Theorem: Lipmaa.soundness
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 : Lipmaa.Stmt
prover : Prover F Lipmaa
t : Polynomial F := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))
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 (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 75
                              | SRS_Elements_G1_Idx.β =>
                                X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25
                              | SRS_Elements_G1_Idx.δ => X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1
                              | SRS_Elements_G1_Idx.x_pow i => X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X none ^ ↑i
                              | SRS_Elements_G1_Idx.x_pow_times_t i =>
                                X (some Vars.y) ^ 5 * X none ^ ↑i *
                                  (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))
                              | SRS_Elements_G1_Idx.y i =>
                                X (some Vars.y) ^ 25 * X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (u_stmt i) +
                                    X (some Vars.y) ^ 75 * X (some Vars.y) ^ 1 *
                                      (to_MvPolynomial_Option Vars) (v_stmt i) +
                                  X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (w_stmt i)
                              | SRS_Elements_G1_Idx.q i =>
                                X (some Vars.y) ^ 25 * X (some Vars.y) ^ 5 * (to_MvPolynomial_Option Vars) (u_wit i) +
                                    X (some Vars.y) ^ 75 * X (some Vars.y) ^ 5 *
                                      (to_MvPolynomial_Option Vars) (v_wit i) +
                                  X (some Vars.y) ^ 5 * (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 (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 75
                      | SRS_Elements_G1_Idx.β => X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25
                      | SRS_Elements_G1_Idx.δ => X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1
                      | SRS_Elements_G1_Idx.x_pow i => X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X none ^ ↑i
                      | SRS_Elements_G1_Idx.x_pow_times_t i =>
                        X (some Vars.y) ^ 5 * X none ^ ↑i *
                          (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))
                      | SRS_Elements_G1_Idx.y i =>
                        X (some Vars.y) ^ 25 * X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (u_stmt i) +
                            X (some Vars.y) ^ 75 * X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (v_stmt i) +
                          X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (w_stmt i)
                      | SRS_Elements_G1_Idx.q i =>
                        X (some Vars.y) ^ 25 * X (some Vars.y) ^ 5 * (to_MvPolynomial_Option Vars) (u_wit i) +
                            X (some Vars.y) ^ 75 * X (some Vars.y) ^ 5 * (to_MvPolynomial_Option Vars) (v_wit i) +
                          X (some Vars.y) ^ 5 * (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 (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25
                              | SRS_Elements_G2_Idx.γ => X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 5
                              | SRS_Elements_G2_Idx.δ => X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1
                              | SRS_Elements_G2_Idx.x_pow i => X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X none ^ ↑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 (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25
                      | SRS_Elements_G2_Idx.γ => X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 5
                      | SRS_Elements_G2_Idx.δ => X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1
                      | SRS_Elements_G2_Idx.x_pow i => X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X none ^ ↑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 (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 [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
r : Fin n_wit → F
stmt : Lipmaa.Stmt
prover : Prover F Lipmaa
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.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 75) +
          (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
              (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25) +
            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1) +
              ((List.map
                    (fun x =>
                      C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                        (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * 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.y) ^ 5 * X none ^ ↑x *
                            (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                      (List.finRange (n_var - 1))).sum +
                  ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                            (X (some Vars.y) ^ 25 * X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (u_stmt x) +
                                X (some Vars.y) ^ 75 * X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (v_stmt x) +
                              X (some Vars.y) ^ 1 * (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 (some Vars.y) ^ 25 * X (some Vars.y) ^ 5 * (to_MvPolynomial_Option Vars) (u_wit x) +
                                X (some Vars.y) ^ 75 * X (some Vars.y) ^ 5 * (to_MvPolynomial_Option Vars) (v_wit x) +
                              X (some Vars.y) ^ 5 * (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 (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X none ^ ↑x))
                    (List.finRange n_var)).sum +
              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                  (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1)) +
            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 5)) +
          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
              (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25))) +
      (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 75 *
          (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25) +
        ((List.map
                (fun x =>
                  C (stmt x) *
                    (X (some Vars.y) ^ 25 * X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (u_stmt x) +
                        X (some Vars.y) ^ 75 * X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (v_stmt x) +
                      X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (w_stmt x)))
                (List.finRange n_stmt)).sum *
            (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 5) +
          (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α) *
                (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 75) +
              (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β) *
                  (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 25) +
                (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ) *
                    (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1) +
                  ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x)) *
                            (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * 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.y) ^ 5 * X none ^ ↑x *
                                (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                          (List.finRange (n_var - 1))).sum +
                      ((List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                (X (some Vars.y) ^ 25 * X (some Vars.y) ^ 1 * (to_MvPolynomial_Option Vars) (u_stmt x) +
                                    X (some Vars.y) ^ 75 * X (some Vars.y) ^ 1 *
                                      (to_MvPolynomial_Option Vars) (v_stmt x) +
                                  X (some Vars.y) ^ 1 * (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 (some Vars.y) ^ 25 * X (some Vars.y) ^ 5 * (to_MvPolynomial_Option Vars) (u_wit x) +
                                    X (some Vars.y) ^ 75 * X (some Vars.y) ^ 5 *
                                      (to_MvPolynomial_Option Vars) (v_wit x) +
                                  X (some Vars.y) ^ 5 * (to_MvPolynomial_Option Vars) (w_wit x)))
                            (List.finRange n_wit)).sum)))))) *
            (X (some Vars.y) ^ 5 * X (some Vars.y) ^ 1 * X (some Vars.y) ^ 1))) =
    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