
   toOrderTop := {
     top := default
     le_top := fun _ => le_of_eq (Subsingleton.elim _ _) }
-  longestChain := RelSeries.singleton _ default
+  height := 0
   chains_bounded := boundedChains_of_subsingleton α 0
 
 end FiniteHeightLattice