Make FiniteHeightLattice extend Lattice and derive Top/Bot

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

lean/Spa/Lattice/Prod.lean

@@ -58,36 +58,23 @@ end Unzip
 
 section FixedHeight
 
-variable {α β : Type*} [Lattice α] [Lattice β]
+variable {α β : Type*}
 
 instance prod [A : FiniteHeightLattice α] [B : FiniteHeightLattice β] :
     FiniteHeightLattice (α × β) where
-  bot := ((⊥ : α), (⊥ : β))
-  top := ((⊤ : α), (⊤ : β))
-  height := A.height + B.height
+  toLattice := inferInstance
   longestChain :=
-    { series :=
-        RelSeries.smash
-          (A.longestChain.series.map (fun a => (a, (⊥ : β)))
-            (fun _ _ h => Prod.mk_lt_mk_iff_left.mpr h))
-          (B.longestChain.series.map (fun b => ((⊤ : α), b))
-            (fun _ _ h => Prod.mk_lt_mk_iff_right.mpr h))
-          (by simp [A.longestChain.last_series, B.longestChain.head_series])
-      head_series :=
-        (RelSeries.head_smash _).trans
-          ((LTSeries.head_map _ _ _).trans
-            (congrArg (·, (⊥ : β)) A.longestChain.head_series))
-      last_series :=
-        (RelSeries.last_smash _).trans
-          ((LTSeries.last_map _ _ _).trans
-            (congrArg ((⊤ : α), ·) B.longestChain.last_series))
-      length_series := by
-        show A.longestChain.series.length + B.longestChain.series.length = _
-        rw [A.longestChain.length_series, B.longestChain.length_series] }
+    RelSeries.smash
+      (A.longestChain.map (fun a => (a, (⊥ : β)))
+        (fun _ _ h => Prod.mk_lt_mk_iff_left.mpr h))
+      (B.longestChain.map (fun b => ((⊤ : α), b))
+        (fun _ _ h => Prod.mk_lt_mk_iff_right.mpr h))
+      rfl
   chains_bounded := fun c => by
     obtain ⟨c₁, c₂, -, -, -, -, hlen⟩ := LTSeries.exists_unzip c
     have h₁ := A.chains_bounded c₁
     have h₂ := B.chains_bounded c₂
+    show c.length ≤ A.longestChain.length + B.longestChain.length
     omega
 
 end FixedHeight