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
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
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 x =>
              t * Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) * Polynomial.X ^ ↑x)
            (List.finRange (n_var - 1))).sum.modByMonic
      t =
    0
Tactic:
simp only [mul_assoc, List.sum_map_mul_right, List.sum_map_mul_left]
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
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
⊢ (t *
          (List.map
              (fun b =>
                Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t b)) * Polynomial.X ^ ↑b)
              (List.finRange (n_var - 1))).sum).modByMonic
      t =
    0