Usw OrderBot / OrderTop for lattice witnesses

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Co-Authored-By: OpenAI Codex <codex@openai.com>

lean/Spa/Lattice/AboveBelow.lean

@@ -93,6 +93,14 @@ lemma bot_le' (a : AboveBelow α) : (bot : AboveBelow α) ≤ a :=
 lemma le_top' (a : AboveBelow α) : a ≤ (top : AboveBelow α) :=
   le_iff.mpr (sup_top a)
 
+instance : OrderBot (AboveBelow α) where
+  bot := bot
+  bot_le := bot_le'
+
+instance : OrderTop (AboveBelow α) where
+  top := top
+  le_top := le_top'
+
 lemma bot_lt_mk (x : α) : (bot : AboveBelow α) < mk x :=
   lt_of_le_of_ne (bot_le' _) (by simp)
 
@@ -224,6 +232,8 @@ lemma boundedChains : BoundedChains (AboveBelow α) 2 := fun c => by
 
 instance [Inhabited α] : FiniteHeightLattice (AboveBelow α) where
   toLattice := inferInstance
+  toOrderBot := inferInstance
+  toOrderTop := inferInstance
   longestChain :=
     ((RelSeries.singleton _ bot).snoc (mk default)
         (by rw [RelSeries.last_singleton]; exact bot_lt_mk default)).snoc top