Proved basic properties of Blanchet protocol

InductiveVerification/NS_Public.lean

@@ -5,6 +5,7 @@ namespace NS_Public
 
 variable [InvKey]
 variable [Bad]
+variable [AgentKeys]
 open Msg
 open Event
 open Bad
@@ -52,10 +53,10 @@ theorem Spy_see_priEK {h : ns_public evs} :
   (Key (priEK A) ∈ parts (spies evs)) ↔ A ∈ bad := by
   constructor
   · induction h with
-    | Nil => simp [ priEK ]
+    | Nil => simp [ priEK, initState ]
     | Fake _ h =>
-      apply Fake_parts_sing at h
-      intro h₁; simp at h₁; apply Fake_parts_sing_helper (h := h) at h₁
+      intro h₁; simp at h₁
+      apply Or.imp_left (f := Fake_parts_sing (h := h)) at h₁
       simp_all
     | NS1 => simp_all
     | NS2 => simp_all
@@ -76,12 +77,11 @@ theorem no_nonce_NS1_NS2 { evs: List Event} { h : ns_public evs  } :
   Nonce NA ∈ analz (spies evs))) := by
   intro h₁ h₂
   induction h with
-  | Nil => simp at h₂
+  | Nil => simp [ initState ] at h₂
   | Fake _ h =>
     simp [*] at *
-    apply Fake_parts_sing at h
-    apply Fake_parts_sing_helper (h := h) at h₁
-    apply Fake_parts_sing_helper (h := h) at h₂
+    apply Or.imp_left (f := Fake_parts_sing (h := h)) at h₁
+    apply Or.imp_left (f := Fake_parts_sing (h := h)) at h₂
     simp_all; grind
   | NS1 =>
     simp [*] at *
@@ -102,13 +102,12 @@ theorem unique_NA { h : ns_public evs } :
   A = A' ∧ B = B'))) := by
   intro h₁ h₂ h₃
   induction h with
-  | Nil => simp_all
+  | Nil => simp_all [ initState ]
   | Fake _ a a_ih =>
-    apply Fake_parts_sing at a;
-    apply analz_spies_mono_neg at h₃;
+    apply mt (h₁ := analz_spies_mono) at h₃;
     simp [*] at *
-    apply Fake_parts_sing_helper (h := a) at h₁
-    apply Fake_parts_sing_helper (h := a) at h₂
+    apply Or.imp_left (f := Fake_parts_sing (h := a)) at h₁
+    apply Or.imp_left (f := Fake_parts_sing (h := a)) at h₂
     simp_all
   | NS1 =>
     simp [*] at *; expand_parts_element at h₁; expand_parts_element at h₂; grind
@@ -160,13 +159,11 @@ theorem A_trusts_NS2 {h : ns_public evs }
   | Nil => simp_all
   | Fake _ a =>
     have snsNA := h₁; apply Spy_not_see_NA at snsNA <;> try assumption
-    apply analz_spies_mono_neg at snsNA
+    apply mt (h₁ := analz_spies_mono) at snsNA;
     simp [*] at *
     cases h₁
     · simp_all
-    · apply Fake_parts_sing at a;
-      apply Fake_parts_sing_helper (h := a) at h₂; simp at h₂
-      grind
+    · apply Or.imp_left (f := Fake_parts_sing (h := a)) at h₂; simp at h₂; grind
     · aapply ns_public.Fake
   | NS1 => simp [*] at *; expand_parts_element at h₂; grind
   | NS2 => simp [*] at *; grind
@@ -180,14 +177,13 @@ lemma B_trusts_NS1 { h : ns_public evs} :
 := by
   intro h₁ h₂
   induction h with
-  | Nil => simp at h₁
+  | Nil => simp [ initState ] at h₁
   | Fake _ a =>
-    apply analz_spies_mono_neg at h₂
-    simp at h₁; apply Fake_parts_sing at a;
-    apply Fake_parts_sing_helper (h := a) at h₁; simp_all
-  | NS1 => apply analz_spies_mono_neg at h₂; simp_all; grind
-  | NS2 => apply analz_spies_mono_neg at h₂; simp_all;
-  | NS3 => apply analz_spies_mono_neg at h₂; simp_all;
+    apply mt (h₁ := analz_spies_mono) at h₂
+    simp at h₁; apply Or.imp_left (f := Fake_parts_sing (h := a)) at h₁; simp_all
+  | NS1 => apply mt (h₁ := analz_spies_mono) at h₂; simp_all; grind
+  | NS2 => apply mt (h₁ := analz_spies_mono) at h₂; simp_all;
+  | NS3 => apply mt (h₁ := analz_spies_mono) at h₂; simp_all;
 
 -- Authenticity Properties obtained from `NS2`
 
@@ -201,14 +197,15 @@ theorem unique_NB { h : ns_public evs } :
   -- Proof closely follows that of unique_NA
   intro h₁ h₂ h₃
   induction h with
-  | Nil => aesop (add safe analz_insertI)
+  | Nil => simp_all [ initState ]
   | Fake _ a =>
-    apply Fake_parts_sing at a; simp [*] at *
-    apply Fake_parts_sing_helper (h := a) at h₁;
-    apply Fake_parts_sing_helper (h := a) at h₂; simp [*] at *
-    apply analz_insert_mono_neg at h₃
+    simp [*] at *
+    apply Or.imp_left (f := Fake_parts_sing (h := a)) at h₁;
+    apply Or.imp_left (f := Fake_parts_sing (h := a)) at h₂
+    simp [*] at *
+    apply mt (h₁ := analz_insert_mono) at h₃
     grind
-  | NS1 => apply analz_spies_mono_neg at h₃; simp_all
+  | NS1 => apply mt (h₁ := analz_spies_mono) at h₃; simp_all
   | NS2 =>
     simp [*] at *; expand_parts_element at h₁; expand_parts_element at h₂; grind
   | NS3 => simp_all; grind
@@ -255,8 +252,7 @@ theorem B_trusts_NS3 { h : ns_public evs }
   | Nil => simp_all
   | Fake _ a =>
     simp [*] at *
-    apply Fake_parts_sing at a
-    apply Fake_parts_sing_helper (h := a) at h₂; simp at h₂
+    apply Or.imp_left (f := Fake_parts_sing (h := a)) at h₂; simp at h₂
     grind
   | NS1 => simp_all
   | NS2 => simp [*] at *; expand_parts_element at h₂; grind
@@ -277,8 +273,8 @@ theorem B_trusts_protocol { h : ns_public evs  }
   | Nil => simp_all
   | Fake _ a =>
     simp [*] at *
-    apply Fake_parts_sing at a
-    apply Fake_parts_sing_helper (h := a) at h₁; expand_parts_element at h₁
+    apply Or.imp_left (f := Fake_parts_sing (h := a)) at h₁
+    expand_parts_element at h₁
     grind
   | NS1 => simp_all
   | NS2 => simp [*] at *; expand_parts_element at h₁; grind