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))
sum_u_stmt : Polynomial F := (List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum
sum_v_stmt : Polynomial F := (List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)).sum
sum_w_stmt : Polynomial F := (List.map (fun i => Polynomial.C (stmt i) * w_stmt i) (List.finRange n_stmt)).sum
A_1 : Polynomial F := Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α)
A_2 : Polynomial F := Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β)
A_3 : Polynomial F := Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ)
sum_A_x : Polynomial F :=
  (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
sum_A_u_stmt : Polynomial F :=
  (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
sum_A_v_stmt : Polynomial F :=
  (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
sum_A_w_stmt : Polynomial F :=
  (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
sum_A_u_wit : Polynomial F :=
  (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
sum_A_v_wit : Polynomial F :=
  (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
sum_A_w_wit : Polynomial F :=
  (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
sum_A_x_t : Polynomial F :=
  (List.map
      (fun x =>
        Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
          (Polynomial.X ^ ↑x * ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))))
      (List.finRange (n_var - 1))).sum
B_1 : Polynomial F := Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)
h1122 : -(A_1 * B_1) + 1 = 0
h0222 : -(A_2 * B_1) = 0
h0212 : -(sum_A_u_stmt * B_1) = 0
h1112 : -(sum_A_v_stmt * B_1) = 0
h0221 : -(sum_A_u_wit * B_1) = 0
h1121 : -(sum_A_v_wit * B_1) = 0
B_2 : Polynomial F := Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)
B_3 : Polynomial F := Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)
sum_B_x : Polynomial F :=
  (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
h0012 : -(sum_A_w_stmt * sum_B_x) = 0
h0021 : -(sum_A_x_t * sum_B_x) + -(sum_A_w_wit * sum_B_x) = 0
h0112 : -(sum_A_u_stmt * sum_B_x) + -(sum_A_w_stmt * B_1) = 0
h0121 : -(sum_A_u_wit * sum_B_x) + (-(sum_A_x_t * B_1) + -(sum_A_w_wit * B_1)) = 0
h0022 :
  -(sum_A_x * sum_B_x) + (-(sum_A_x_t * B_3) + -(sum_A_w_wit * B_3)) + -(sum_A_w_stmt * B_2) +
      (sum_w_stmt +
        ((List.map
              (fun x =>
                Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                  (Polynomial.X ^ ↑x * ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))))
              (List.finRange (n_var - 1))).sum +
          (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)) =
    0
C_1 : Polynomial F := Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α)
C_2 : Polynomial F := Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β)
C_3 : Polynomial F := Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ)
sum_C_u_wit : Polynomial F :=
  (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
h0122 :
  -(A_2 * sum_B_x) + -(sum_A_u_wit * B_3) + -(sum_A_u_stmt * B_2) + -(sum_A_x * B_1) + (sum_u_stmt + sum_C_u_wit) = 0
sum_C_v_wit : Polynomial F :=
  (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
h1022 : -(A_1 * sum_B_x) + -(sum_A_v_wit * B_3) + -(sum_A_v_stmt * B_2) + (sum_v_stmt + sum_C_v_wit) = 0
⊢ (sum_u_stmt + sum_C_u_wit) * (sum_v_stmt + sum_C_v_wit) =
    sum_w_stmt +
        (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:
set sum_C_w_wit := List.sum
      (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))
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))
sum_u_stmt : Polynomial F := (List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum
sum_v_stmt : Polynomial F := (List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)).sum
sum_w_stmt : Polynomial F := (List.map (fun i => Polynomial.C (stmt i) * w_stmt i) (List.finRange n_stmt)).sum
A_1 : Polynomial F := Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α)
A_2 : Polynomial F := Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β)
A_3 : Polynomial F := Polynomial.C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ)
sum_A_x : Polynomial F :=
  (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
sum_A_u_stmt : Polynomial F :=
  (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
sum_A_v_stmt : Polynomial F :=
  (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
sum_A_w_stmt : Polynomial F :=
  (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
sum_A_u_wit : Polynomial F :=
  (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
sum_A_v_wit : Polynomial F :=
  (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
sum_A_w_wit : Polynomial F :=
  (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
sum_A_x_t : Polynomial F :=
  (List.map
      (fun x =>
        Polynomial.C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
          (Polynomial.X ^ ↑x * ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))))
      (List.finRange (n_var - 1))).sum
B_1 : Polynomial F := Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)
h1122 : -(A_1 * B_1) + 1 = 0
h0222 : -(A_2 * B_1) = 0
h0212 : -(sum_A_u_stmt * B_1) = 0
h1112 : -(sum_A_v_stmt * B_1) = 0
h0221 : -(sum_A_u_wit * B_1) = 0
h1121 : -(sum_A_v_wit * B_1) = 0
B_2 : Polynomial F := Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)
B_3 : Polynomial F := Polynomial.C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)
sum_B_x : Polynomial F :=
  (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
h0012 : -(sum_A_w_stmt * sum_B_x) = 0
h0021 : -(sum_A_x_t * sum_B_x) + -(sum_A_w_wit * sum_B_x) = 0
h0112 : -(sum_A_u_stmt * sum_B_x) + -(sum_A_w_stmt * B_1) = 0
h0121 : -(sum_A_u_wit * sum_B_x) + (-(sum_A_x_t * B_1) + -(sum_A_w_wit * B_1)) = 0
C_1 : Polynomial F := Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α)
C_2 : Polynomial F := Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β)
C_3 : Polynomial F := Polynomial.C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ)
sum_C_u_wit : Polynomial F :=
  (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
h0122 :
  -(A_2 * sum_B_x) + -(sum_A_u_wit * B_3) + -(sum_A_u_stmt * B_2) + -(sum_A_x * B_1) + (sum_u_stmt + sum_C_u_wit) = 0
sum_C_v_wit : Polynomial F :=
  (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
h1022 : -(A_1 * sum_B_x) + -(sum_A_v_wit * B_3) + -(sum_A_v_stmt * B_2) + (sum_v_stmt + sum_C_v_wit) = 0
sum_C_w_wit : Polynomial F :=
  (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
h0022 :
  -(sum_A_x * sum_B_x) + (-(sum_A_x_t * B_3) + -(sum_A_w_wit * B_3)) + -(sum_A_w_stmt * B_2) +
      (sum_w_stmt +
        ((List.map
              (fun x =>
                Polynomial.C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                  (Polynomial.X ^ ↑x * ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))))
              (List.finRange (n_var - 1))).sum +
          sum_C_w_wit)) =
    0
⊢ (sum_u_stmt + sum_C_u_wit) * (sum_v_stmt + sum_C_v_wit) =
    sum_w_stmt + sum_C_w_wit +
      (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