Start working on notation for formalization

Per convention, create a new instance for 'interpretable' thing,
with an fundep'ed semantic domain. I feel at peace with this notation
even though it conflicts with Mathlib's quotients.

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

lean/Spa/Analysis/Constant.lean

@@ -26,6 +26,7 @@ Correspondence:
 -/
 import Spa.Analysis.Forward
 import Spa.Analysis.Utils
+import Spa.Interp
 import Spa.Showable
 
 namespace Spa
@@ -86,9 +87,12 @@ def interpConst : ConstLattice → Value → Prop
   | .top, _ => True
   | .mk z, v => v = .int z
 
+/-- Agda: `⟦_⟧ᶜ` is registered for the `⟦_⟧` interpretation notation. -/
+instance : Interp ConstLattice (Value → Prop) := ⟨interpConst⟩
+
 /-- Agda: `s₁≢s₂⇒¬s₁∧s₂`. -/
 theorem interpConst_mk_disjoint {z₁ z₂ : ℤ} (hne : z₁ ≠ z₂) {v : Value} :
-    ¬(interpConst (.mk z₁) v ∧ interpConst (.mk z₂) v) := by
+    ¬(⟦(.mk z₁ : ConstLattice)⟧ v ∧ ⟦(.mk z₂ : ConstLattice)⟧ v) := by
   rintro ⟨h₁, h₂⟩
   rw [h₁] at h₂
   injection h₂ with hz
@@ -96,12 +100,12 @@ theorem interpConst_mk_disjoint {z₁ z₂ : ℤ} (hne : z₁ ≠ z₂) {v : Val
 
 /-- Agda: `⟦⟧ᶜ-⊔ᶜ-∨` (via the factored flat-lattice lemma). -/
 theorem interpConst_sup {s₁ s₂ : ConstLattice} (v : Value)
-    (h : interpConst s₁ v ∨ interpConst s₂ v) : interpConst (s₁ ⊔ s₂) v :=
+    (h : ⟦s₁⟧ v ∨ ⟦s₂⟧ v) : ⟦s₁ ⊔ s₂⟧ v :=
   AboveBelow.interp_sup_of (fun _ h => h) (fun _ => trivial) v h
 
 /-- Agda: `⟦⟧ᶜ-⊓ᶜ-∧` (via the factored flat-lattice lemma). -/
 theorem interpConst_inf {s₁ s₂ : ConstLattice} (v : Value)
-    (h : interpConst s₁ v ∧ interpConst s₂ v) : interpConst (s₁ ⊓ s₂) v :=
+    (h : ⟦s₁⟧ v ∧ ⟦s₂⟧ v) : ⟦s₁ ⊓ s₂⟧ v :=
   AboveBelow.interp_inf_of (fun hne _ => interpConst_mk_disjoint hne) v h
 
 /-- Agda: `latticeInterpretationᶜ` (an instance there too). -/
@@ -153,8 +157,8 @@ def output : String :=
 
 /-- Agda: `plus-valid`. -/
 theorem plus_valid {g₁ g₂ : ConstLattice} {z₁ z₂ : ℤ}
-    (h₁ : interpConst g₁ (.int z₁)) (h₂ : interpConst g₂ (.int z₂)) :
-    interpConst (plus g₁ g₂) (.int (z₁ + z₂)) := by
+    (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
@@ -171,8 +175,8 @@ theorem plus_valid {g₁ g₂ : ConstLattice} {z₁ z₂ : ℤ}
 
 /-- Agda: `minus-valid`. -/
 theorem minus_valid {g₁ g₂ : ConstLattice} {z₁ z₂ : ℤ}
-    (h₁ : interpConst g₁ (.int z₁)) (h₂ : interpConst g₂ (.int z₂)) :
-    interpConst (minus g₁ g₂) (.int (z₁ - z₂)) := by
+    (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
@@ -194,11 +198,11 @@ instance eval_valid : ValidExprEvaluator ConstLattice prog := by
   induction hev with
   | num n =>
     intro _
-    show interpConst (eval prog (.num n) vs) (.int n)
+    show ⟦eval prog (.num n) vs⟧ (.int n)
     rfl
   | var x v hxv =>
     intro hvs
-    show interpConst (eval prog (.var x) vs) v
+    show ⟦eval prog (.var x) vs⟧ v
     simp only [eval]
     by_cases hk : FiniteMap.MemKey x vs
     · rw [dif_pos hk]
@@ -207,15 +211,15 @@ instance eval_valid : ValidExprEvaluator ConstLattice prog := by
       exact trivial
   | add e₁ e₂ z₁ z₂ _ _ ih₁ ih₂ =>
     intro hvs
-    have h₁ : interpConst (eval prog e₁ vs) (.int z₁) := ih₁ hvs
-    have h₂ : interpConst (eval prog e₂ vs) (.int z₂) := ih₂ hvs
-    show interpConst (eval prog (.add e₁ e₂) vs) (.int (z₁ + z₂))
+    have h₁ : ⟦eval prog e₁ vs⟧ (.int z₁) := ih₁ hvs
+    have h₂ : ⟦eval prog e₂ vs⟧ (.int z₂) := ih₂ hvs
+    show ⟦eval prog (.add e₁ e₂) vs⟧ (.int (z₁ + z₂))
     exact plus_valid h₁ h₂
   | sub e₁ e₂ z₁ z₂ _ _ ih₁ ih₂ =>
     intro hvs
-    have h₁ : interpConst (eval prog e₁ vs) (.int z₁) := ih₁ hvs
-    have h₂ : interpConst (eval prog e₂ vs) (.int z₂) := ih₂ hvs
-    show interpConst (eval prog (.sub e₁ e₂) vs) (.int (z₁ - z₂))
+    have h₁ : ⟦eval prog e₁ vs⟧ (.int z₁) := ih₁ hvs
+    have h₂ : ⟦eval prog e₂ vs⟧ (.int z₂) := ih₂ hvs
+    show ⟦eval prog (.sub e₁ e₂) vs⟧ (.int (z₁ - z₂))
     exact minus_valid h₁ h₂
 
 /-- Agda: `WithProg.analyze-correct`. -/