Theorem: BabySNARK.is_sound
Proof state before:
F : Type
inst✝ : Field F
n_stmt n_wit n_var : ℕ
u_stmt : Fin n_stmt → Polynomial F
u_wit : Fin n_wit → Polynomial F
v_stmt : Fin n_stmt → Polynomial F
v_wit : Fin n_wit → Polynomial F
w_stmt : Fin n_stmt → Polynomial F
w_wit : Fin n_wit → Polynomial F
t : Polynomial F
ht : (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum = t
ht0 : t.Monic
⊢ soundness F BabySNARK (Fin n_wit → F)
    (fun stmt wit =>
      (((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
                  (List.map (fun i => Polynomial.C (wit i) * u_wit i) (List.finRange n_wit)).sum) ^
                2 -
              1).modByMonic
          t =
        0)
    fun prover i => prover.1 Proof_Idx.B (SRS_Elements_Idx.βu i)
Tactic:
unfold soundness verify check_poly pairing_poly proof_element_G1_as_poly proof_element_G2_as_poly
Proof state after:
F : Type
inst✝ : Field F
n_stmt n_wit n_var : ℕ
u_stmt : Fin n_stmt → Polynomial F
u_wit : Fin n_wit → Polynomial F
v_stmt : Fin n_stmt → Polynomial F
v_wit : Fin n_wit → Polynomial F
w_stmt : Fin n_stmt → Polynomial F
w_wit : Fin n_wit → Polynomial F
t : Polynomial F
ht : (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum = t
ht0 : t.Monic
⊢ ∀ (stmt : BabySNARK.Stmt) (prover : Prover F BabySNARK),
    ((∀ (check_idx : BabySNARK.EqualityChecks),
          (List.map
                (fun pairing =>
                  ((List.map
                          (fun pf_elem =>
                            C (BabySNARK.verificationPairingProof_G1 stmt check_idx pairing pf_elem) *
                              (List.map
                                  (fun SRS_elem =>
                                    C (prover.1 pf_elem SRS_elem) * BabySNARK.SRSElementValue_G1 SRS_elem)
                                  BabySNARK.ListSRSElements_G1).sum)
                          BabySNARK.ListProof_G1).sum +
                      (List.map
                          (fun SRS_elem =>
                            C (BabySNARK.verificationPairingSRS_G1 stmt check_idx pairing SRS_elem) *
                              BabySNARK.SRSElementValue_G1 SRS_elem)
                          BabySNARK.ListSRSElements_G1).sum) *
                    ((List.map
                          (fun pf_elem =>
                            C (BabySNARK.verificationPairingProof_G2 stmt check_idx pairing pf_elem) *
                              (List.map
                                  (fun SRS_elem =>
                                    C (prover.2 pf_elem SRS_elem) * BabySNARK.SRSElementValue_G2 SRS_elem)
                                  BabySNARK.ListSRSElements_G2).sum)
                          BabySNARK.ListProof_G2).sum +
                      (List.map
                          (fun SRS_elem =>
                            C (BabySNARK.verificationPairingSRS_G2 stmt check_idx pairing SRS_elem) *
                              BabySNARK.SRSElementValue_G2 SRS_elem)
                          BabySNARK.ListSRSElements_G2).sum))
                (BabySNARK.ListPairings check_idx)).sum =
            0) ∧
        ∀ pfs ∈ BabySNARK.Identified_Proof_Elems,
          (List.map (fun SRS_elem => C (prover.1 pfs.1 SRS_elem) * BabySNARK.SRSElementValue_G1 SRS_elem)
                BabySNARK.ListSRSElements_G1).sum =
            (List.map (fun SRS_elem => C (prover.2 pfs.2 SRS_elem) * BabySNARK.SRSElementValue_G2 SRS_elem)
                BabySNARK.ListSRSElements_G2).sum) →
      (fun stmt wit =>
          (((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
                      (List.map (fun i => Polynomial.C (wit i) * u_wit i) (List.finRange n_wit)).sum) ^
                    2 -
                  1).modByMonic
              t =
            0)
        stmt ((fun prover i => prover.1 Proof_Idx.B (SRS_Elements_Idx.βu i)) prover)