Make FiniteHeightLattice extend Lattice and derive Top/Bot

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

lean/Spa/Lattice.lean

@@ -67,34 +67,44 @@ end Folds
 def BoundedChains (α : Type*) [Preorder α] (n : ℕ) : Prop :=
   ∀ c : LTSeries α, c.length ≤ n
 
-/-- Wrapper over `LTSeries` that exposes its beginning and end points, as well as its length, as part of the type. -/
-structure PointedLTSeries (α : Type*) (f t : α) (n : ℕ) [Preorder α] where
-  series : LTSeries α
-  head_series : series.head = f
-  last_series : series.last = t
-  length_series : series.length = n
-
 /-- A finite height lattice is a lattice in which all chains $a < \ldots < z$ have a maximum height `height`. -/
-class FiniteHeightLattice (α : Type*) [Lattice α] extends Bot α, Top α where
-  height : ℕ
-  longestChain : PointedLTSeries α ⊥ ⊤ height
-  chains_bounded : BoundedChains α height
+class FiniteHeightLattice (α : Type*) extends Lattice α where
+  longestChain : LTSeries α
+  chains_bounded : BoundedChains α longestChain.length
+
+-- a < ... < z
+-- ----------- length <= height
 
 namespace FiniteHeightLattice
 
-variable (α : Type*) [Lattice α] [FiniteHeightLattice α]
+def height (α : Type*) [FiniteHeightLattice α] : ℕ :=
+  (longestChain (α := α)).length
+
+variable (α : Type*) [FiniteHeightLattice α]
+
+instance (priority := 100) : Bot α := ⟨(longestChain (α := α)).head⟩
+instance (priority := 100) : Top α := ⟨(longestChain (α := α)).last⟩
 
 /-- The bottom element `⊥` of a finite height lattice is _actually_ the least element. -/
 lemma bot_le (a : α) : (⊥ : α) ≤ a := by
   by_cases heq : ⊥ ⊓ a = ⊥
   · exact inf_eq_left.mp heq
   · exfalso
-    have lc := FiniteHeightLattice.longestChain (α := α)
-    have hlt : ⊥ ⊓ a < lc.series.head := by
-      rw [lc.head_series]
-      exact lt_of_le_of_ne inf_le_left heq
-    have hbound := FiniteHeightLattice.chains_bounded (lc.series.cons (⊥ ⊓ a) hlt)
-    rw [RelSeries.cons_length, lc.length_series] at hbound
+    have hlt : ⊥ ⊓ a < (longestChain (α := α)).head :=
+      lt_of_le_of_ne inf_le_left heq
+    have hbound := chains_bounded ((longestChain (α := α)).cons (⊥ ⊓ a) hlt)
+    rw [RelSeries.cons_length] at hbound
+    omega
+
+/-- The top element `⊤` of a finite height lattice is _actually_ the greatest element. -/
+lemma le_top (a : α) : a ≤ (⊤ : α) := by
+  by_cases heq : a ⊔ ⊤ = ⊤
+  · exact sup_eq_right.mp heq
+  · exfalso
+    have hlt : (longestChain (α := α)).last < a ⊔ ⊤ :=
+      lt_of_le_of_ne le_sup_right (Ne.symm heq)
+    have hbound := chains_bounded ((longestChain (α := α)).snoc (a ⊔ ⊤) hlt)
+    rw [RelSeries.snoc_length] at hbound
     omega
 
 end FiniteHeightLattice