Register cases rules on lattice carriers for aesop automation

Tag the finite lattice carrier types with `@[aesop safe cases]`
(`AboveBelow`, `Sign`) so aesop performs the dominant proof step in this
framework -- case-splitting a lattice element -- automatically. Combined
with the existing `@[simp]` operation lemmas, this collapses the recurring
"case-split then reduce" proofs to a bare `aesop`:

  * AboveBelow's six lattice axioms drop their explicit `rcases`
  * Sign/Constant `plus_mono₂`/`minus_mono₂` become `by aesop`
  * Constant `plus_valid`/`minus_valid` shrink to a 2-line `rcases <;> simp_all`
  * `not_mk_lt_mk` is reexpressed via `le_cases`

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

lean/Spa/Analysis/Constant.lean

@@ -29,15 +29,13 @@ def minus : ConstLattice → ConstLattice → ConstLattice
 
 lemma plus_mono₂ : Monotone₂ plus :=
   AboveBelow.monotone₂_of_strict plus
-    (fun y => by cases y <;> rfl) (fun x => by cases x <;> rfl)
-    (fun y hy => by cases y <;> first | exact absurd rfl hy | rfl)
-    (fun x hx => by cases x <;> first | exact absurd rfl hx | rfl)
+    (fun y => by aesop) (fun x => by aesop)
+    (fun y hy => by aesop) (fun x hx => by aesop)
 
 lemma minus_mono₂ : Monotone₂ minus :=
   AboveBelow.monotone₂_of_strict minus
-    (fun y => by cases y <;> rfl) (fun x => by cases x <;> rfl)
-    (fun y hy => by cases y <;> first | exact absurd rfl hy | rfl)
-    (fun x hx => by cases x <;> first | exact absurd rfl hx | rfl)
+    (fun y => by aesop) (fun x => by aesop)
+    (fun y hy => by aesop) (fun x hx => by aesop)
 
 def interpConst : ConstLattice → Value → Prop
   | .bot, _ => False
@@ -96,36 +94,14 @@ def output : String :=
 lemma plus_valid {g₁ g₂ : ConstLattice} {z₁ z₂ : ℤ}
     (h₁ : ⟦g₁⟧ (.int z₁)) (h₂ : ⟦g₂⟧ (.int z₂)) :
     ⟦plus g₁ g₂⟧ (.int (z₁ + z₂)) := by
-  rcases g₁ with _ | _ | c₁
-  · exact h₁.elim
-  · rcases g₂ with _ | _ | c₂
-    · exact h₂.elim
-    · exact trivial
-    · exact trivial
-  · rcases g₂ with _ | _ | c₂
-    · exact h₂.elim
-    · exact trivial
-    · injection h₁ with hz₁
-      injection h₂ with hz₂
-      show Value.int (z₁ + z₂) = Value.int (c₁ + c₂)
-      rw [hz₁, hz₂]
+  rcases g₁ with _ | _ | c₁ <;> rcases g₂ with _ | _ | c₂ <;>
+    simp_all [plus, constInterpretation, interpConst]
 
 lemma minus_valid {g₁ g₂ : ConstLattice} {z₁ z₂ : ℤ}
     (h₁ : ⟦g₁⟧ (.int z₁)) (h₂ : ⟦g₂⟧ (.int z₂)) :
     ⟦minus g₁ g₂⟧ (.int (z₁ - z₂)) := by
-  rcases g₁ with _ | _ | c₁
-  · exact h₁.elim
-  · rcases g₂ with _ | _ | c₂
-    · exact h₂.elim
-    · exact trivial
-    · exact trivial
-  · rcases g₂ with _ | _ | c₂
-    · exact h₂.elim
-    · exact trivial
-    · injection h₁ with hz₁
-      injection h₂ with hz₂
-      show Value.int (z₁ - z₂) = Value.int (c₁ - c₂)
-      rw [hz₁, hz₂]
+  rcases g₁ with _ | _ | c₁ <;> rcases g₂ with _ | _ | c₂ <;>
+    simp_all [minus, constInterpretation, interpConst]
 
 instance eval_valid : ValidExprEvaluator ConstLattice prog := by
   constructor

lean/Spa/Analysis/Sign.lean

@@ -13,6 +13,8 @@ inductive Sign where
   | zero
   deriving DecidableEq
 
+attribute [aesop safe cases] Sign
+
 instance : Showable Sign :=
   ⟨fun
     | .plus => "+"
@@ -57,21 +59,13 @@ def minus : SignLattice → SignLattice → SignLattice
 
 lemma plus_mono₂ : Monotone₂ plus :=
   AboveBelow.monotone₂_of_strict plus
-    (fun y => by cases y <;> rfl)
-    (fun x => by rcases x with _ | _ | s <;> first | rfl | (cases s <;> rfl))
-    (fun y hy => by cases y <;> first | exact absurd rfl hy | rfl)
-    (fun x hx => by
-      rcases x with _ | _ | s <;>
-        first | exact absurd rfl hx | rfl | (cases s <;> rfl))
+    (fun y => by aesop) (fun x => by aesop)
+    (fun y hy => by aesop) (fun x hx => by aesop)
 
 lemma minus_mono₂ : Monotone₂ minus :=
   AboveBelow.monotone₂_of_strict minus
-    (fun y => by cases y <;> rfl)
-    (fun x => by rcases x with _ | _ | s <;> first | rfl | (cases s <;> rfl))
-    (fun y hy => by cases y <;> first | exact absurd rfl hy | rfl)
-    (fun x hx => by
-      rcases x with _ | _ | s <;>
-        first | exact absurd rfl hx | rfl | (cases s <;> rfl))
+    (fun y => by aesop) (fun x => by aesop)
+    (fun y hy => by aesop) (fun x hx => by aesop)
 
 def interpSign : SignLattice → Value → Prop
   | .bot, _ => False