Theorem: Groth16TypeIII.is_sound_quotient_version
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
t : Polynomial F
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
eqn :
  (List.map
        (fun pairing =>
          ((List.map
                  (fun pf_elem =>
                    C
                        (match pairing with
                        | PairingsIdx.ab =>
                          match pf_elem with
                          | Proof_G1_Idx.A => 1
                          | Proof_G1_Idx.C => 0
                        | PairingsIdx.αβ =>
                          match pf_elem with
                          | Proof_G1_Idx.A => 0
                          | Proof_G1_Idx.C => 0
                        | PairingsIdx.stmtγ =>
                          match pf_elem with
                          | Proof_G1_Idx.A => 0
                          | Proof_G1_Idx.C => 0
                        | PairingsIdx.cδ =>
                          match pf_elem with
                          | Proof_G1_Idx.A => 0
                          | Proof_G1_Idx.C => 1) *
                      (List.map
                          (fun SRS_elem =>
                            C (prover.1 pf_elem SRS_elem) *
                              match SRS_elem with
                              | SRS_Elements_G1_Idx.α => X Vars_γ * X Vars_δ * X Vars_α
                              | SRS_Elements_G1_Idx.β => X Vars_γ * X Vars_δ * X Vars_β
                              | SRS_Elements_G1_Idx.δ => X Vars_γ * X Vars_δ * X Vars_δ
                              | SRS_Elements_G1_Idx.x_pow i => X Vars_γ * X Vars_δ * X Vars_x ^ ↑i
                              | SRS_Elements_G1_Idx.x_pow_times_t i =>
                                X Vars_γ * X Vars_x ^ ↑i * (to_MvPolynomial_Option Vars) t
                              | SRS_Elements_G1_Idx.y i =>
                                X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i) +
                                    X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i) +
                                  X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)
                              | SRS_Elements_G1_Idx.q i =>
                                X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i) +
                                    X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i) +
                                  X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i))
                          ([SRS_Elements_G1_Idx.α] ++ [SRS_Elements_G1_Idx.β] ++ [SRS_Elements_G1_Idx.δ] ++
                                  List.map (fun i => SRS_Elements_G1_Idx.x_pow i) (List.finRange n_var) ++
                                List.map (fun i => SRS_Elements_G1_Idx.x_pow_times_t i) (List.finRange (n_var - 1)) ++
                              List.map (fun i => SRS_Elements_G1_Idx.y i) (List.finRange n_stmt) ++
                            List.map (fun i => SRS_Elements_G1_Idx.q i) (List.finRange n_wit))).sum)
                  [Proof_G1_Idx.A, Proof_G1_Idx.C]).sum +
              (List.map
                  (fun SRS_elem =>
                    C
                        (match pairing with
                        | PairingsIdx.ab =>
                          match SRS_elem with
                          | SRS_Elements_G1_Idx.α => 0
                          | SRS_Elements_G1_Idx.β => 0
                          | SRS_Elements_G1_Idx.δ => 0
                          | SRS_Elements_G1_Idx.x_pow a => 0
                          | SRS_Elements_G1_Idx.x_pow_times_t a => 0
                          | SRS_Elements_G1_Idx.y a => 0
                          | SRS_Elements_G1_Idx.q a => 0
                        | PairingsIdx.αβ =>
                          match SRS_elem with
                          | SRS_Elements_G1_Idx.α => 1
                          | SRS_Elements_G1_Idx.β => 0
                          | SRS_Elements_G1_Idx.δ => 0
                          | SRS_Elements_G1_Idx.x_pow a => 0
                          | SRS_Elements_G1_Idx.x_pow_times_t a => 0
                          | SRS_Elements_G1_Idx.y a => 0
                          | SRS_Elements_G1_Idx.q a => 0
                        | PairingsIdx.stmtγ =>
                          match SRS_elem with
                          | SRS_Elements_G1_Idx.α => 0
                          | SRS_Elements_G1_Idx.β => 0
                          | SRS_Elements_G1_Idx.δ => 0
                          | SRS_Elements_G1_Idx.x_pow a => 0
                          | SRS_Elements_G1_Idx.x_pow_times_t a => 0
                          | SRS_Elements_G1_Idx.y i => stmt i
                          | SRS_Elements_G1_Idx.q a => 0
                        | PairingsIdx.cδ =>
                          match SRS_elem with
                          | SRS_Elements_G1_Idx.α => 0
                          | SRS_Elements_G1_Idx.β => 0
                          | SRS_Elements_G1_Idx.δ => 0
                          | SRS_Elements_G1_Idx.x_pow a => 0
                          | SRS_Elements_G1_Idx.x_pow_times_t a => 0
                          | SRS_Elements_G1_Idx.y a => 0
                          | SRS_Elements_G1_Idx.q a => 0) *
                      match SRS_elem with
                      | SRS_Elements_G1_Idx.α => X Vars_γ * X Vars_δ * X Vars_α
                      | SRS_Elements_G1_Idx.β => X Vars_γ * X Vars_δ * X Vars_β
                      | SRS_Elements_G1_Idx.δ => X Vars_γ * X Vars_δ * X Vars_δ
                      | SRS_Elements_G1_Idx.x_pow i => X Vars_γ * X Vars_δ * X Vars_x ^ ↑i
                      | SRS_Elements_G1_Idx.x_pow_times_t i =>
                        X Vars_γ * X Vars_x ^ ↑i * (to_MvPolynomial_Option Vars) t
                      | SRS_Elements_G1_Idx.y i =>
                        X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i) +
                            X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i) +
                          X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)
                      | SRS_Elements_G1_Idx.q i =>
                        X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i) +
                            X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i) +
                          X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i))
                  ([SRS_Elements_G1_Idx.α] ++ [SRS_Elements_G1_Idx.β] ++ [SRS_Elements_G1_Idx.δ] ++
                          List.map (fun i => SRS_Elements_G1_Idx.x_pow i) (List.finRange n_var) ++
                        List.map (fun i => SRS_Elements_G1_Idx.x_pow_times_t i) (List.finRange (n_var - 1)) ++
                      List.map (fun i => SRS_Elements_G1_Idx.y i) (List.finRange n_stmt) ++
                    List.map (fun i => SRS_Elements_G1_Idx.q i) (List.finRange n_wit))).sum) *
            ((List.map
                  (fun pf_elem =>
                    C
                        (match pairing with
                        | PairingsIdx.ab => -1
                        | PairingsIdx.αβ => 0
                        | PairingsIdx.stmtγ => 0
                        | PairingsIdx.cδ => 0) *
                      (List.map
                          (fun SRS_elem =>
                            C (prover.2 pf_elem SRS_elem) *
                              match SRS_elem with
                              | SRS_Elements_G2_Idx.β => X Vars_γ * X Vars_δ * X Vars_β
                              | SRS_Elements_G2_Idx.γ => X Vars_γ * X Vars_δ * X Vars_γ
                              | SRS_Elements_G2_Idx.δ => X Vars_γ * X Vars_δ * X Vars_δ
                              | SRS_Elements_G2_Idx.x_pow i => X Vars_γ * X Vars_δ * X Vars_x ^ ↑i)
                          ([SRS_Elements_G2_Idx.β] ++ [SRS_Elements_G2_Idx.γ] ++ [SRS_Elements_G2_Idx.δ] ++
                            List.map (fun i => SRS_Elements_G2_Idx.x_pow i) (List.finRange n_var))).sum)
                  [Proof_G2_Idx.B]).sum +
              (List.map
                  (fun SRS_elem =>
                    C
                        (match pairing with
                        | PairingsIdx.ab =>
                          match SRS_elem with
                          | SRS_Elements_G2_Idx.β => 0
                          | SRS_Elements_G2_Idx.γ => 0
                          | SRS_Elements_G2_Idx.δ => 0
                          | SRS_Elements_G2_Idx.x_pow a => 0
                        | PairingsIdx.αβ =>
                          match SRS_elem with
                          | SRS_Elements_G2_Idx.β => 1
                          | SRS_Elements_G2_Idx.γ => 0
                          | SRS_Elements_G2_Idx.δ => 0
                          | SRS_Elements_G2_Idx.x_pow a => 0
                        | PairingsIdx.stmtγ =>
                          match SRS_elem with
                          | SRS_Elements_G2_Idx.β => 0
                          | SRS_Elements_G2_Idx.γ => 1
                          | SRS_Elements_G2_Idx.δ => 0
                          | SRS_Elements_G2_Idx.x_pow a => 0
                        | PairingsIdx.cδ =>
                          match SRS_elem with
                          | SRS_Elements_G2_Idx.β => 0
                          | SRS_Elements_G2_Idx.γ => 0
                          | SRS_Elements_G2_Idx.δ => 1
                          | SRS_Elements_G2_Idx.x_pow a => 0) *
                      match SRS_elem with
                      | SRS_Elements_G2_Idx.β => X Vars_γ * X Vars_δ * X Vars_β
                      | SRS_Elements_G2_Idx.γ => X Vars_γ * X Vars_δ * X Vars_γ
                      | SRS_Elements_G2_Idx.δ => X Vars_γ * X Vars_δ * X Vars_δ
                      | SRS_Elements_G2_Idx.x_pow i => X Vars_γ * X Vars_δ * X Vars_x ^ ↑i)
                  ([SRS_Elements_G2_Idx.β] ++ [SRS_Elements_G2_Idx.γ] ++ [SRS_Elements_G2_Idx.δ] ++
                    List.map (fun i => SRS_Elements_G2_Idx.x_pow i) (List.finRange n_var))).sum))
        [PairingsIdx.ab, PairingsIdx.αβ, PairingsIdx.stmtγ, PairingsIdx.cδ]).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
Tactic:
simp only [monomial_zero', List.singleton_append, List.cons_append, List.append_assoc,
  List.map_cons, Sum.elim_inl, Sum.elim_inr, List.map_append, List.map_map, List.sum_cons,
  List.sum_append, List.map_nil, List.sum_nil, add_zero, Sum.elim_lam_const_lam_const, map_one,
  one_mul, map_zero, zero_mul, map_neg, neg_mul, neg_add_rev, zero_add, mul_zero,
  Function.comp, List.sum_map_zero] 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
t : Polynomial F
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
eqn :
  (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) +
          (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) +
            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) +
              ((List.map
                    (fun x =>
                      C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) * (X Vars_γ * X Vars_δ * X Vars_x ^ ↑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 Vars_γ * X Vars_x ^ ↑x * (to_MvPolynomial_Option Vars) t))
                      (List.finRange (n_var - 1))).sum +
                  ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                            (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt x) +
                                X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt x) +
                              X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt x)))
                        (List.finRange n_stmt)).sum +
                    (List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                            (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit x) +
                                X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit x) +
                              X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit x)))
                        (List.finRange n_wit)).sum)))))) *
        (-(List.map
                    (fun x =>
                      C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) * (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                    (List.finRange n_var)).sum +
              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β))) +
      (X Vars_γ * X Vars_δ * X Vars_α * (X Vars_γ * X Vars_δ * X Vars_β) +
        ((List.map
                (fun x =>
                  C (stmt x) *
                    (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt x) +
                        X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt x) +
                      X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt x)))
                (List.finRange n_stmt)).sum *
            (X Vars_γ * X Vars_δ * X Vars_γ) +
          (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) +
              (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) +
                (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) +
                  ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x)) *
                            (X Vars_γ * X Vars_δ * X Vars_x ^ ↑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 Vars_γ * X Vars_x ^ ↑x * (to_MvPolynomial_Option Vars) t))
                          (List.finRange (n_var - 1))).sum +
                      ((List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt x) +
                                    X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt x) +
                                  X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt)).sum +
                        (List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit x) +
                                    X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit x) +
                                  X Vars_γ * (to_MvPolynomial_Option Vars) (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 ((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