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Added expand_parts_element macro
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Further simplified proofs in NS_public
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Your Name
2026-03-04 18:44:21 +01:00
parent 7367681bc6
commit 2c84c4a975
2 changed files with 55 additions and 60 deletions
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InductiveVerification/Message.lean
+11
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@@ -1,5 +1,6 @@
import Init.Data.Nat.Lemmas import Init.Data.Nat.Lemmas
import Init.Prelude import Init.Prelude
import Lean
import Mathlib.Data.Nat.Basic import Mathlib.Data.Nat.Basic
import Mathlib.Data.Nat.Dist import Mathlib.Data.Nat.Dist
import Mathlib.Data.Set.Basic import Mathlib.Data.Set.Basic
@@ -12,6 +13,8 @@ import Mathlib.Order.Lattice
import Mathlib.Tactic.ApplyAt import Mathlib.Tactic.ApplyAt
import Mathlib.Tactic.SimpIntro import Mathlib.Tactic.SimpIntro
import Mathlib.Tactic.NthRewrite import Mathlib.Tactic.NthRewrite
open Lean Elab Command Term Meta
open Parser.Tactic
-- Keys are integers -- Keys are integers
abbrev Key := Nat 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; apply_rules [ parts_subset_iff.mp, Set.singleton_subset_iff.mpr ]
· intro h; aapply parts_subset_iff.mpr; simp · 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] @[simp]
lemma parts_insert_Agent {H : Set Msg} {agt : Agent} : lemma parts_insert_Agent {H : Set Msg} {agt : Agent} :
parts (insert (Agent agt) H) = insert (Agent agt) (parts H) := parts (insert (Agent agt) H) = insert (Agent agt) (parts H) :=
InductiveVerification/NS_Public.lean
+44 -60
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@@ -69,10 +69,10 @@ theorem Spy_see_priEK {h : ns_public evs} :
| Fake _ h ih => | Fake _ h ih =>
apply Fake_parts_sing at 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 Fake_parts_sing_helper (h := h) at h₁
simp at h₁; aapply ih; simp_all
| NS1 _ _ ih => simp; assumption | NS1 => simp_all
| NS2 _ _ _ ih => simp; assumption | NS2 => simp_all
| NS3 _ _ _ ih => simp; assumption | NS3 => simp_all
· intro h₁; apply parts_increasing; aapply Spy_spies_bad_privateKey · intro h₁; apply parts_increasing; aapply Spy_spies_bad_privateKey
@[simp] @[simp]
@@ -89,7 +89,7 @@ theorem no_nonce_NS1_NS2 { evs: List Event} { h : ns_public evs } :
Nonce NA ∈ analz (spies evs))) := by Nonce NA ∈ analz (spies evs))) := by
intro h₁ h₂ intro h₁ h₂
induction h with induction h with
| Nil => rw[spies, knows] at h₂; simp[initState] at h₂ | Nil => simp[spies, knows] at h₂
| Fake _ h ih => | Fake _ h ih =>
simp; apply analz_insert; right simp; apply analz_insert; right
apply Fake_parts_sing at h apply Fake_parts_sing at h
@@ -101,14 +101,14 @@ theorem no_nonce_NS1_NS2 { evs: List Event} { h : ns_public evs } :
all_goals (aapply ih <;> aapply analz_subset_parts) all_goals (aapply ih <;> aapply analz_subset_parts)
| NS1 _ nonce_not_used => | NS1 _ nonce_not_used =>
apply parts_knows_Spy_subset_used_neg at nonce_not_used; apply parts_knows_Spy_subset_used_neg at nonce_not_used;
simp[spies] at h₁; rw[parts_element, Set.subset_def] at h₁; simp at h₁; simp[spies] at h₁; expand_parts_element at h₁;
simp[spies] at h₂; rw[parts_element, Set.subset_def] at h₂; simp at h₂; simp[spies] at h₂; expand_parts_element at h₂;
apply analz_mono; apply Set.subset_insert apply analz_mono; apply Set.subset_insert
cases h₂ <;> simp_all cases h₂ <;> simp_all
| NS2 _ nonce_not_used => | NS2 _ nonce_not_used =>
apply parts_knows_Spy_subset_used_neg at nonce_not_used; apply parts_knows_Spy_subset_used_neg at nonce_not_used;
simp[spies] at h₁; rw[parts_element, Set.subset_def] at h₁; simp at h₁; simp[spies] at h₁; expand_parts_element at h₁;
simp[spies] at h₂; rw[parts_element, Set.subset_def] at h₂; simp at h₂; simp[spies] at h₂; expand_parts_element at h₂;
apply analz_mono; apply Set.subset_insert apply analz_mono; apply Set.subset_insert
cases h₁ <;> simp_all cases h₁ <;> simp_all
| NS3 _ _ _ a_ih => simp at h₁; simp at h₂; apply analz_mono | NS3 _ _ _ a_ih => simp at h₁; simp at h₂; apply analz_mono
@@ -135,11 +135,9 @@ theorem unique_NA { h : ns_public evs } :
| Nil => aesop (add norm spies, norm knows, safe analz_insertI) | Nil => aesop (add norm spies, norm knows, safe analz_insertI)
| Fake _ a a_ih => | Fake _ a a_ih =>
apply Fake_parts_sing at a; intro h₁ h₂ h₃; apply Fake_parts_sing at a; intro h₁ h₂ h₃;
simp[spies, knows] at h₁; apply Fake_parts_sing_helper (h := a) at h₁ simp at h₁; apply Fake_parts_sing_helper (h := a) at h₁; simp at h₁
simp at h₁ simp at h₂; apply Fake_parts_sing_helper (h := a) at h₂; simp at h₂
simp[spies, knows] at h₂; apply Fake_parts_sing_helper (h := a) at h₂ simp at h₃;
simp at h₂
simp[spies, knows] at h₃;
rcases h₁ with ((_ | _) | _) <;> rcases h₁ with ((_ | _) | _) <;>
rcases h₂ with ((_ | _) | _) <;> rcases h₂ with ((_ | _) | _) <;>
try ( try (
@@ -152,11 +150,12 @@ theorem unique_NA { h : ns_public evs } :
)) ))
| NS1 _ nonce_not_used a_ih => | NS1 _ nonce_not_used a_ih =>
intro h₁ h₂ h₃ intro h₁ h₂ h₃
simp at h₁; rw[parts_element, Set.subset_def] at h₁; simp at h₁ simp at h₁; expand_parts_element at h₁
simp at h₂; rw[parts_element, Set.subset_def] at h₂; simp at h₂ simp at h₂; expand_parts_element at h₂
apply parts_knows_Spy_subset_used_neg at nonce_not_used apply parts_knows_Spy_subset_used_neg at nonce_not_used
cases h₁ <;> cases h₂ <;> simp_all cases h₁ <;> cases h₂ <;> simp_all
aapply a_ih; intro h; apply h₃; apply_rules[analz_mono, Set.subset_insert] -- TODO make an analz_mono_neg lemma for these cases
aapply a_ih; intro _; apply h₃; apply_rules[analz_mono, Set.subset_insert]
| NS2 _ _ _ a_ih => intro h₁ h₂ h₃; simp_all; apply a_ih; intro h; apply h₃ | NS2 _ _ _ a_ih => intro h₁ h₂ h₃; simp_all; apply a_ih; intro h; apply h₃
apply_rules [analz_mono, Set.subset_insert] apply_rules [analz_mono, Set.subset_insert]
| NS3 _ _ _ a_ih => intro h₁ h₂ h₃; simp at h₁; simp at h₂; aapply a_ih | NS3 _ _ _ a_ih => intro h₁ h₂ h₃; simp at h₁; simp at h₂; aapply a_ih
@@ -199,7 +198,7 @@ theorem Spy_not_see_NA { h : ns_public evs }
cases h₁ with | tail _ b => cases h₁ with | tail _ b =>
have _ := h₄ have _ := h₄
simp at h₄; apply analz_insert_Crypt_subset at h₄ simp at h₄; apply analz_insert_Crypt_subset at h₄
simp at h₄; rcases h₄ with ( h | h | h) simp at h₄; rcases h₄ with ( h | h )
· have _ := b; have _ := a₁; have _ := a₂ · have _ := b; have _ := a₁; have _ := a₂
rw[h] at b; apply Says_imp_parts_knows_Spy at b rw[h] at b; apply Says_imp_parts_knows_Spy at b
apply Says_imp_parts_knows_Spy at a₂ apply Says_imp_parts_knows_Spy at a₂
@@ -207,10 +206,7 @@ theorem Spy_not_see_NA { h : ns_public evs }
· assumption · assumption
· rw[h]; exact a₂ · rw[h]; exact a₂
· rw[h]; exact b · rw[h]; exact b
· aapply a_ih; aapply analz.inj · aapply a_ih
· 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. -- 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 } theorem A_trusts_NS2 {h : ns_public evs }
@@ -236,23 +232,16 @@ theorem A_trusts_NS2 {h : ns_public evs }
· aapply analz_subset_parts · aapply analz_subset_parts
· apply False.elim; apply snsNA; apply analz_spies_mono; tauto; · apply False.elim; apply snsNA; apply analz_spies_mono; tauto;
· aapply ns_public.Fake · aapply ns_public.Fake
| NS1 _ a a_ih => right; simp at h₂; cases h₁ | NS1 _ a a_ih =>
· apply False.elim; apply a simp at h₂; expand_parts_element at h₂
apply parts_knows_Spy_subset_used; apply parts.fst apply parts_knows_Spy_subset_used_neg at a; cases h₁ <;> simp_all
aapply parts.body aapply a_ih
· aapply a_ih;
| NS2 _ _ a a_ih => | NS2 _ _ a a_ih =>
simp at h₁; have b := h₁; have snsNA := h₁ simp at h₁; have b := h₁; have snsNA := h₁
apply Spy_not_see_NA at snsNA <;> try assumption apply Spy_not_see_NA at snsNA <;> try assumption
simp at h₂; rcases h₂ with (⟨_ , e₂ , _, e₄⟩ | _) simp at h₂; cases h₂ <;> simp_all
· apply Says_imp_parts_knows_Spy at a apply Says_imp_parts_knows_Spy at a; apply unique_NA at a;
apply Says_imp_parts_knows_Spy at b apply Says_imp_parts_knows_Spy at h₁; apply a at h₁; all_goals simp_all
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
· right; aapply a_ih
| NS3 _ _ a a_ih => simp at h₁; simp at h₂; 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` -- If the encrypted message appears then it originated with Alice in `NS1`
@@ -263,7 +252,7 @@ lemma B_trusts_NS1 { h : ns_public evs} :
:= by := by
intro h₁ h₂ intro h₁ h₂
induction h with induction h with
| Nil => simp[spies] at h₁; rw[knows] at h₁; simp[initState] at h₁ | Nil => simp[spies, knows] at h₁
| Fake _ a a_ih => | Fake _ a a_ih =>
simp at h₁; apply Fake_parts_sing at a; simp at h₁; apply Fake_parts_sing at a;
apply Fake_parts_sing_helper (h := a) at h₁; simp at h₁ apply Fake_parts_sing_helper (h := a) at h₁; simp at h₁
@@ -289,25 +278,20 @@ theorem unique_NB { h : ns_public evs } :
induction h with induction h with
| Nil => aesop (add norm spies, norm knows, safe analz_insertI) | Nil => aesop (add norm spies, norm knows, safe analz_insertI)
| Fake _ a a_ih => | Fake _ a a_ih =>
intro h₁ h₂ h₃; apply Fake_parts_sing at a; intro h₁ h₂ h₃;
apply Fake_parts_sing at a simp at h₁; apply Fake_parts_sing_helper (h := a) at h₁; simp at h₁;
simp[spies, knows] at h₁; apply Fake_parts_sing_helper (h := a) at h₁
simp at h₁;
simp at h₂; apply Fake_parts_sing_helper (h := a) at h₂; simp at h₂; simp at h₂; apply Fake_parts_sing_helper (h := a) at h₂; simp at h₂;
apply analz_spies_mono_neg at h₃ apply analz_spies_mono_neg at h₃
rcases h₁ with ((h₁ | h₁) | h₁) <;> rcases h₁ with ((_ | _) | _) <;>
rcases h₂ with ((h₂ | h₂) | h₂) <;> rcases h₂ with ((_ | _) | _) <;>
simp_all simp_all
all_goals (aapply a_ih; repeat aapply analz_subset_parts) all_goals (aapply a_ih; repeat aapply analz_subset_parts)
| NS1 _ _ a_ih => intro h₁ h₂ h₃; simp at h₁; simp at h₂; aapply a_ih | NS1 _ _ a_ih => intro h₁ h₂ h₃; simp at h₁; simp at h₂; aapply a_ih
aapply analz_spies_mono_neg aapply analz_spies_mono_neg
| NS2 _ nonce_not_used _ a_ih => | NS2 _ nonce_not_used _ a_ih =>
intro h₁ h₂ h₃; intro h₁ h₂ h₃;
-- This is how to rewrite `M ∈ parts` terms into something useful simp at h₁; expand_parts_element at h₁
-- TODO create a macro for this simp at h₂; expand_parts_element at h₂
-- TODO this should work with analz as well
simp at h₁; rw[parts_element, Set.subset_def] at h₁; simp at h₁
simp at h₂; rw[parts_element, Set.subset_def] at h₂; simp at h₂
apply analz_spies_mono_neg at h₃; apply analz_spies_mono_neg at h₃;
apply parts_knows_Spy_subset_used_neg at nonce_not_used apply parts_knows_Spy_subset_used_neg at nonce_not_used
rcases h₁ with (_ | h₁) <;> rcases h₁ with (_ | h₁) <;>
@@ -333,7 +317,7 @@ theorem Spy_not_see_NB { h : ns_public evs }
apply parts_knows_Spy_subset_used_neg at nonce_not_used apply parts_knows_Spy_subset_used_neg at nonce_not_used
cases h₄ with cases h₄ with
| inl e => apply Says_imp_parts_knows_Spy at h₁; | inl e => apply Says_imp_parts_knows_Spy at h₁;
rw[parts_element, Set.subset_def] at h₁; simp_all expand_parts_element at h₁; simp_all
| inr => aapply a_ih | inr => aapply a_ih
| NS2 _ not_used_NB a a_ih => | NS2 _ not_used_NB a a_ih =>
simp at h₁; simp at h₁;
@@ -345,7 +329,7 @@ theorem Spy_not_see_NB { h : ns_public evs }
· aapply a_ih; apply Says_imp_parts_knows_Spy at a; · aapply a_ih; apply Says_imp_parts_knows_Spy at a;
apply Says_imp_parts_knows_Spy at h₁; simp_all; aapply no_nonce_NS1_NS2 apply Says_imp_parts_knows_Spy at h₁; simp_all; aapply no_nonce_NS1_NS2
· apply Says_imp_parts_knows_Spy at h₁; · apply Says_imp_parts_knows_Spy at h₁;
rw[parts_element, Set.subset_def] at h₁; simp_all expand_parts_element at h₁; simp_all
· aapply a_ih · aapply a_ih
| NS3 _ _ a a_ih => | NS3 _ _ a a_ih =>
simp at h₁; simp[analz_insert_Crypt_element] at h₄; simp at h₁; simp[analz_insert_Crypt_element] at h₄;
@@ -370,7 +354,7 @@ theorem B_trusts_NS3 { h : ns_public evs }
right; simp at h₁ right; simp at h₁
apply Fake_parts_sing at a apply Fake_parts_sing at a
simp at h₂; apply Fake_parts_sing_helper (h := a) at h₂; simp at h₂ simp at h₂; apply Fake_parts_sing_helper (h := a) at h₂; simp at h₂
rw[parts_element, Set.subset_def] at h₂; simp at h₂ expand_parts_element at h₂;
have _ := Spy_in_bad have _ := Spy_in_bad
rcases h₁ with (h₁ | h₁) <;> rcases h₂ with ((h₂ | h₂) | h₂) <;> simp_all rcases h₁ with (h₁ | h₁) <;> rcases h₂ with ((h₂ | h₂) | h₂) <;> simp_all
· aapply a_ih; aapply analz_subset_parts · aapply a_ih; aapply analz_subset_parts
@@ -380,11 +364,11 @@ theorem B_trusts_NS3 { h : ns_public evs }
| NS2 _ nonce_not_used a a_ih => | NS2 _ nonce_not_used a a_ih =>
right right
apply parts_knows_Spy_subset_used_neg at nonce_not_used; apply parts_knows_Spy_subset_used_neg at nonce_not_used;
simp at h₂; rw[parts_element, Set.subset_def] at h₂; simp at h₂ simp at h₂; expand_parts_element at h₂;
simp at h₁; cases h₁ <;> simp_all; aapply a_ih simp at h₁; cases h₁ <;> simp_all; aapply a_ih
| NS3 _ _ a₂ a_ih => | NS3 _ _ a₂ a_ih =>
simp at h₁ simp at h₁
simp at h₂; rw[parts_element, Set.subset_def] at h₂; simp at h₂ simp at h₂; expand_parts_element at h₂;
cases h₂ <;> simp_all cases h₂ <;> simp_all
have h₁c := h₁ have h₁c := h₁
apply Spy_not_see_NB at h₁c apply Spy_not_see_NB at h₁c
@@ -409,21 +393,21 @@ theorem B_trusts_protocol { h : ns_public evs }
right right
apply Fake_parts_sing at a apply Fake_parts_sing at a
simp at h₁; apply Fake_parts_sing_helper (h := a) at h₁; simp at h₁; apply Fake_parts_sing_helper (h := a) at h₁;
rw[parts_element, Set.subset_def] at h₁; simp at h₁ expand_parts_element at h₁
have _ := Spy_in_bad have _ := Spy_in_bad
simp at h₂; rcases h₂ with (_ | h₂) <;> simp_all simp at h₂; rcases h₂ with (_ | h₂) <;> simp_all
rcases h₁ with (((_ |_ ) | _) | _) <;> try (aapply a_ih) rcases h₁ with (((_ |_ ) | _) | _) <;> try (aapply a_ih)
· aapply analz_subset_parts · aapply analz_subset_parts
· apply Spy_not_see_NB at h₂ <;> simp_all · apply Spy_not_see_NB at h₂ <;> simp_all
· simp_all · simp_all
| NS1 _ a a_ih => right; simp at h₂; simp at h₁; aapply a_ih; | NS1 _ a a_ih => simp_all
| NS2 _ _ a a_ih => right; simp at h₁; simp at h₂; cases h₂ with | NS2 _ nonce_not_used a a_ih =>
| inl => apply parts.body at h₁; apply parts_knows_Spy_subset_used at h₁ simp at h₁; simp at h₂;
simp_all apply parts_knows_Spy_subset_used_neg at nonce_not_used;
| inr => aapply a_ih expand_parts_element at h₁; cases h₂ <;> simp_all
| NS3 _ _ a₂ a_ih => | NS3 _ _ a₂ a_ih =>
simp at h₂ simp at h₂
simp at h₁; rw[parts_element, Set.subset_def] at h₁; simp at h₁ simp at h₁; expand_parts_element at h₁
cases h₁ <;> simp_all cases h₁ <;> simp_all
have h₂c := h₂ have h₂c := h₂
apply Spy_not_see_NB at h₂c apply Spy_not_see_NB at h₂c