Theorem: Groth16TypeIII.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
r : Fin n_wit → F
⊢ soundness F Groth16TypeIII (Fin n_wit → F)
    (fun stmt wit =>
      let t := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i));
      (((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) *
                ((List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)).sum +
                  (List.map (fun i => Polynomial.C (wit i) * v_wit i) (List.finRange n_wit)).sum) -
              ((List.map (fun i => Polynomial.C (stmt i) * w_stmt i) (List.finRange n_stmt)).sum +
                (List.map (fun i => Polynomial.C (wit i) * w_wit i) (List.finRange n_wit)).sum)).modByMonic
          t =
        0)
    fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q 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
r : Fin n_wit → F
⊢ ∀ (stmt : Groth16TypeIII.Stmt) (prover : Prover F Groth16TypeIII),
    ((∀ (check_idx : Groth16TypeIII.EqualityChecks),
          (List.map
                (fun pairing =>
                  ((List.map
                          (fun pf_elem =>
                            C (Groth16TypeIII.verificationPairingProof_G1 stmt check_idx pairing pf_elem) *
                              (List.map
                                  (fun SRS_elem =>
                                    C (prover.1 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                                  Groth16TypeIII.ListSRSElements_G1).sum)
                          Groth16TypeIII.ListProof_G1).sum +
                      (List.map
                          (fun SRS_elem =>
                            C (Groth16TypeIII.verificationPairingSRS_G1 stmt check_idx pairing SRS_elem) *
                              Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                          Groth16TypeIII.ListSRSElements_G1).sum) *
                    ((List.map
                          (fun pf_elem =>
                            C (Groth16TypeIII.verificationPairingProof_G2 stmt check_idx pairing pf_elem) *
                              (List.map
                                  (fun SRS_elem =>
                                    C (prover.2 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                                  Groth16TypeIII.ListSRSElements_G2).sum)
                          Groth16TypeIII.ListProof_G2).sum +
                      (List.map
                          (fun SRS_elem =>
                            C (Groth16TypeIII.verificationPairingSRS_G2 stmt check_idx pairing SRS_elem) *
                              Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                          Groth16TypeIII.ListSRSElements_G2).sum))
                (Groth16TypeIII.ListPairings check_idx)).sum =
            0) ∧
        ∀ pfs ∈ Groth16TypeIII.Identified_Proof_Elems,
          (List.map (fun SRS_elem => C (prover.1 pfs.1 SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                Groth16TypeIII.ListSRSElements_G1).sum =
            (List.map (fun SRS_elem => C (prover.2 pfs.2 SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                Groth16TypeIII.ListSRSElements_G2).sum) →
      (fun stmt wit =>
          let t := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i));
          (((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) *
                    ((List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)).sum +
                      (List.map (fun i => Polynomial.C (wit i) * v_wit i) (List.finRange n_wit)).sum) -
                  ((List.map (fun i => Polynomial.C (stmt i) * w_stmt i) (List.finRange n_stmt)).sum +
                    (List.map (fun i => Polynomial.C (wit i) * w_wit i) (List.finRange n_wit)).sum)).modByMonic
              t =
            0)
        stmt ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover)