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

lean/Spa/Lattice/Bool.lean

@@ -29,6 +29,8 @@ lemma boundedChains : BoundedChains Bool 1 := fun c => by
 
 instance : FiniteHeightLattice Bool where
   toLattice := inferInstance
+  toOrderBot := inferInstance
+  toOrderTop := inferInstance
   longestChain := (RelSeries.singleton _ (⊥ : Bool)).snoc (⊤ : Bool)
     (by rw [RelSeries.last_singleton]; exact bot_lt_top)
   chains_bounded := boundedChains

lean/Spa/Lattice/Tuple.lean

@@ -141,10 +141,12 @@ private def stdChain : (n : ℕ) →
           ((FiniteHeightLattice.longestChain (α := β)).map
             (fun b => (Fin.cons b (⊥ : Fin n → β) : Fin (n + 1) → β)) consBot_strictMono)
           (prev.1.map (fun f => (Fin.cons (⊤ : β) f : Fin (n + 1) → β)) consTop_strictMono)
-          (by rw [LTSeries.last_map, LTSeries.head_map, prev.2.1]; rfl),
+          (by
+            rw [LTSeries.last_map, LTSeries.head_map,
+              FiniteHeightLattice.longestChain_last, prev.2.1]),
         by
           simp only [RelSeries.head_smash, LTSeries.head_map]
-          rw [show (FiniteHeightLattice.longestChain (α := β)).head = (⊥ : β) from rfl]
+          rw [FiniteHeightLattice.longestChain_head]
           funext i
           refine Fin.cases ?_ (fun j => ?_) i <;> simp [Pi.bot_apply],
         by
@@ -154,6 +156,8 @@ private def stdChain : (n : ℕ) →
 
 instance instFiniteHeight {n : ℕ} : FiniteHeightLattice (Fin n → β) where
   toLattice := inferInstance
+  toOrderBot := inferInstance
+  toOrderTop := inferInstance
   longestChain := (stdChain n).1
   chains_bounded := fun c => by
     obtain ⟨cs, _, _, hbound⟩ := exists_unzip c