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Proof for soundness #24

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z-tech merged 1 commit into from
May 31, 2026
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Proof for soundness #24

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60 changes: 18 additions & 42 deletions SumcheckProtocol/Properties/Theorems/Soundness.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -24,54 +24,30 @@ theorem soundness {𝔽 : Type _} {n : ℕ} [Field 𝔽] [Fintype 𝔽] [Decidab
probOverChallenges (E := AcceptsAndBadTranscriptOnChallenges ⟨n, Nat.lt_succ_self n⟩ st P)
≤ soundnessError st.polynomial := by
classical

-- Keep AcceptsAndBad in the per-round event.
let E : Fin n → (Fin n → 𝔽) → Prop :=
fun i r =>
AcceptsAndBadTranscriptOnChallenges ⟨n, Nat.lt_succ_self n⟩ st P r ∧
RoundDisagreeButAgreeAtChallenge ⟨n, Nat.lt_succ_self n⟩ st P r i

-- Step 1: Accepts∧Bad implies ∃ i, (Accepts∧Bad ∧ RoundDisagreeButAgreeAtChallenge i).
let E : Fin n → (Fin n → 𝔽) → Prop := fun i r =>
AcceptsAndBadTranscriptOnChallenges ⟨n, Nat.lt_succ_self n⟩ st P r ∧
RoundDisagreeButAgreeAtChallenge ⟨n, Nat.lt_succ_self n⟩ st P r i
have hImp :
∀ r : (Fin n → 𝔽),
∀ r,
AcceptsAndBadTranscriptOnChallenges ⟨n, Nat.lt_succ_self n⟩ st P r →
∃ i : Fin n, E i r := by
∃ i, E i r := by
intro r hAB
rcases
accepts_and_bad_implies_exists_round_disagree_but_agree
(st := st) (P := P) (r := r) hAB
with ⟨i, hi⟩
exact ⟨i, ⟨hAB, hi⟩⟩

have hmono :
probOverChallenges (𝔽 := 𝔽) (n := n)
(fun r => AcceptsAndBadTranscriptOnChallenges ⟨n, Nat.lt_succ_self n⟩ st P r)
≤
probOverChallenges (𝔽 := 𝔽) (n := n)
(fun r => ∃ i : Fin n, E i r) :=
prob_over_challenges_mono (𝔽 := 𝔽) (n := n) hImp

-- Step 2: union bound over i.
have hunion :
probOverChallenges (𝔽 := 𝔽) (n := n)
(fun r => ∃ i : Fin n, E i r)
≤
(∑ i : Fin n,
probOverChallenges (𝔽 := 𝔽) (n := n)
(fun r => E i r)) :=
rcases accepts_and_bad_implies_exists_round_disagree_but_agree
(st := st) (P := P) (r := r) hAB with ⟨i, hi⟩
exact ⟨i, hAB, hi⟩
have hmono :=
prob_over_challenges_mono
(E := AcceptsAndBadTranscriptOnChallenges ⟨n, Nat.lt_succ_self n⟩ st P)
(F := fun r => ∃ i, E i r)
hImp
have hunion :=
prob_over_challenges_exists_le_sum (𝔽 := 𝔽) (n := n) E

-- Step 3: use the (now-lemma) sumcheck-specific bound.
have hround :
(∑ i : Fin n,
probOverChallenges (𝔽 := 𝔽) (n := n) (fun r => E i r))
≤ soundnessError st.polynomial := by
simpa [E, soundnessError] using
sum_accepts_and_round_disagree_but_agree_bound
(st := st) (P := P)

have hround := by
simpa only [E, soundnessError] using
sum_accepts_and_round_disagree_but_agree_bound (st := st) (P := P)
exact le_trans (le_trans hmono hunion) hround


-- Prob verifier accepts transcript when claim is not honest claim
theorem soundness_dishonest {𝔽 : Type _} {n : ℕ} [Field 𝔽] [Fintype 𝔽] [DecidableEq 𝔽]
(st : SumcheckProtocolStatement 𝔽 n)
Expand Down
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