Theorem: Groth16TypeIII.is_sound
Proof state before:
case intro
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
eqns :
  ∀ (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
null :
  ∀ 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
t : Polynomial F := ∏ 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 ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover 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 ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover 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 ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover i) * w_wit i)
                (List.finRange n_wit)).sum)).modByMonic
      t =
    0
Tactic:
have eqn := eqns ()
Proof state after:
case intro
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
eqns :
  ∀ (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
null :
  ∀ 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
t : Polynomial F := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))
eqn :
  (List.map
        (fun pairing =>
          ((List.map
                  (fun pf_elem =>
                    C (Groth16TypeIII.verificationPairingProof_G1 stmt () 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 () pairing SRS_elem) *
                      Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                  Groth16TypeIII.ListSRSElements_G1).sum) *
            ((List.map
                  (fun pf_elem =>
                    C (Groth16TypeIII.verificationPairingProof_G2 stmt () 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 () pairing SRS_elem) *
                      Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                  Groth16TypeIII.ListSRSElements_G2).sum))
        (Groth16TypeIII.ListPairings ())).sum =
    0
⊢ (((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
              (List.map
                  (fun i =>
                    Polynomial.C ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover 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 ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover 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 ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover i) * w_wit i)
                (List.finRange n_wit)).sum)).modByMonic
      t =
    0