diff --git a/InductiveVerification.lean b/InductiveVerification.lean
index 06a9158..1861a8f 100644
--- a/InductiveVerification.lean
+++ b/InductiveVerification.lean
@@ -1,17 +1,4 @@
 -- This module serves as the root of the `InductiveVerification` library.
 -- Import modules here that should be built as part of the library.
-import InductiveVerification.Basic
-
-inductive Agent
-  | Server : Agent
-  | Friend : Nat → Agent
-  | Spy : Agent
-
-inductive Msg
-  | Agent : Agent → Msg -- Agent names
-  | Number : Nat → Msg -- Ordinary integers, timestamps, ...
-  | Nonce : Nat → Msg -- Unguessable nonces
-  | Key : Key → Msg -- Crypto keys
-  | Hash : Msg → Msg -- Hashing
-  | MPair : Msg → Msg → Msg -- Compound messages
-  | Crypt : Key → Msg → Msg -- Encryption, public- or shared-key
+import InductiveVerification.Message
+import InductiveVerification.Event
diff --git a/InductiveVerification/Basic.lean b/InductiveVerification/Basic.lean
deleted file mode 100644
index 99415d9..0000000
--- a/InductiveVerification/Basic.lean
+++ /dev/null
@@ -1 +0,0 @@
-def hello := "world"
diff --git a/InductiveVerification/Message.lean b/InductiveVerification/Message.lean
index d6e1d9d..3588166 100644
--- a/InductiveVerification/Message.lean
+++ b/InductiveVerification/Message.lean
@@ -1,4 +1,17 @@
-import Mathlib
+import Init.Data.Nat.Lemmas
+import Init.Prelude
+import Mathlib.Data.Nat.Basic
+import Mathlib.Data.Nat.Dist
+import Mathlib.Data.Set.Basic
+import Mathlib.Data.Set.Defs
+import Mathlib.Data.Set.Image
+import Mathlib.Data.Set.Insert
+import Mathlib.Data.Set.Lattice
+import Mathlib.Order.Closure
+import Mathlib.Order.Lattice
+import Mathlib.Tactic.ApplyAt
+import Mathlib.Tactic.SimpIntro
+import Mathlib.Tactic.NthRewrite
 
 -- Keys are integers
 abbrev Key := Nat
@@ -121,9 +134,11 @@ lemma keysFor_union (H H' : Set Msg) : keysFor (H ∪ H') = keysFor H ∪ keysFo
 
 -- Monotonicity
 @[simp]
-lemma keysFor_mono {G H : Set Msg} (h : G ⊆ H) : keysFor G ⊆ keysFor H := by
-  simp
-  grind
+lemma keysFor_mono: Monotone keysFor := by
+  simp_intro _ _ sub _ h
+  rcases h with ⟨K, ⟨⟨X, _⟩, _⟩⟩ ; exists K; apply And.intro
+  · exists X; aapply sub
+  · assumption
 
 -- Lemmas for `keysFor` with specific message types
 @[simp]
@@ -321,7 +336,7 @@ lemma parts_idem {H : Set Msg} : parts (parts H) = parts H :=
 lemma parts_subset_iff {G H : Set Msg} : (G ⊆ parts H) ↔ (parts G ⊆ parts H) :=
 by apply partsClosureOperator.le_closure_iff
 
-@[simp, grind]
+@[simp]
 lemma parts_trans {G H : Set Msg} {X : Msg} :
   X ∈ parts G → G ⊆ parts H → X ∈ parts H :=
 by intro a b; apply parts_mono at b; rw[parts_idem] at b; apply b; apply a;
@@ -331,9 +346,12 @@ by intro a b; apply parts_mono at b; rw[parts_idem] at b; apply b; apply a;
 lemma parts_cut {G H : Set Msg} {X Y : Msg} :
   Y ∈ parts (insert X G) → X ∈ parts H → Y ∈ parts (G ∪ H) :=
 by
-  intro a b; rw[parts_union]; rw[parts_insert] at a; cases a <;> grind
-
--- Rewrite rules for pulling out atomic messages
+  intro a b; rw[parts_union]; rw[parts_insert] at a; cases a <;> grind[parts_trans]
+  
+@[simp]
+lemma parts_cut_mono {G H : Set Msg} {X : Msg} :
+  X ∈ parts H → parts (insert X G) ⊆ parts (G ∪ H) :=
+by grind[parts_cut]
 
 @[simp]
 lemma parts_insert_Agent {H : Set Msg} {agt : Agent} :
@@ -1320,7 +1338,7 @@ lemma pushKeysMPair {K : Key} {X Y : Msg} {H : Set Msg}:
   by simp [Set.insert_comm]
 
 lemma pushKeysCrypt {K K' : Key} {X : Msg} {H : Set Msg}:
-  insert (Key K) (insert (Crypt K' N) H) = insert (Crypt K' N) (insert (Key K) H) :=
+  insert (Key K) (insert (Crypt K' X) H) = insert (Crypt K' X) (insert (Key K) H) :=
   by simp [Set.insert_comm]
 
 lemma pushCryptsAgent {X : Msg} {K : Key} {C : Agent} {H : Set Msg} :
diff --git a/Main.lean b/Main.lean
index 9da78ea..c707829 100644
--- a/Main.lean
+++ b/Main.lean
@@ -1,4 +1,4 @@
 import InductiveVerification
 
 def main : IO Unit :=
-  IO.println s!"Hello, {hello}!"
+  IO.println "Hello, world!"
diff --git a/lakefile.toml b/lakefile.toml
index a836890..09d8c33 100644
--- a/lakefile.toml
+++ b/lakefile.toml
@@ -2,6 +2,10 @@ name = "inductive-verification"
 version = "0.1.0"
 defaultTargets = ["inductive-verification"]
 
+[[require]]
+name = "mathlib"
+git = "https://github.com/leanprover-community/mathlib4.git"
+
 [[lean_lib]]
 name = "InductiveVerification"
 
diff --git a/lean-toolchain b/lean-toolchain
index 7f254a9..fb18a7f 100644
--- a/lean-toolchain
+++ b/lean-toolchain
@@ -1 +1 @@
-leanprover/lean4:v4.23.0
+leanprover/lean4:v4.27.0-rc1
\ No newline at end of file