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
t : Polynomial F := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))
eqn :
  (X Vars.γ * (X Vars.δ * (X Vars.α * C (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α)))) +
          (X Vars.γ * (X Vars.δ * (X Vars.β * C (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β)))) +
            (X Vars.γ * (X Vars.δ * (X Vars.δ * C (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ)))) +
              (X Vars.γ *
                  (X Vars.δ *
                    (List.map
                        (fun x =>
                          C (Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x))) *
                            C Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum) +
                (X Vars.γ *
                    (List.map
                        (fun x =>
                          C (Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x))) *
                            (C Polynomial.X ^ ↑x *
                              (optionEquivRight F Vars)
                                (∏ x : Fin n_wit,
                                  ((to_MvPolynomial_Option Vars) Polynomial.X -
                                    (to_MvPolynomial_Option Vars) (Polynomial.C (r x))))))
                        (List.finRange (n_var - 1))).sum +
                  (X Vars.β *
                          (X Vars.δ *
                            (List.map
                                (fun x =>
                                  C (Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x))) * C (u_stmt x))
                                (List.finRange n_stmt)).sum) +
                        X Vars.α *
                          (X Vars.δ *
                            (List.map
                                (fun x =>
                                  C (Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x))) * C (v_stmt x))
                                (List.finRange n_stmt)).sum) +
                      X Vars.δ *
                        (List.map
                            (fun x =>
                              C (Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x))) * C (w_stmt x))
                            (List.finRange n_stmt)).sum +
                    (X Vars.β *
                          (X Vars.γ *
                            (List.map
                                (fun x =>
                                  C (Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x))) * C (u_wit x))
                                (List.finRange n_wit)).sum) +
                        X Vars.α *
                          (X Vars.γ *
                            (List.map
                                (fun x =>
                                  C (Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x))) * C (v_wit x))
                                (List.finRange n_wit)).sum) +
                      X Vars.γ *
                        (List.map
                            (fun x =>
                              C (Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x))) * C (w_wit x))
                            (List.finRange n_wit)).sum))))))) *
        (-(X Vars.γ *
                  (X Vars.δ *
                    (List.map
                        (fun x =>
                          C (Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x))) *
                            C Polynomial.X ^ ↑x)
                        (List.finRange n_var)).sum)) +
              -(X Vars.γ * (X Vars.δ * (X Vars.δ * C (Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))) +
            -(X Vars.γ * (X Vars.δ * (X Vars.γ * C (Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))) +
          -(X Vars.γ * (X Vars.δ * (X Vars.β * C (Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))) +
      (X Vars.γ * (X Vars.δ * (X Vars.α * (X Vars.γ * (X Vars.δ * X Vars.β)))) +
        ((X Vars.β *
                  (X Vars.δ *
                    (List.map (fun x => C (Polynomial.C (stmt x)) * C (u_stmt x)) (List.finRange n_stmt)).sum) +
                X Vars.α *
                  (X Vars.δ *
                    (List.map (fun x => C (Polynomial.C (stmt x)) * C (v_stmt x)) (List.finRange n_stmt)).sum) +
              X Vars.δ * (List.map (fun x => C (Polynomial.C (stmt x)) * C (w_stmt x)) (List.finRange n_stmt)).sum) *
            (X Vars.γ * (X Vars.δ * X Vars.γ)) +
          (X Vars.γ * (X Vars.δ * (X Vars.α * C (Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α)))) +
              (X Vars.γ * (X Vars.δ * (X Vars.β * C (Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β)))) +
                (X Vars.γ * (X Vars.δ * (X Vars.δ * C (Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ)))) +
                  (X Vars.γ *
                      (X Vars.δ *
                        (List.map
                            (fun x =>
                              C (Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x))) *
                                C Polynomial.X ^ ↑x)
                            (List.finRange n_var)).sum) +
                    (X Vars.γ *
                        (List.map
                            (fun x =>
                              C (Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x))) *
                                (C Polynomial.X ^ ↑x *
                                  (optionEquivRight F Vars)
                                    (∏ x : Fin n_wit,
                                      ((to_MvPolynomial_Option Vars) Polynomial.X -
                                        (to_MvPolynomial_Option Vars) (Polynomial.C (r x))))))
                            (List.finRange (n_var - 1))).sum +
                      (X Vars.β *
                              (X Vars.δ *
                                (List.map
                                    (fun x =>
                                      C (Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x))) *
                                        C (u_stmt x))
                                    (List.finRange n_stmt)).sum) +
                            X Vars.α *
                              (X Vars.δ *
                                (List.map
                                    (fun x =>
                                      C (Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x))) *
                                        C (v_stmt x))
                                    (List.finRange n_stmt)).sum) +
                          X Vars.δ *
                            (List.map
                                (fun x =>
                                  C (Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x))) * C (w_stmt x))
                                (List.finRange n_stmt)).sum +
                        (X Vars.β *
                              (X Vars.γ *
                                (List.map
                                    (fun x =>
                                      C (Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x))) *
                                        C (u_wit x))
                                    (List.finRange n_wit)).sum) +
                            X Vars.α *
                              (X Vars.γ *
                                (List.map
                                    (fun x =>
                                      C (Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x))) *
                                        C (v_wit x))
                                    (List.finRange n_wit)).sum) +
                          X Vars.γ *
                            (List.map
                                (fun x =>
                                  C (Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x))) * C (w_wit x))
                                (List.finRange n_wit)).sum))))))) *
            (X Vars.γ * (X Vars.δ * X Vars.δ)))) =
    0
⊢ ((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q 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 (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q 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 (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) * w_wit i)
            (List.finRange n_wit)).sum +
      (List.map
          (fun x =>
            Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) * (Polynomial.X ^ ↑x * t))
          (List.finRange (n_var - 1))).sum
Tactic:
simp only [←C_mul, ←C_pow, ←C_add,
  sum_map_C] at eqn
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
t : Polynomial F := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))
eqn :
  (X Vars.γ * (X Vars.δ * (X Vars.α * C (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α)))) +
          (X Vars.γ * (X Vars.δ * (X Vars.β * C (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β)))) +
            (X Vars.γ * (X Vars.δ * (X Vars.δ * C (Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ)))) +
              (X Vars.γ *
                  (X Vars.δ *
                    C
                      (List.map
                          (fun x =>
                            Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum) +
                (X Vars.γ *
                    (List.map
                        (fun x =>
                          C (Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x))) *
                            (C (Polynomial.X ^ ↑x) *
                              (optionEquivRight F Vars)
                                (∏ x : Fin n_wit,
                                  ((to_MvPolynomial_Option Vars) Polynomial.X -
                                    (to_MvPolynomial_Option Vars) (Polynomial.C (r x))))))
                        (List.finRange (n_var - 1))).sum +
                  (X Vars.β *
                          (X Vars.δ *
                            C
                              (List.map
                                  (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * u_stmt x)
                                  (List.finRange n_stmt)).sum) +
                        X Vars.α *
                          (X Vars.δ *
                            C
                              (List.map
                                  (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * v_stmt x)
                                  (List.finRange n_stmt)).sum) +
                      X Vars.δ *
                        C
                          (List.map
                              (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) * w_stmt x)
                              (List.finRange n_stmt)).sum +
                    (X Vars.β *
                          (X Vars.γ *
                            C
                              (List.map
                                  (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * u_wit x)
                                  (List.finRange n_wit)).sum) +
                        X Vars.α *
                          (X Vars.γ *
                            C
                              (List.map
                                  (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * v_wit x)
                                  (List.finRange n_wit)).sum) +
                      X Vars.γ *
                        C
                          (List.map
                              (fun x => Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) * w_wit x)
                              (List.finRange n_wit)).sum))))))) *
        (-(X Vars.γ *
                  (X Vars.δ *
                    C
                      (List.map
                          (fun x =>
                            Polynomial.C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) * Polynomial.X ^ ↑x)
                          (List.finRange n_var)).sum)) +
              -(X Vars.γ * (X Vars.δ * (X Vars.δ * C (Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))) +
            -(X Vars.γ * (X Vars.δ * (X Vars.γ * C (Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))) +
          -(X Vars.γ * (X Vars.δ * (X Vars.β * C (Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))) +
      (X Vars.γ * (X Vars.δ * (X Vars.α * (X Vars.γ * (X Vars.δ * X Vars.β)))) +
        ((X Vars.β * (X Vars.δ * C (List.map (fun x => Polynomial.C (stmt x) * u_stmt x) (List.finRange n_stmt)).sum) +
                X Vars.α *
                  (X Vars.δ * C (List.map (fun x => Polynomial.C (stmt x) * v_stmt x) (List.finRange n_stmt)).sum) +
              X Vars.δ * C (List.map (fun x => Polynomial.C (stmt x) * w_stmt x) (List.finRange n_stmt)).sum) *
            (X Vars.γ * (X Vars.δ * X Vars.γ)) +
          (X Vars.γ * (X Vars.δ * (X Vars.α * C (Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α)))) +
              (X Vars.γ * (X Vars.δ * (X Vars.β * C (Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β)))) +
                (X Vars.γ * (X Vars.δ * (X Vars.δ * C (Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ)))) +
                  (X Vars.γ *
                      (X Vars.δ *
                        C
                          (List.map
                              (fun x =>
                                Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x)) *
                                  Polynomial.X ^ ↑x)
                              (List.finRange n_var)).sum) +
                    (X Vars.γ *
                        (List.map
                            (fun x =>
                              C (Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x))) *
                                (C (Polynomial.X ^ ↑x) *
                                  (optionEquivRight F Vars)
                                    (∏ x : Fin n_wit,
                                      ((to_MvPolynomial_Option Vars) Polynomial.X -
                                        (to_MvPolynomial_Option Vars) (Polynomial.C (r x))))))
                            (List.finRange (n_var - 1))).sum +
                      (X Vars.β *
                              (X Vars.δ *
                                C
                                  (List.map
                                      (fun x =>
                                        Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) * u_stmt x)
                                      (List.finRange n_stmt)).sum) +
                            X Vars.α *
                              (X Vars.δ *
                                C
                                  (List.map
                                      (fun x =>
                                        Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) * v_stmt x)
                                      (List.finRange n_stmt)).sum) +
                          X Vars.δ *
                            C
                              (List.map
                                  (fun x => Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) * w_stmt x)
                                  (List.finRange n_stmt)).sum +
                        (X Vars.β *
                              (X Vars.γ *
                                C
                                  (List.map
                                      (fun x =>
                                        Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) * u_wit x)
                                      (List.finRange n_wit)).sum) +
                            X Vars.α *
                              (X Vars.γ *
                                C
                                  (List.map
                                      (fun x =>
                                        Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) * v_wit x)
                                      (List.finRange n_wit)).sum) +
                          X Vars.γ *
                            C
                              (List.map
                                  (fun x => Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) * w_wit x)
                                  (List.finRange n_wit)).sum))))))) *
            (X Vars.γ * (X Vars.δ * X Vars.δ)))) =
    0
⊢ ((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
        (List.map (fun i => Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q 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 (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q 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 (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) * w_wit i)
            (List.finRange n_wit)).sum +
      (List.map
          (fun x =>
            Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) * (Polynomial.X ^ ↑x * t))
          (List.finRange (n_var - 1))).sum