Added expand_parts_element macro

Further simplified proofs in NS_public

InductiveVerification/Message.lean

@@ -1,5 +1,6 @@
 import Init.Data.Nat.Lemmas
 import Init.Prelude
+import Lean
 import Mathlib.Data.Nat.Basic
 import Mathlib.Data.Nat.Dist
 import Mathlib.Data.Set.Basic
@@ -12,6 +13,8 @@ import Mathlib.Order.Lattice
 import Mathlib.Tactic.ApplyAt
 import Mathlib.Tactic.SimpIntro
 import Mathlib.Tactic.NthRewrite
+open Lean Elab Command Term Meta
+open Parser.Tactic
 
 -- Keys are integers
 abbrev Key := Nat
@@ -351,6 +354,14 @@ lemma parts_element:
   · intro h; apply_rules [ parts_subset_iff.mp, Set.singleton_subset_iff.mpr ]
   · intro h; aapply parts_subset_iff.mpr; simp
 
+/--
+A tactic that expands terms like `X ∈ parts H`
+-/
+syntax (name := expandPartsElement) "expand_parts_element" (ppSpace location) : tactic
+macro_rules
+  | `(tactic| expand_parts_element at $loc) =>
+  `(tactic| rw[parts_element, Set.subset_def] at $loc; simp at $loc)
+
 @[simp]
 lemma parts_insert_Agent {H : Set Msg} {agt : Agent} :
   parts (insert (Agent agt) H) = insert (Agent agt) (parts H) :=