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 :
  (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α)) +
          (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β)) +
            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ)) +
              ((List.map
                    (fun x =>
                      C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                        (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                    (List.finRange n_var)).sum +
                ((List.map
                      (fun x =>
                        C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                          (X (some Vars.γ) * X none ^ ↑x *
                            ∏ x : Fin n_wit,
                              ((to_MvPolynomial_Option Vars) Polynomial.X -
                                (to_MvPolynomial_Option Vars) (Polynomial.C (r x)))))
                      (List.finRange (n_var - 1))).sum +
                  ((List.map
                            (fun i =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt i)))
                            (List.finRange n_stmt)).sum +
                        (List.map
                            (fun i =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt i)))
                            (List.finRange n_stmt)).sum +
                      (List.map
                          (fun i =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                              (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt i)))
                          (List.finRange n_stmt)).sum +
                    ((List.map
                            (fun i =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit i)))
                            (List.finRange n_wit)).sum +
                        (List.map
                            (fun i =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit i)))
                            (List.finRange n_wit)).sum +
                      (List.map
                          (fun i =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                              (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit i)))
                          (List.finRange n_wit)).sum))))))) *
        (-(List.map
                    (fun x =>
                      C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                        (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                    (List.finRange n_var)).sum +
              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                  (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))) +
            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))) +
          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
              (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β)))) +
      (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α) * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β)) +
        (((List.map
                    (fun i =>
                      C (stmt i) * (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt i)))
                    (List.finRange n_stmt)).sum +
                (List.map
                    (fun i =>
                      C (stmt i) * (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt i)))
                    (List.finRange n_stmt)).sum +
              (List.map (fun i => C (stmt i) * (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt i)))
                  (List.finRange n_stmt)).sum) *
            (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ)) +
          (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α) * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α)) +
              (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β) *
                  (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β)) +
                (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ) *
                    (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ)) +
                  ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x)) *
                            (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                        (List.finRange n_var)).sum +
                    ((List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                              (X (some Vars.γ) * X none ^ ↑x *
                                ∏ x : Fin n_wit,
                                  ((to_MvPolynomial_Option Vars) Polynomial.X -
                                    (to_MvPolynomial_Option Vars) (Polynomial.C (r x)))))
                          (List.finRange (n_var - 1))).sum +
                      ((List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y i)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt i)))
                                (List.finRange n_stmt)).sum +
                            (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y i)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt i)))
                                (List.finRange n_stmt)).sum +
                          (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y i)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt i)))
                              (List.finRange n_stmt)).sum +
                        ((List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) *
                                    (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit i)))
                                (List.finRange n_wit)).sum +
                            (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) *
                                    (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit i)))
                                (List.finRange n_wit)).sum +
                          (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) *
                                  (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit i)))
                              (List.finRange n_wit)).sum))))))) *
            (X (some Vars.γ) * X (some Vars.δ) * X (some 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 [
  mul_assoc,
  mul_left_comm (C _) (X (some _)) _, mul_left_comm (List.sum _) (X (some _)) _,
  mul_comm (C _) (X (some _)), mul_comm (List.sum _) (X (some _)),
  neg_mul, mul_neg,
  List.sum_map_mul_right, List.sum_map_mul_left] 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 (some Vars.γ) * (X (some Vars.δ) * (X (some Vars.α) * C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α))) +
          (X (some Vars.γ) * (X (some Vars.δ) * (X (some Vars.β) * C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β))) +
            (X (some Vars.γ) *
                (X (some Vars.δ) * (X (some Vars.δ) * C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ))) +
              (X (some Vars.γ) *
                  (X (some Vars.δ) *
                    (List.map (fun b => C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow b)) * X none ^ ↑b)
                        (List.finRange n_var)).sum) +
                (X (some Vars.γ) *
                    (List.map
                        (fun b =>
                          C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t b)) *
                            (X none ^ ↑b *
                              ∏ 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 (some Vars.β) *
                          (X (some Vars.δ) *
                            (List.map
                                (fun b =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y b)) *
                                    (to_MvPolynomial_Option Vars) (u_stmt b))
                                (List.finRange n_stmt)).sum) +
                        X (some Vars.α) *
                          (X (some Vars.δ) *
                            (List.map
                                (fun b =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y b)) *
                                    (to_MvPolynomial_Option Vars) (v_stmt b))
                                (List.finRange n_stmt)).sum) +
                      X (some Vars.δ) *
                        (List.map
                            (fun b =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y b)) *
                                (to_MvPolynomial_Option Vars) (w_stmt b))
                            (List.finRange n_stmt)).sum +
                    (X (some Vars.β) *
                          (X (some Vars.γ) *
                            (List.map
                                (fun b =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q b)) *
                                    (to_MvPolynomial_Option Vars) (u_wit b))
                                (List.finRange n_wit)).sum) +
                        X (some Vars.α) *
                          (X (some Vars.γ) *
                            (List.map
                                (fun b =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q b)) *
                                    (to_MvPolynomial_Option Vars) (v_wit b))
                                (List.finRange n_wit)).sum) +
                      X (some Vars.γ) *
                        (List.map
                            (fun b =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q b)) *
                                (to_MvPolynomial_Option Vars) (w_wit b))
                            (List.finRange n_wit)).sum))))))) *
        (-(X (some Vars.γ) *
                  (X (some Vars.δ) *
                    (List.map (fun b => C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow b)) * X none ^ ↑b)
                        (List.finRange n_var)).sum)) +
              -(X (some Vars.γ) *
                  (X (some Vars.δ) * (X (some Vars.δ) * C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))) +
            -(X (some Vars.γ) *
                (X (some Vars.δ) * (X (some Vars.γ) * C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))) +
          -(X (some Vars.γ) *
              (X (some Vars.δ) * (X (some Vars.β) * C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))) +
      (X (some Vars.γ) *
          (X (some Vars.δ) * (X (some Vars.α) * (X (some Vars.γ) * (X (some Vars.δ) * X (some Vars.β))))) +
        ((X (some Vars.β) *
                  (X (some Vars.δ) *
                    (List.map (fun b => C (stmt b) * (to_MvPolynomial_Option Vars) (u_stmt b))
                        (List.finRange n_stmt)).sum) +
                X (some Vars.α) *
                  (X (some Vars.δ) *
                    (List.map (fun b => C (stmt b) * (to_MvPolynomial_Option Vars) (v_stmt b))
                        (List.finRange n_stmt)).sum) +
              X (some Vars.δ) *
                (List.map (fun b => C (stmt b) * (to_MvPolynomial_Option Vars) (w_stmt b))
                    (List.finRange n_stmt)).sum) *
            (X (some Vars.γ) * (X (some Vars.δ) * X (some Vars.γ))) +
          (X (some Vars.γ) * (X (some Vars.δ) * (X (some Vars.α) * C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α))) +
              (X (some Vars.γ) *
                  (X (some Vars.δ) * (X (some Vars.β) * C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β))) +
                (X (some Vars.γ) *
                    (X (some Vars.δ) * (X (some Vars.δ) * C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ))) +
                  (X (some Vars.γ) *
                      (X (some Vars.δ) *
                        (List.map (fun b => C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow b)) * X none ^ ↑b)
                            (List.finRange n_var)).sum) +
                    (X (some Vars.γ) *
                        (List.map
                            (fun b =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t b)) *
                                (X none ^ ↑b *
                                  ∏ 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 (some Vars.β) *
                              (X (some Vars.δ) *
                                (List.map
                                    (fun b =>
                                      C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y b)) *
                                        (to_MvPolynomial_Option Vars) (u_stmt b))
                                    (List.finRange n_stmt)).sum) +
                            X (some Vars.α) *
                              (X (some Vars.δ) *
                                (List.map
                                    (fun b =>
                                      C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y b)) *
                                        (to_MvPolynomial_Option Vars) (v_stmt b))
                                    (List.finRange n_stmt)).sum) +
                          X (some Vars.δ) *
                            (List.map
                                (fun b =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y b)) *
                                    (to_MvPolynomial_Option Vars) (w_stmt b))
                                (List.finRange n_stmt)).sum +
                        (X (some Vars.β) *
                              (X (some Vars.γ) *
                                (List.map
                                    (fun b =>
                                      C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q b)) *
                                        (to_MvPolynomial_Option Vars) (u_wit b))
                                    (List.finRange n_wit)).sum) +
                            X (some Vars.α) *
                              (X (some Vars.γ) *
                                (List.map
                                    (fun b =>
                                      C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q b)) *
                                        (to_MvPolynomial_Option Vars) (v_wit b))
                                    (List.finRange n_wit)).sum) +
                          X (some Vars.γ) *
                            (List.map
                                (fun b =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q b)) *
                                    (to_MvPolynomial_Option Vars) (w_wit b))
                                (List.finRange n_wit)).sum))))))) *
            (X (some Vars.γ) * (X (some Vars.δ) * X (some 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