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 (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
Tactic:
simp only [←(EquivLike.apply_eq_iff_eq (optionEquivRight _ _))] 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 :
  (optionEquivRight F Vars)
      ((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)))))) =
    (optionEquivRight F Vars) 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