Finished proff of NS

InductiveVerification/Event.lean

@@ -185,10 +185,10 @@ lemma knows_Spy_partsEs [Bad] :
     Event.Says A B X ∈ evs → X ∈ parts (knows Agent.Spy evs) := by
   exact Says_imp_parts_knows_Spy
 
-lemma Says_imp_analz_Spy [InvKey] [Bad] :
-  ∀ {A B : Agent} {X : Msg} {evs : List Event},
+lemma Says_imp_analz_Spy [InvKey] [Bad]
+  {A B : Agent} {X : Msg} {evs : List Event} :
     Event.Says A B X ∈ evs → X ∈ analz (knows Agent.Spy evs) := by
-  intro A B X evs h
+  intro h
   apply analz.inj
   apply Says_imp_knows_Spy
   exact h
@@ -209,6 +209,18 @@ lemma parts_insert_spies [Bad] :
 by
   apply parts_insert
 
+lemma analz_spies_mono [InvKey] [Bad]
+  { h : M ∈ analz (knows Agent.Spy evs) } :
+  M ∈ analz (knows Agent.Spy (ev :: evs))
+:= by
+  aapply analz_mono; exact knows_subset_knows_Cons
+
+lemma analz_spies_mono_neg [InvKey] [Bad]
+  { h : M ∉ analz (knows Agent.Spy (ev :: evs)) } :
+  M ∉ analz (knows Agent.Spy evs)
+:= by
+  intro h₁; apply h; aapply analz_spies_mono
+
 -- Knowledge of Agents
 lemma knows_subset_knows_Says [Bad] :
   ∀ {A A' B : Agent} {X : Msg} {evs : List Event},

InductiveVerification/Message.lean

@@ -911,7 +911,6 @@ by
 lemma analz_subset_iff {G H : Set Msg} [InvKey] : (G ⊆ analz H) ↔ (analz G ⊆ analz H) :=
   by apply analzClosureOperator.le_closure_iff
 
-@[simp]
 lemma analz_trans {G H : Set Msg} {X : Msg} [InvKey] :
   X ∈ analz G → G ⊆ analz H → X ∈ analz H :=
 by

InductiveVerification/NS_Public.lean

@@ -108,7 +108,7 @@ lemma no_nonce_NS1_NS2_helper2
   | inr => aapply h₁
 
 -- It is impossible to re-use a nonce in both NS1 and NS2, provided the nonce is secret
-theorem no_nonce_NS1_NS2 { h : ns_public evs  } :
+theorem no_nonce_NS1_NS2 { evs: List Event} { h : ns_public evs  } {A B C D : Agent} :
   (Crypt (pubEK C) ⦃NA', Nonce NA, Agent D⦄ ∈ parts (spies evs) →
   (Crypt (pubEK B) ⦃Nonce NA, Agent A⦄ ∈ parts (spies evs) →
   Nonce NA ∈ analz (spies evs))) := by
@@ -143,6 +143,18 @@ theorem no_nonce_NS1_NS2 { h : ns_public evs  } :
     | inr => rw[spies, knows]; apply analz_mono; apply Set.subset_insert; simp_all
   | NS3 => rw[spies, knows]; apply analz_mono; apply Set.subset_insert; simp_all
 
+lemma unique_Nonce_apply_ih {P : Prop}
+{h₃ : Nonce NA ∉ analz (spies (Says Agent.Spy C X :: evsf))}
+{h₁ : M₁ ∈ parts (spies evsf)}
+{h₂ : M₂ ∈ parts (spies evsf)}
+{a_ih : M₁ ∈ parts (spies evsf)
+  → M₂ ∈ parts (spies evsf)
+  → Nonce NA ∉ analz (spies evsf) → P}
+: P := by
+  simp[spies, knows] at h₃; apply Set.notMem_subset at h₃
+  · aapply a_ih;
+  · apply analz_mono; apply Set.subset_insert    
+
 lemma unique_NA_apply_ih {P : Prop}
 {a_ih : Crypt (pubEK B) ⦃Nonce NA, Msg.Agent A⦄ ∈ parts (spies evsf)
   → Crypt (pubEK B') ⦃Nonce NA, Msg.Agent A'⦄ ∈ parts (spies evsf)
@@ -151,9 +163,7 @@ lemma unique_NA_apply_ih {P : Prop}
 {h₁ : Crypt (pubEK B) ⦃Nonce NA, Msg.Agent A⦄ ∈ parts (spies evsf)}
 {h₂ : Crypt (pubEK B') ⦃Nonce NA, Msg.Agent A'⦄ ∈ parts (spies evsf)}
 : P := by
-  simp[spies, knows] at h₃; apply Set.notMem_subset at h₃
-  · aapply a_ih;
-  · apply analz_mono; apply Set.subset_insert    
+  aapply unique_Nonce_apply_ih (h₁ := h₁) (h₂ := h₂)
 
 lemma unique_NA_contradict
 {h₃ : Nonce NA ∉ analz (spies (Says Agent.Spy B X :: evsf))}
@@ -207,31 +217,368 @@ theorem unique_NA { h : ns_public evs } :
         | inr => aapply unique_NA_apply_ih
       | inr => aapply unique_NA_apply_ih
   | NS1 _ _ a_ih => intro h₁ h₂ h₃; simp at h₁; cases h₁ with
-    | inl => sorry
-    | inr => simp at h₂; cases h₂ with
-      | inl h₂ => simp at h₃; rw[analz_Crypt] at h₃
-                  rcases h₂ with ⟨_, ⟨nonce_eq, _⟩⟩; rw[nonce_eq] at h₃; simp at h₃;
+    | inl h₁ => simp at h₂; cases h₂ with
+      | inl h₂ => rcases h₁ with ⟨h₁, _⟩; rcases h₂ with ⟨h₂, _⟩;
+                  apply injective_publicKey at h₁;
+                  apply injective_publicKey at h₂;
+                  simp_all;
+      | inr h₂ => apply parts.body at h₂; apply parts.fst at h₂; 
+                  apply parts_knows_Spy_subset_used at h₂; simp_all;
+    | inr h₁ => simp at h₂; cases h₂ with
+      | inl h₂ => apply parts.body at h₁; apply parts.fst at h₁;
+                  apply parts_knows_Spy_subset_used at h₁; simp_all;
       | inr => aapply unique_NA_apply_ih;
-  | NS2 => sorry
-  | NS3 => sorry
+  | NS2 _ _ _ a_ih => intro h₁ h₂ h₃; simp_all; apply a_ih
+                      apply Set.notMem_subset at h₃
+                      · apply h₃;
+                      · apply_rules [analz_mono, Set.subset_insert]
+  | NS3 _ _ _ a_ih => intro h₁ h₂ h₃; simp_all; apply a_ih; 
+                      apply Set.notMem_subset at h₃
+                      · apply h₃;
+                      · apply_rules [analz_mono, Set.subset_insert]
 
 -- Spy does not see the nonce sent in NS1 if A and B are secure
-theorem Spy_not_see_NA { h : ns_public evs  }:
+theorem Spy_not_see_NA { h : ns_public evs  }
+  { not_bad_A : A ∉ bad }
+  { not_bad_B : B ∉ bad } :
   Says A B (Crypt (pubEK B) ⦃Nonce NA, Agent A⦄) ∈ evs →
-  A ∉ bad →
-  B ∉ bad →
   Nonce NA ∉ analz (spies evs) := by
-  intro h
-  induction h <;> simp_all [analz_insertI, no_nonce_NS1_NS2]
+  intro h₁ h₄
+  induction h with
+  | Nil => simp_all
+  | Fake _ a a_ih => 
+    apply Fake_analz_insert at a; apply a at h₄; simp_all; cases h₁ with
+    | inl h => rcases h with ⟨l, _⟩; simp_all [Spy_in_bad];
+    | inr h => cases h₄ with
+      | inl h₄ => cases h₄; apply a_ih at h; contradiction;
+      | inr => apply a_ih at h; contradiction;
+  | NS1 _ a a_ih => simp_all; cases h₁ with
+    | inl h => rcases h with ⟨_, ⟨_, ⟨_, ⟨h, _⟩⟩⟩⟩; simp_all; apply a
+               apply parts_knows_Spy_subset_used
+               aapply analz_subset_parts
+    | inr h => apply analz_insert_Crypt_subset at h₄; cases h₄ with
+      | inl h₄ => contradiction; 
+      | inr h₄ => simp at h₄; cases h₄
+                  · simp_all; apply a; apply Says_imp_used at h;
+                    apply used_parts_subset_parts at h; simp at h; apply h;
+                    tauto;
+                  · simp_all;
+  | NS2 _ not_used_NB a a_ih =>
+    cases h₁ with | tail _ b =>
+      have _ := h₄
+      simp at h₄; apply analz_insert_Crypt_subset at h₄
+      simp at h₄; rcases h₄ with ( h | ( h | h ))
+      · simp at a_ih; have c := b; apply a_ih at c; rw[h] at b;
+        have _ := c; rw[h] at c;
+        apply Says_imp_parts_knows_Spy at b
+        apply Says_imp_parts_knows_Spy at a
+        apply unique_NA at b; apply b at a; apply a at c; simp_all
+        assumption
+      · rw [h] at b
+        apply not_used_NB; apply parts_knows_Spy_subset_used; apply parts.fst;
+        apply parts.body; apply Says_imp_parts_knows_Spy; assumption
+      · aapply a_ih
+  | NS3 _ a₁ a₂ a_ih => 
+    cases h₁ with | tail _ b =>
+      have _ := h₄
+      simp at h₄; apply analz_insert_Crypt_subset at h₄
+      simp at h₄; rcases h₄ with ( h | ( h | h ))
+      · have c := b; have d := a₁; have e := a₂
+        rw[h] at b; apply Says_imp_parts_knows_Spy at b
+        apply Says_imp_parts_knows_Spy at a₂
+        aapply a_ih; apply no_nonce_NS1_NS2
+        · assumption
+        · rw[h]; exact a₂
+        · rw[h]; exact b
+      · aapply a_ih; aapply analz.inj
+      · aapply a_ih; aapply analz.fst
+      · aapply a_ih; aapply analz.snd
+      · aapply a_ih; aapply analz.decrypt
+      
+-- Authentication for `A`: if she receives message 2 and has used `NA` to start a run, then `B` has sent message 2.
+theorem A_trusts_NS2 {h : ns_public evs }
+  { not_bad_A : A ∉ bad }
+  { not_bad_B : B ∉ bad } :
+  Says A B (Crypt (pubEK B) ⦃Nonce NA, Agent A⦄) ∈ evs →
+  Says B' A (Crypt (pubEK B) ⦃Nonce NA, Nonce NB, Agent B⦄) ∈ evs →
+  Says B A (Crypt (pubEK B) ⦃Nonce NA, Nonce NB, Agent B⦄) ∈ evs
+:= by
+   intro h₁ h₂;
+   -- have snsNA := h₁; apply Spy_not_see_NA at snsNA <;> try assumption
+   apply Says_imp_parts_knows_Spy at h₂
+   -- use unique_NA to show that B' = B
+   induction h with
+  | Nil => simp_all
+  | Fake _ a a_ih => 
+    have snsNA := h₁; apply Spy_not_see_NA at snsNA <;> try assumption
+    cases h₁
+    · have _ := Spy_in_bad; contradiction
+    · right; simp at h₂; cases h₂ with
+    | inl h₂ => apply Fake_parts_sing at a; apply a at h₂; cases h₂ with
+      | inl h₂ => aapply a_ih; cases h₂ with
+        | inj => aapply analz_subset_parts
+        | crypt h => apply False.elim; apply snsNA
+                     apply MPair_synth_analz.mp at h; rcases h with ⟨⟨l⟩ , _⟩; 
+                     aapply analz_spies_mono
+      | inr h₂ => aapply a_ih;
+    | inr => aapply a_ih;
+    · aapply ns_public.Fake
+  | NS1 _ a a_ih => right; simp at h₂; cases h₁
+                    · apply False.elim; apply a
+                      apply parts_knows_Spy_subset_used; apply parts.fst
+                      aapply parts.body
+                    · aapply a_ih; 
+  | NS2 _ _ a a_ih => simp at h₁; have b := h₁; have snsNA := h₁
+                      apply Spy_not_see_NA at snsNA <;> try assumption
+                      simp at h₂; cases h₂ with
+      | inl h => apply Says_imp_parts_knows_Spy at a
+                 apply Says_imp_parts_knows_Spy at b
+                 rcases h with ⟨e₁ , ⟨e₂ , ⟨e₃, e₄⟩⟩⟩
+                 apply unique_NA at a
+                 rw[e₂] at b; rw[e₂] at snsNA
+                 apply a at b
+                 apply b at snsNA
+                 simp_all[-e₄]; assumption
+      | inr => right; aapply a_ih
+  | NS3 _ _ a a_ih => simp at h₁; simp at h₂; right; aapply a_ih
+  
+-- If the encrypted message appears then it originated with Alice in `NS1`
+lemma B_trusts_NS1 { h : ns_public evs} :
+  Crypt (pubEK B) ⦃Nonce NA, Agent A⦄ ∈ parts (spies evs) →
+  Nonce NA ∉ analz (spies evs) →
+  Says A B (Crypt (pubEK B) ⦃Nonce NA, Agent A⦄) ∈ evs
+:= by
+  intro h₁ h₂
+  induction h with
+  | Nil => simp[spies] at h₁; rw[knows] at h₁; simp[initState] at h₁
+  | Fake _ a a_ih => simp at h₁; cases h₁ with
+    | inl h₁ => apply Fake_parts_sing at a; apply a at h₁; cases h₁ with
+      | inl h₁ => cases h₁ with
+        | inj h₁ => right; apply analz_subset_parts at h₁; aapply a_ih
+                    aapply analz_spies_mono_neg
+        | crypt h₁ => apply False.elim;
+                      apply MPair_synth_analz.mp at h₁
+                      rcases h₁ with ⟨⟨h₁⟩, _⟩; aapply analz_spies_mono_neg
+      | inr => right; aapply a_ih; aapply analz_spies_mono_neg
+    | inr h₁ => right; aapply a_ih; aapply analz_spies_mono_neg
+  | NS1 _ _ a_ih => simp at h₁; cases h₁ with
+    | inl h₁ => rcases h₁ with ⟨e₁, ⟨e₂, e₃⟩⟩; apply injective_publicKey at e₁
+                simp_all
+    | inr => right; aapply a_ih; aapply analz_spies_mono_neg
+  | NS2 _ _ _ a_ih => simp at h₁; right; aapply a_ih; aapply analz_spies_mono_neg
+  | NS3 _ _ _ a_ih => simp at h₁; right; aapply a_ih; aapply analz_spies_mono_neg
+  
+-- Authenticity Properties obtained from `NS2`
+
+-- Helper lemmas for unique_NB
+lemma unique_NB_apply_ih {P : Prop}
+{ a_ih :
+  Crypt (pubEK A) ⦃Nonce NA, ⦃Nonce NB, Msg.Agent B⦄⦄ ∈ parts (spies evsf) →
+    Crypt (pubEK A') ⦃Nonce NA', ⦃Nonce NB, Msg.Agent B'⦄⦄ ∈ parts (spies evsf) →
+      Nonce NB ∉ analz (spies evsf) → P }
+{ h₁ : Crypt (pubEK A) ⦃Nonce NA, ⦃Nonce NB, Msg.Agent B⦄⦄ ∈ parts (spies evsf) }
+{ h₂ : Crypt (pubEK A') ⦃Nonce NA', ⦃Nonce NB, Msg.Agent B'⦄⦄ ∈ parts (spies evsf) }
+{ h₃ : Nonce NB ∉ analz (spies (Says Agent.Spy B X :: evsf)) }
+: P := by
+  aapply unique_Nonce_apply_ih (h₁ := h₁) (h₂ := h₂) (h₃ := h₃)
+
+lemma unique_NB_contradict
+{ h₃ : Nonce NB ∉ analz (spies (Says Agent.Spy B X :: evsf)) }
+{ h₂ : synth (analz (spies evsf)) ⦃Nonce NA', ⦃Nonce NB, Msg.Agent B'⦄⦄ }
+{P : Prop}
+: P := by
+  apply MPair_synth_analz.mp at h₂; apply False.elim
+  rcases h₂ with ⟨_, r⟩; cases r with
+    | inj r => aapply analz_spies_mono_neg; aapply analz.fst
+    | mpair r₁ _ => cases r₁; aapply analz_spies_mono_neg
+
+-- Unicity for `NS2`: nonce `NB` identifies nonce `NA` and agent `A`
+theorem unique_NB { h : ns_public evs } :
+  (Crypt (pubEK A) ⦃Nonce NA, Nonce NB, Agent B⦄ ∈ parts (spies evs) →
+  (Crypt (pubEK A') ⦃Nonce NA', Nonce NB, Agent B'⦄ ∈ parts (spies evs) →
+  (Nonce NB ∉ analz (spies evs) →
+  A = A' ∧ NA = NA' ∧ B = B'))) := by
+  -- Proof closely follows that of unique_NA
+  induction h with
+  | Nil => aesop (add norm spies, norm knows, safe analz_insertI)
+  | Fake _ a a_ih =>
+    apply Fake_parts_sing at a; intro h₁ h₂ h₃;
+    simp[spies, knows] at h₁; cases h₁ with
+    | inl h₁ => apply a at h₁; cases h₁ with
+      | inl h₁ => cases h₁ with
+        | inj h₁ => simp[spies, knows] at h₂; cases h₂ with
+          | inl h₂ => apply a at h₂; cases h₂ with
+            | inl h₂ => cases h₂ with
+              | inj h₂ => apply analz_subset_parts at h₁
+                          apply analz_subset_parts at h₂
+                          aapply unique_NB_apply_ih
+              | crypt h₂ => aapply unique_NB_contradict
+            | inr h₂ => apply analz_subset_parts at h₁
+                        aapply unique_NB_apply_ih
+          | inr h₂ => apply analz_subset_parts at h₁
+                      aapply unique_NB_apply_ih
+        | crypt h₁ => aapply unique_NB_contradict
+      | inr h₁ => simp[spies] at h₁; simp[spies, knows] at h₂; cases h₂ with
+        | inl h₂ => apply a at h₂; cases h₂ with
+          | inl h₂ => cases h₂ with
+            | inj h₂ => apply analz_subset_parts at h₂
+                        aapply unique_NB_apply_ih
+            | crypt => aapply unique_NB_contradict
+          | inr => aapply unique_NB_apply_ih
+        | inr => aapply unique_NB_apply_ih
+    | inr => simp[spies, knows] at h₂; cases h₂ with
+      | inl h₂ => apply a at h₂; cases h₂ with
+        | inl h₂ => cases h₂ with
+          | inj h₂ => apply analz_subset_parts at h₂
+                      aapply unique_NB_apply_ih
+          | crypt => aapply unique_NB_contradict
+        | inr => aapply unique_NB_apply_ih
+      | inr => aapply unique_NB_apply_ih
+  | NS1 _ _ a_ih => intro h₁ h₂ h₃; simp at h₁; simp at h₂; aapply a_ih
+                    aapply analz_spies_mono_neg
+  | NS2 _ _ _ a_ih => intro h₁ h₂ h₃; simp at h₁; simp at h₂; cases h₁ with
+    | inl h₁ => rcases h₁ with ⟨e₁, _⟩; apply injective_publicKey at e₁
+                cases h₂ with
+      | inl h₂ => rcases h₂ with ⟨e₂, _⟩; apply injective_publicKey at e₂
+                  simp_all
+      | inr h₂ => apply parts.body at h₂; apply parts.snd at h₂
+                  apply parts.fst at h₂; apply parts_knows_Spy_subset_used at h₂;
+                  simp_all;
+    | inr h₁ => cases h₂ with
+      | inl h₂ => apply parts.body at h₁; apply parts.snd at h₁
+                  apply parts.fst at h₁; apply parts_knows_Spy_subset_used at h₁;
+                  simp_all
+      | inr => aapply a_ih; aapply analz_spies_mono_neg
+  | NS3 _ _ _ a_ih => intro h₁ h₂ h₃; simp_all; apply a_ih; 
+                      apply Set.notMem_subset at h₃
+                      · apply h₃;
+                      · apply_rules [analz_mono, Set.subset_insert]
+
+-- `NB` remains secret 
+theorem Spy_not_see_NB { h : ns_public evs }
+  { not_bad_A : A ∉ bad }
+  { not_bad_B : B ∉ bad } :
+  Says B A (Crypt (pubEK A) ⦃Nonce NA, Nonce NB, Agent B⦄) ∈ evs →
+  Nonce NB ∉ analz (spies evs)
+:= by
+  intro h₁ h₄
+  induction h with
+  | Nil => simp_all
+  | Fake _ a a_ih => 
+    apply Fake_analz_insert at a; apply a at h₄; simp_all; cases h₁ with
+    | inl h => rcases h with ⟨l, _⟩; simp_all [Spy_in_bad];
+    | inr h => cases h₄ with
+      | inl h₄ => cases h₄; apply a_ih at h; contradiction;
+      | inr => apply a_ih at h; contradiction;
+  | NS1 _ a a_ih => simp at h₁; simp[spies, knows] at h₄
+                    apply analz_insert_Crypt_subset at h₄; simp at h₄
+                    cases h₄ with
+    | inl e => rw[e] at h₁; apply a; apply parts_knows_Spy_subset_used
+               apply parts.fst; apply parts.snd; apply parts.body
+               aapply Says_imp_parts_knows_Spy
+    | inr => aapply a_ih
+  | NS2 _ not_used_NB a a_ih => simp at h₁; simp[spies, knows] at h₄; 
+                                cases h₁ with
+    | inl h => rcases h with ⟨_, ⟨_, ⟨e₃, _⟩⟩⟩; apply injective_publicKey at e₃;
+               simp_all; apply not_used_NB; apply parts_knows_Spy_subset_used;
+               aapply analz_subset_parts
+    | inr h => apply analz_insert_Crypt_subset at h₄; simp at h₄; cases h₄ with
+      | inl e => aapply a_ih; rw[e]; apply Says_imp_parts_knows_Spy at a
+                 apply Says_imp_parts_knows_Spy at h; rw[e] at h
+                 aapply no_nonce_NS1_NS2
+      | inr h => cases h
+                 · simp_all; apply not_used_NB; apply parts_knows_Spy_subset_used
+                   apply parts.fst; apply parts.snd; apply parts.body
+                   aapply Says_imp_parts_knows_Spy
+                 · aapply a_ih
+  | @NS3 evs3 _ B' _ _ _ _ a₁ a₂ a_ih => 
+    cases h₁ with | tail _ b =>
+      simp at h₄; by_cases bad_B' : Key (invKey (pubEK B')) ∈ analz (spies evs3)
+      · have aC := bad_B'; apply analz_subset_parts at bad_B'
+        apply Spy_see_priEK.mp at bad_B'; have c := b; apply a_ih at c; 
+        apply analz_insert_Decrypt at aC; rw[aC] at h₄; simp at h₄; cases h₄ with
+    | inl h₄ =>
+        apply Says_imp_parts_knows_Spy at a₂
+        apply Says_imp_parts_knows_Spy at b; rw[h₄] at b
+        apply unique_NB at a₂; apply a₂ at b;
+        rw[h₄] at c; simp_all; assumption
+    | inr h₄ => aapply a_ih
+      · apply analz_Crypt at aC; rw[aC] at h₄; simp at h₄; aapply a_ih;
+
+-- Authentication for `B`: if he receives message 3 and has used `NB` in message 2, then `A` has sent message 3.
+theorem B_trusts_NS3 { h : ns_public evs } 
+  { not_bad_A : A ∉ bad }
+  { not_bad_B : B ∉ bad } :
+  Says B A (Crypt (pubEK A) ⦃Nonce NA, Nonce NB, Agent B⦄) ∈ evs →
+  Says A' B (Crypt (pubEK B) (Nonce NB)) ∈ evs →
+  Says A B (Crypt (pubEK B) (Nonce NB)) ∈ evs
+:= by 
+  intro h₁ h₂
+  apply Says_imp_parts_knows_Spy at h₂
+  induction h with
+  | Nil => simp_all
+  | Fake _ a a_ih => right; simp at h₁; simp at h₂; cases h₁ with
+    | inl => simp_all[Spy_in_bad]
+    | inr h₁ => cases h₂ with
+      | inl h₂ => apply Fake_parts_sing at a; apply a at h₂; cases h₂ with
+        | inl h₂ => simp at h₂; cases h₂ with
+          | inj => aapply a_ih; aapply analz_subset_parts;
+          | crypt h₂ => cases h₂; apply Spy_not_see_NB at h₁ <;> simp_all
+        | inr => aapply a_ih
+      | inr => aapply a_ih
+  | NS1 _ a a_ih => right; simp at h₂; simp at h₁; aapply a_ih;
+  | NS2 _ _ a a_ih => right; simp at h₁; simp at h₂; cases h₁ with
+    | inl => apply parts.body at h₂; apply parts_knows_Spy_subset_used at h₂
+             simp_all
+    | inr => aapply a_ih
+  | NS3 _ a₁ a₂ a_ih => simp at h₁; simp at h₂; cases h₂ with
+    | inl h₂ => simp_all; left; rcases h₂ with ⟨e₁, _⟩
+                apply injective_publicKey at e₁; simp_all
+                have h₁c := h₁
+                apply Says_imp_parts_knows_Spy at h₁
+                apply Says_imp_parts_knows_Spy at a₂
+                apply unique_NB at h₁; apply h₁ at a₂
+                apply Spy_not_see_NB at h₁c
+                apply a₂ at h₁c
+                all_goals simp_all
+    | inr => right; aapply a_ih
+
+-- Overall guarantee for `B`
   
 -- If NS3 has been sent and the nonce NB agrees with the nonce B joined with NA, then A initiated the run using NA
-theorem B_trusts_protocol { h : ns_public evs  }:
-  A ∉ bad →
-  B ∉ bad →
+theorem B_trusts_protocol { h : ns_public evs  }
+  { not_bad_A : A ∉ bad }
+  { not_bad_B : B ∉ bad } :
   Crypt (pubEK B) (Nonce NB) ∈ parts (spies evs) →
   Says B A (Crypt (pubEK A) ⦃Nonce NA, Nonce NB, Agent B⦄) ∈ evs →
   Says A B (Crypt (pubEK B) ⦃Nonce NA, Agent A⦄) ∈ evs := by
-  intro h
-  induction h <;> simp_all [analz_insertI, no_nonce_NS1_NS2]
+  intro h₁ h₂
+  induction h with
+  | Nil => simp_all
+  | Fake _ a a_ih => right; simp at h₁; simp at h₂; cases h₂ with
+    | inl => simp_all[Spy_in_bad]
+    | inr h₂ => cases h₁ with
+      | inl h₁ => apply Fake_parts_sing at a; apply a at h₁; cases h₁ with
+        | inl h₁ => simp at h₁; cases h₁ with
+          | inj => aapply a_ih; aapply analz_subset_parts
+          | crypt h₁ => cases h₁; apply Spy_not_see_NB at h₂ <;> simp_all
+        | inr => aapply a_ih
+      | inr => aapply a_ih
+  | NS1 _ a a_ih => right; simp at h₂; simp at h₁; aapply a_ih;
+  | NS2 _ _ a a_ih => right; simp at h₁; simp at h₂; cases h₂ with
+    | inl => apply parts.body at h₁; apply parts_knows_Spy_subset_used at h₁
+             simp_all
+    | inr => aapply a_ih
+  | NS3 _ a₁ a₂ a_ih => simp at h₁; simp at h₂; cases h₁ with
+    | inl h₁ => simp_all; rcases h₁ with ⟨e₁, _⟩
+                apply injective_publicKey at e₁; simp_all
+                have h₂c := h₂
+                apply Says_imp_parts_knows_Spy at h₂
+                apply Says_imp_parts_knows_Spy at a₂
+                apply unique_NB at h₂; apply h₂ at a₂
+                apply Spy_not_see_NB at h₂c
+                apply a₂ at h₂c
+                all_goals simp_all
+    | inr => right; aapply a_ih
 
 end NS_Public