Theorem: Groth16TypeIII.is_sound (FormalSnarksProject/SNARKs/Groth16TypeIII/Soundness.lean)

State before:
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
⊢ soundness F Groth16TypeIII (Fin n_wit → F)
    (fun stmt wit =>
      let t := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i));
      (((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
                  (List.map (fun i => Polynomial.C (wit 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 (wit 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 (wit i) * w_wit i) (List.finRange n_wit)).sum)).modByMonic
          t =
        0)
    fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)
Tactic:
unfold soundness verify check_poly pairing_poly proof_element_G1_as_poly proof_element_G2_as_poly
State after:
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),
    ((∀ (check_idx : Groth16TypeIII.EqualityChecks),
          (List.map
                (fun pairing =>
                  ((List.map
                          (fun pf_elem =>
                            C (Groth16TypeIII.verificationPairingProof_G1 stmt check_idx pairing pf_elem) *
                              (List.map
                                  (fun SRS_elem =>
                                    C (prover.1 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                                  Groth16TypeIII.ListSRSElements_G1).sum)
                          Groth16TypeIII.ListProof_G1).sum +
                      (List.map
                          (fun SRS_elem =>
                            C (Groth16TypeIII.verificationPairingSRS_G1 stmt check_idx pairing SRS_elem) *
                              Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                          Groth16TypeIII.ListSRSElements_G1).sum) *
                    ((List.map
                          (fun pf_elem =>
                            C (Groth16TypeIII.verificationPairingProof_G2 stmt check_idx pairing pf_elem) *
                              (List.map
                                  (fun SRS_elem =>
                                    C (prover.2 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                                  Groth16TypeIII.ListSRSElements_G2).sum)
                          Groth16TypeIII.ListProof_G2).sum +
                      (List.map
                          (fun SRS_elem =>
                            C (Groth16TypeIII.verificationPairingSRS_G2 stmt check_idx pairing SRS_elem) *
                              Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                          Groth16TypeIII.ListSRSElements_G2).sum))
                (Groth16TypeIII.ListPairings check_idx)).sum =
            0) ∧
        ∀ pfs ∈ Groth16TypeIII.Identified_Proof_Elems,
          (List.map (fun SRS_elem => C (prover.1 pfs.1 SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                Groth16TypeIII.ListSRSElements_G1).sum =
            (List.map (fun SRS_elem => C (prover.2 pfs.2 SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                Groth16TypeIII.ListSRSElements_G2).sum) →
      (fun stmt wit =>
          let t := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i));
          (((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
                      (List.map (fun i => Polynomial.C (wit 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 (wit 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 (wit i) * w_wit i) (List.finRange n_wit)).sum)).modByMonic
              t =
            0)
        stmt ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover)

State before:
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),
    ((∀ (check_idx : Groth16TypeIII.EqualityChecks),
          (List.map
                (fun pairing =>
                  ((List.map
                          (fun pf_elem =>
                            C (Groth16TypeIII.verificationPairingProof_G1 stmt check_idx pairing pf_elem) *
                              (List.map
                                  (fun SRS_elem =>
                                    C (prover.1 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                                  Groth16TypeIII.ListSRSElements_G1).sum)
                          Groth16TypeIII.ListProof_G1).sum +
                      (List.map
                          (fun SRS_elem =>
                            C (Groth16TypeIII.verificationPairingSRS_G1 stmt check_idx pairing SRS_elem) *
                              Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                          Groth16TypeIII.ListSRSElements_G1).sum) *
                    ((List.map
                          (fun pf_elem =>
                            C (Groth16TypeIII.verificationPairingProof_G2 stmt check_idx pairing pf_elem) *
                              (List.map
                                  (fun SRS_elem =>
                                    C (prover.2 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                                  Groth16TypeIII.ListSRSElements_G2).sum)
                          Groth16TypeIII.ListProof_G2).sum +
                      (List.map
                          (fun SRS_elem =>
                            C (Groth16TypeIII.verificationPairingSRS_G2 stmt check_idx pairing SRS_elem) *
                              Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                          Groth16TypeIII.ListSRSElements_G2).sum))
                (Groth16TypeIII.ListPairings check_idx)).sum =
            0) ∧
        ∀ pfs ∈ Groth16TypeIII.Identified_Proof_Elems,
          (List.map (fun SRS_elem => C (prover.1 pfs.1 SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                Groth16TypeIII.ListSRSElements_G1).sum =
            (List.map (fun SRS_elem => C (prover.2 pfs.2 SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                Groth16TypeIII.ListSRSElements_G2).sum) →
      (fun stmt wit =>
          let t := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i));
          (((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
                      (List.map (fun i => Polynomial.C (wit 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 (wit 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 (wit i) * w_wit i) (List.finRange n_wit)).sum)).modByMonic
              t =
            0)
        stmt ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover)
Tactic:
intros stmt prover eqns'
State after:
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
eqns' :
  (∀ (check_idx : Groth16TypeIII.EqualityChecks),
      (List.map
            (fun pairing =>
              ((List.map
                      (fun pf_elem =>
                        C (Groth16TypeIII.verificationPairingProof_G1 stmt check_idx pairing pf_elem) *
                          (List.map
                              (fun SRS_elem =>
                                C (prover.1 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                              Groth16TypeIII.ListSRSElements_G1).sum)
                      Groth16TypeIII.ListProof_G1).sum +
                  (List.map
                      (fun SRS_elem =>
                        C (Groth16TypeIII.verificationPairingSRS_G1 stmt check_idx pairing SRS_elem) *
                          Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                      Groth16TypeIII.ListSRSElements_G1).sum) *
                ((List.map
                      (fun pf_elem =>
                        C (Groth16TypeIII.verificationPairingProof_G2 stmt check_idx pairing pf_elem) *
                          (List.map
                              (fun SRS_elem =>
                                C (prover.2 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                              Groth16TypeIII.ListSRSElements_G2).sum)
                      Groth16TypeIII.ListProof_G2).sum +
                  (List.map
                      (fun SRS_elem =>
                        C (Groth16TypeIII.verificationPairingSRS_G2 stmt check_idx pairing SRS_elem) *
                          Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                      Groth16TypeIII.ListSRSElements_G2).sum))
            (Groth16TypeIII.ListPairings check_idx)).sum =
        0) ∧
    ∀ pfs ∈ Groth16TypeIII.Identified_Proof_Elems,
      (List.map (fun SRS_elem => C (prover.1 pfs.1 SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
            Groth16TypeIII.ListSRSElements_G1).sum =
        (List.map (fun SRS_elem => C (prover.2 pfs.2 SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
            Groth16TypeIII.ListSRSElements_G2).sum
⊢ let t := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i));
  (((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

State before:
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
eqns' :
  (∀ (check_idx : Groth16TypeIII.EqualityChecks),
      (List.map
            (fun pairing =>
              ((List.map
                      (fun pf_elem =>
                        C (Groth16TypeIII.verificationPairingProof_G1 stmt check_idx pairing pf_elem) *
                          (List.map
                              (fun SRS_elem =>
                                C (prover.1 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                              Groth16TypeIII.ListSRSElements_G1).sum)
                      Groth16TypeIII.ListProof_G1).sum +
                  (List.map
                      (fun SRS_elem =>
                        C (Groth16TypeIII.verificationPairingSRS_G1 stmt check_idx pairing SRS_elem) *
                          Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                      Groth16TypeIII.ListSRSElements_G1).sum) *
                ((List.map
                      (fun pf_elem =>
                        C (Groth16TypeIII.verificationPairingProof_G2 stmt check_idx pairing pf_elem) *
                          (List.map
                              (fun SRS_elem =>
                                C (prover.2 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                              Groth16TypeIII.ListSRSElements_G2).sum)
                      Groth16TypeIII.ListProof_G2).sum +
                  (List.map
                      (fun SRS_elem =>
                        C (Groth16TypeIII.verificationPairingSRS_G2 stmt check_idx pairing SRS_elem) *
                          Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                      Groth16TypeIII.ListSRSElements_G2).sum))
            (Groth16TypeIII.ListPairings check_idx)).sum =
        0) ∧
    ∀ pfs ∈ Groth16TypeIII.Identified_Proof_Elems,
      (List.map (fun SRS_elem => C (prover.1 pfs.1 SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
            Groth16TypeIII.ListSRSElements_G1).sum =
        (List.map (fun SRS_elem => C (prover.2 pfs.2 SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
            Groth16TypeIII.ListSRSElements_G2).sum
⊢ let t := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i));
  (((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:
rcases eqns' with ⟨eqns, null⟩
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
eqns :
  ∀ (check_idx : Groth16TypeIII.EqualityChecks),
    (List.map
          (fun pairing =>
            ((List.map
                    (fun pf_elem =>
                      C (Groth16TypeIII.verificationPairingProof_G1 stmt check_idx pairing pf_elem) *
                        (List.map
                            (fun SRS_elem => C (prover.1 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                            Groth16TypeIII.ListSRSElements_G1).sum)
                    Groth16TypeIII.ListProof_G1).sum +
                (List.map
                    (fun SRS_elem =>
                      C (Groth16TypeIII.verificationPairingSRS_G1 stmt check_idx pairing SRS_elem) *
                        Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                    Groth16TypeIII.ListSRSElements_G1).sum) *
              ((List.map
                    (fun pf_elem =>
                      C (Groth16TypeIII.verificationPairingProof_G2 stmt check_idx pairing pf_elem) *
                        (List.map
                            (fun SRS_elem => C (prover.2 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                            Groth16TypeIII.ListSRSElements_G2).sum)
                    Groth16TypeIII.ListProof_G2).sum +
                (List.map
                    (fun SRS_elem =>
                      C (Groth16TypeIII.verificationPairingSRS_G2 stmt check_idx pairing SRS_elem) *
                        Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                    Groth16TypeIII.ListSRSElements_G2).sum))
          (Groth16TypeIII.ListPairings check_idx)).sum =
      0
null :
  ∀ pfs ∈ Groth16TypeIII.Identified_Proof_Elems,
    (List.map (fun SRS_elem => C (prover.1 pfs.1 SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
          Groth16TypeIII.ListSRSElements_G1).sum =
      (List.map (fun SRS_elem => C (prover.2 pfs.2 SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
          Groth16TypeIII.ListSRSElements_G2).sum
⊢ let t := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i));
  (((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

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
eqns :
  ∀ (check_idx : Groth16TypeIII.EqualityChecks),
    (List.map
          (fun pairing =>
            ((List.map
                    (fun pf_elem =>
                      C (Groth16TypeIII.verificationPairingProof_G1 stmt check_idx pairing pf_elem) *
                        (List.map
                            (fun SRS_elem => C (prover.1 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                            Groth16TypeIII.ListSRSElements_G1).sum)
                    Groth16TypeIII.ListProof_G1).sum +
                (List.map
                    (fun SRS_elem =>
                      C (Groth16TypeIII.verificationPairingSRS_G1 stmt check_idx pairing SRS_elem) *
                        Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                    Groth16TypeIII.ListSRSElements_G1).sum) *
              ((List.map
                    (fun pf_elem =>
                      C (Groth16TypeIII.verificationPairingProof_G2 stmt check_idx pairing pf_elem) *
                        (List.map
                            (fun SRS_elem => C (prover.2 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                            Groth16TypeIII.ListSRSElements_G2).sum)
                    Groth16TypeIII.ListProof_G2).sum +
                (List.map
                    (fun SRS_elem =>
                      C (Groth16TypeIII.verificationPairingSRS_G2 stmt check_idx pairing SRS_elem) *
                        Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                    Groth16TypeIII.ListSRSElements_G2).sum))
          (Groth16TypeIII.ListPairings check_idx)).sum =
      0
null :
  ∀ pfs ∈ Groth16TypeIII.Identified_Proof_Elems,
    (List.map (fun SRS_elem => C (prover.1 pfs.1 SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
          Groth16TypeIII.ListSRSElements_G1).sum =
      (List.map (fun SRS_elem => C (prover.2 pfs.2 SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
          Groth16TypeIII.ListSRSElements_G2).sum
⊢ let t := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i));
  (((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:
intro t
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
eqns :
  ∀ (check_idx : Groth16TypeIII.EqualityChecks),
    (List.map
          (fun pairing =>
            ((List.map
                    (fun pf_elem =>
                      C (Groth16TypeIII.verificationPairingProof_G1 stmt check_idx pairing pf_elem) *
                        (List.map
                            (fun SRS_elem => C (prover.1 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                            Groth16TypeIII.ListSRSElements_G1).sum)
                    Groth16TypeIII.ListProof_G1).sum +
                (List.map
                    (fun SRS_elem =>
                      C (Groth16TypeIII.verificationPairingSRS_G1 stmt check_idx pairing SRS_elem) *
                        Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                    Groth16TypeIII.ListSRSElements_G1).sum) *
              ((List.map
                    (fun pf_elem =>
                      C (Groth16TypeIII.verificationPairingProof_G2 stmt check_idx pairing pf_elem) *
                        (List.map
                            (fun SRS_elem => C (prover.2 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                            Groth16TypeIII.ListSRSElements_G2).sum)
                    Groth16TypeIII.ListProof_G2).sum +
                (List.map
                    (fun SRS_elem =>
                      C (Groth16TypeIII.verificationPairingSRS_G2 stmt check_idx pairing SRS_elem) *
                        Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                    Groth16TypeIII.ListSRSElements_G2).sum))
          (Groth16TypeIII.ListPairings check_idx)).sum =
      0
null :
  ∀ pfs ∈ Groth16TypeIII.Identified_Proof_Elems,
    (List.map (fun SRS_elem => C (prover.1 pfs.1 SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
          Groth16TypeIII.ListSRSElements_G1).sum =
      (List.map (fun SRS_elem => C (prover.2 pfs.2 SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
          Groth16TypeIII.ListSRSElements_G2).sum
t : Polynomial F := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))
⊢ (((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

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
eqns :
  ∀ (check_idx : Groth16TypeIII.EqualityChecks),
    (List.map
          (fun pairing =>
            ((List.map
                    (fun pf_elem =>
                      C (Groth16TypeIII.verificationPairingProof_G1 stmt check_idx pairing pf_elem) *
                        (List.map
                            (fun SRS_elem => C (prover.1 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                            Groth16TypeIII.ListSRSElements_G1).sum)
                    Groth16TypeIII.ListProof_G1).sum +
                (List.map
                    (fun SRS_elem =>
                      C (Groth16TypeIII.verificationPairingSRS_G1 stmt check_idx pairing SRS_elem) *
                        Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                    Groth16TypeIII.ListSRSElements_G1).sum) *
              ((List.map
                    (fun pf_elem =>
                      C (Groth16TypeIII.verificationPairingProof_G2 stmt check_idx pairing pf_elem) *
                        (List.map
                            (fun SRS_elem => C (prover.2 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                            Groth16TypeIII.ListSRSElements_G2).sum)
                    Groth16TypeIII.ListProof_G2).sum +
                (List.map
                    (fun SRS_elem =>
                      C (Groth16TypeIII.verificationPairingSRS_G2 stmt check_idx pairing SRS_elem) *
                        Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                    Groth16TypeIII.ListSRSElements_G2).sum))
          (Groth16TypeIII.ListPairings check_idx)).sum =
      0
null :
  ∀ pfs ∈ Groth16TypeIII.Identified_Proof_Elems,
    (List.map (fun SRS_elem => C (prover.1 pfs.1 SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
          Groth16TypeIII.ListSRSElements_G1).sum =
      (List.map (fun SRS_elem => C (prover.2 pfs.2 SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
          Groth16TypeIII.ListSRSElements_G2).sum
t : Polynomial F := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))
⊢ (((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:
have eqn := eqns ()
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
eqns :
  ∀ (check_idx : Groth16TypeIII.EqualityChecks),
    (List.map
          (fun pairing =>
            ((List.map
                    (fun pf_elem =>
                      C (Groth16TypeIII.verificationPairingProof_G1 stmt check_idx pairing pf_elem) *
                        (List.map
                            (fun SRS_elem => C (prover.1 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                            Groth16TypeIII.ListSRSElements_G1).sum)
                    Groth16TypeIII.ListProof_G1).sum +
                (List.map
                    (fun SRS_elem =>
                      C (Groth16TypeIII.verificationPairingSRS_G1 stmt check_idx pairing SRS_elem) *
                        Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                    Groth16TypeIII.ListSRSElements_G1).sum) *
              ((List.map
                    (fun pf_elem =>
                      C (Groth16TypeIII.verificationPairingProof_G2 stmt check_idx pairing pf_elem) *
                        (List.map
                            (fun SRS_elem => C (prover.2 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                            Groth16TypeIII.ListSRSElements_G2).sum)
                    Groth16TypeIII.ListProof_G2).sum +
                (List.map
                    (fun SRS_elem =>
                      C (Groth16TypeIII.verificationPairingSRS_G2 stmt check_idx pairing SRS_elem) *
                        Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                    Groth16TypeIII.ListSRSElements_G2).sum))
          (Groth16TypeIII.ListPairings check_idx)).sum =
      0
null :
  ∀ pfs ∈ Groth16TypeIII.Identified_Proof_Elems,
    (List.map (fun SRS_elem => C (prover.1 pfs.1 SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
          Groth16TypeIII.ListSRSElements_G1).sum =
      (List.map (fun SRS_elem => C (prover.2 pfs.2 SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
          Groth16TypeIII.ListSRSElements_G2).sum
t : Polynomial F := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))
eqn :
  (List.map
        (fun pairing =>
          ((List.map
                  (fun pf_elem =>
                    C (Groth16TypeIII.verificationPairingProof_G1 stmt () pairing pf_elem) *
                      (List.map
                          (fun SRS_elem => C (prover.1 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                          Groth16TypeIII.ListSRSElements_G1).sum)
                  Groth16TypeIII.ListProof_G1).sum +
              (List.map
                  (fun SRS_elem =>
                    C (Groth16TypeIII.verificationPairingSRS_G1 stmt () pairing SRS_elem) *
                      Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                  Groth16TypeIII.ListSRSElements_G1).sum) *
            ((List.map
                  (fun pf_elem =>
                    C (Groth16TypeIII.verificationPairingProof_G2 stmt () pairing pf_elem) *
                      (List.map
                          (fun SRS_elem => C (prover.2 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                          Groth16TypeIII.ListSRSElements_G2).sum)
                  Groth16TypeIII.ListProof_G2).sum +
              (List.map
                  (fun SRS_elem =>
                    C (Groth16TypeIII.verificationPairingSRS_G2 stmt () pairing SRS_elem) *
                      Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                  Groth16TypeIII.ListSRSElements_G2).sum))
        (Groth16TypeIII.ListPairings ())).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

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
eqns :
  ∀ (check_idx : Groth16TypeIII.EqualityChecks),
    (List.map
          (fun pairing =>
            ((List.map
                    (fun pf_elem =>
                      C (Groth16TypeIII.verificationPairingProof_G1 stmt check_idx pairing pf_elem) *
                        (List.map
                            (fun SRS_elem => C (prover.1 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                            Groth16TypeIII.ListSRSElements_G1).sum)
                    Groth16TypeIII.ListProof_G1).sum +
                (List.map
                    (fun SRS_elem =>
                      C (Groth16TypeIII.verificationPairingSRS_G1 stmt check_idx pairing SRS_elem) *
                        Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                    Groth16TypeIII.ListSRSElements_G1).sum) *
              ((List.map
                    (fun pf_elem =>
                      C (Groth16TypeIII.verificationPairingProof_G2 stmt check_idx pairing pf_elem) *
                        (List.map
                            (fun SRS_elem => C (prover.2 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                            Groth16TypeIII.ListSRSElements_G2).sum)
                    Groth16TypeIII.ListProof_G2).sum +
                (List.map
                    (fun SRS_elem =>
                      C (Groth16TypeIII.verificationPairingSRS_G2 stmt check_idx pairing SRS_elem) *
                        Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                    Groth16TypeIII.ListSRSElements_G2).sum))
          (Groth16TypeIII.ListPairings check_idx)).sum =
      0
null :
  ∀ pfs ∈ Groth16TypeIII.Identified_Proof_Elems,
    (List.map (fun SRS_elem => C (prover.1 pfs.1 SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
          Groth16TypeIII.ListSRSElements_G1).sum =
      (List.map (fun SRS_elem => C (prover.2 pfs.2 SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
          Groth16TypeIII.ListSRSElements_G2).sum
t : Polynomial F := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))
eqn :
  (List.map
        (fun pairing =>
          ((List.map
                  (fun pf_elem =>
                    C (Groth16TypeIII.verificationPairingProof_G1 stmt () pairing pf_elem) *
                      (List.map
                          (fun SRS_elem => C (prover.1 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                          Groth16TypeIII.ListSRSElements_G1).sum)
                  Groth16TypeIII.ListProof_G1).sum +
              (List.map
                  (fun SRS_elem =>
                    C (Groth16TypeIII.verificationPairingSRS_G1 stmt () pairing SRS_elem) *
                      Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                  Groth16TypeIII.ListSRSElements_G1).sum) *
            ((List.map
                  (fun pf_elem =>
                    C (Groth16TypeIII.verificationPairingProof_G2 stmt () pairing pf_elem) *
                      (List.map
                          (fun SRS_elem => C (prover.2 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                          Groth16TypeIII.ListSRSElements_G2).sum)
                  Groth16TypeIII.ListProof_G2).sum +
              (List.map
                  (fun SRS_elem =>
                    C (Groth16TypeIII.verificationPairingSRS_G2 stmt () pairing SRS_elem) *
                      Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                  Groth16TypeIII.ListSRSElements_G2).sum))
        (Groth16TypeIII.ListPairings ())).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:
clear eqns null
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 :
  (List.map
        (fun pairing =>
          ((List.map
                  (fun pf_elem =>
                    C (Groth16TypeIII.verificationPairingProof_G1 stmt () pairing pf_elem) *
                      (List.map
                          (fun SRS_elem => C (prover.1 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                          Groth16TypeIII.ListSRSElements_G1).sum)
                  Groth16TypeIII.ListProof_G1).sum +
              (List.map
                  (fun SRS_elem =>
                    C (Groth16TypeIII.verificationPairingSRS_G1 stmt () pairing SRS_elem) *
                      Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                  Groth16TypeIII.ListSRSElements_G1).sum) *
            ((List.map
                  (fun pf_elem =>
                    C (Groth16TypeIII.verificationPairingProof_G2 stmt () pairing pf_elem) *
                      (List.map
                          (fun SRS_elem => C (prover.2 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                          Groth16TypeIII.ListSRSElements_G2).sum)
                  Groth16TypeIII.ListProof_G2).sum +
              (List.map
                  (fun SRS_elem =>
                    C (Groth16TypeIII.verificationPairingSRS_G2 stmt () pairing SRS_elem) *
                      Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                  Groth16TypeIII.ListSRSElements_G2).sum))
        (Groth16TypeIII.ListPairings ())).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

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 :
  (List.map
        (fun pairing =>
          ((List.map
                  (fun pf_elem =>
                    C (Groth16TypeIII.verificationPairingProof_G1 stmt () pairing pf_elem) *
                      (List.map
                          (fun SRS_elem => C (prover.1 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                          Groth16TypeIII.ListSRSElements_G1).sum)
                  Groth16TypeIII.ListProof_G1).sum +
              (List.map
                  (fun SRS_elem =>
                    C (Groth16TypeIII.verificationPairingSRS_G1 stmt () pairing SRS_elem) *
                      Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                  Groth16TypeIII.ListSRSElements_G1).sum) *
            ((List.map
                  (fun pf_elem =>
                    C (Groth16TypeIII.verificationPairingProof_G2 stmt () pairing pf_elem) *
                      (List.map
                          (fun SRS_elem => C (prover.2 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                          Groth16TypeIII.ListSRSElements_G2).sum)
                  Groth16TypeIII.ListProof_G2).sum +
              (List.map
                  (fun SRS_elem =>
                    C (Groth16TypeIII.verificationPairingSRS_G2 stmt () pairing SRS_elem) *
                      Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                  Groth16TypeIII.ListSRSElements_G2).sum))
        (Groth16TypeIII.ListPairings ())).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:
suffices
    ((List.sum (List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)))
    + (List.sum (List.map (fun i => Polynomial.C (prover.fst Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) * u_wit i) (List.finRange n_wit))))
    *
    ((List.sum (List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)))
    + (List.sum (List.map (fun i => Polynomial.C (prover.fst Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) * v_wit i) (List.finRange n_wit))))
    =
    ((List.sum (List.map (fun i => Polynomial.C (stmt i) * w_stmt i) (List.finRange n_stmt)))
    + (List.sum (List.map (fun i => Polynomial.C (prover.fst Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) * w_wit i) (List.finRange n_wit))))
    +
    List.sum (List.map (fun x : Fin (n_var - 1) => Polynomial.C (prover.fst Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) * (Polynomial.X ^ (x : ℕ) * t)) (List.finRange (n_var - 1))) by
  rw [<-sub_eq_iff_eq_add'] at this
  have h := congr_arg (fun x => x %ₘ t) this
  simp only at h
  simp
  rw [h]
  clear this h
  simp only [mul_comm _ (t), <-mul_assoc]
  simp only [mul_assoc, List.sum_map_mul_right, List.sum_map_mul_left]
  apply Polynomial.self_mul_modByMonic
  apply Polynomial.monic_prod_of_monic
  intro i hi
  exact Polynomial.monic_X_sub_C (r i)
  done
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 :
  (List.map
        (fun pairing =>
          ((List.map
                  (fun pf_elem =>
                    C (Groth16TypeIII.verificationPairingProof_G1 stmt () pairing pf_elem) *
                      (List.map
                          (fun SRS_elem => C (prover.1 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                          Groth16TypeIII.ListSRSElements_G1).sum)
                  Groth16TypeIII.ListProof_G1).sum +
              (List.map
                  (fun SRS_elem =>
                    C (Groth16TypeIII.verificationPairingSRS_G1 stmt () pairing SRS_elem) *
                      Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                  Groth16TypeIII.ListSRSElements_G1).sum) *
            ((List.map
                  (fun pf_elem =>
                    C (Groth16TypeIII.verificationPairingProof_G2 stmt () pairing pf_elem) *
                      (List.map
                          (fun SRS_elem => C (prover.2 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                          Groth16TypeIII.ListSRSElements_G2).sum)
                  Groth16TypeIII.ListProof_G2).sum +
              (List.map
                  (fun SRS_elem =>
                    C (Groth16TypeIII.verificationPairingSRS_G2 stmt () pairing SRS_elem) *
                      Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                  Groth16TypeIII.ListSRSElements_G2).sum))
        (Groth16TypeIII.ListPairings ())).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

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 :
  (List.map
        (fun pairing =>
          ((List.map
                  (fun pf_elem =>
                    C (Groth16TypeIII.verificationPairingProof_G1 stmt () pairing pf_elem) *
                      (List.map
                          (fun SRS_elem => C (prover.1 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                          Groth16TypeIII.ListSRSElements_G1).sum)
                  Groth16TypeIII.ListProof_G1).sum +
              (List.map
                  (fun SRS_elem =>
                    C (Groth16TypeIII.verificationPairingSRS_G1 stmt () pairing SRS_elem) *
                      Groth16TypeIII.SRSElementValue_G1 SRS_elem)
                  Groth16TypeIII.ListSRSElements_G1).sum) *
            ((List.map
                  (fun pf_elem =>
                    C (Groth16TypeIII.verificationPairingProof_G2 stmt () pairing pf_elem) *
                      (List.map
                          (fun SRS_elem => C (prover.2 pf_elem SRS_elem) * Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                          Groth16TypeIII.ListSRSElements_G2).sum)
                  Groth16TypeIII.ListProof_G2).sum +
              (List.map
                  (fun SRS_elem =>
                    C (Groth16TypeIII.verificationPairingSRS_G2 stmt () pairing SRS_elem) *
                      Groth16TypeIII.SRSElementValue_G2 SRS_elem)
                  Groth16TypeIII.ListSRSElements_G2).sum))
        (Groth16TypeIII.ListPairings ())).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_rw [Groth16TypeIII] at eqn
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 :
  (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.γ) * X (some Vars.δ) * X (some Vars.α)
                              | SRS_Elements_G1_Idx.β => X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β)
                              | SRS_Elements_G1_Idx.δ => X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ)
                              | SRS_Elements_G1_Idx.x_pow i => X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑i
                              | SRS_Elements_G1_Idx.x_pow_times_t i =>
                                X (some Vars.γ) * 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.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt i) +
                                    X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt i) +
                                  X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt i)
                              | SRS_Elements_G1_Idx.q i =>
                                X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit i) +
                                    X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit i) +
                                  X (some 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 (some Vars.γ) * X (some Vars.δ) * X (some Vars.α)
                      | SRS_Elements_G1_Idx.β => X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β)
                      | SRS_Elements_G1_Idx.δ => X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ)
                      | SRS_Elements_G1_Idx.x_pow i => X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑i
                      | SRS_Elements_G1_Idx.x_pow_times_t i =>
                        X (some Vars.γ) * 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.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt i) +
                            X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt i) +
                          X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt i)
                      | SRS_Elements_G1_Idx.q i =>
                        X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit i) +
                            X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit i) +
                          X (some 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 (some Vars.γ) * X (some Vars.δ) * X (some Vars.β)
                              | SRS_Elements_G2_Idx.γ => X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ)
                              | SRS_Elements_G2_Idx.δ => X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ)
                              | SRS_Elements_G2_Idx.x_pow i => X (some Vars.γ) * X (some Vars.δ) * 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.γ) * X (some Vars.δ) * X (some Vars.β)
                      | SRS_Elements_G2_Idx.γ => X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ)
                      | SRS_Elements_G2_Idx.δ => X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ)
                      | SRS_Elements_G2_Idx.x_pow i => X (some Vars.γ) * X (some Vars.δ) * 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

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 :
  (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.γ) * X (some Vars.δ) * X (some Vars.α)
                              | SRS_Elements_G1_Idx.β => X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β)
                              | SRS_Elements_G1_Idx.δ => X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ)
                              | SRS_Elements_G1_Idx.x_pow i => X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑i
                              | SRS_Elements_G1_Idx.x_pow_times_t i =>
                                X (some Vars.γ) * 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.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt i) +
                                    X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt i) +
                                  X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt i)
                              | SRS_Elements_G1_Idx.q i =>
                                X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit i) +
                                    X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit i) +
                                  X (some 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 (some Vars.γ) * X (some Vars.δ) * X (some Vars.α)
                      | SRS_Elements_G1_Idx.β => X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β)
                      | SRS_Elements_G1_Idx.δ => X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ)
                      | SRS_Elements_G1_Idx.x_pow i => X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑i
                      | SRS_Elements_G1_Idx.x_pow_times_t i =>
                        X (some Vars.γ) * 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.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt i) +
                            X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt i) +
                          X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt i)
                      | SRS_Elements_G1_Idx.q i =>
                        X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit i) +
                            X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit i) +
                          X (some 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 (some Vars.γ) * X (some Vars.δ) * X (some Vars.β)
                              | SRS_Elements_G2_Idx.γ => X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ)
                              | SRS_Elements_G2_Idx.δ => X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ)
                              | SRS_Elements_G2_Idx.x_pow i => X (some Vars.γ) * X (some Vars.δ) * 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.γ) * X (some Vars.δ) * X (some Vars.β)
                      | SRS_Elements_G2_Idx.γ => X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ)
                      | SRS_Elements_G2_Idx.δ => X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ)
                      | SRS_Elements_G2_Idx.x_pow i => X (some Vars.γ) * X (some Vars.δ) * 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 [
  Function.comp, List.sum_map_zero,
  mul_add, add_mul, List.sum_map_add,
  List.map_append, List.map_map, List.map_cons, List.map_nil,
  ] at eqn
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 :
  [[C 1 *
                    ([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) ++
                            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 *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum,
                  C 0 *
                    ([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) ++
                            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 *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum].sum *
              [C (-1) *
                    ([C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                            [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.δ) *
                              (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        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].sum +
            ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                            [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                          [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var) ++
                      List.map
                        (fun x =>
                          C 0 *
                            (X (some Vars.γ) * X none ^ ↑x *
                              (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                        (List.finRange (n_var - 1)) ++
                    List.map
                      (fun x =>
                        C 0 * (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                            C 0 * (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                          C 0 * (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                      (List.finRange n_stmt) ++
                  List.map
                    (fun x =>
                      C 0 * (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                          C 0 * (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                        C 0 * (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                    (List.finRange n_wit)).sum *
              [C (-1) *
                    ([C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                            [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.δ) *
                              (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        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].sum +
          ([C 1 *
                    ([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) ++
                            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 *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum,
                  C 0 *
                    ([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) ++
                            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 *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum].sum *
              ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                      [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                    [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                  List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                    (List.finRange n_var)).sum +
            ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                            [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                          [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var) ++
                      List.map
                        (fun x =>
                          C 0 *
                            (X (some Vars.γ) * X none ^ ↑x *
                              (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                        (List.finRange (n_var - 1)) ++
                    List.map
                      (fun x =>
                        C 0 * (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                            C 0 * (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                          C 0 * (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                      (List.finRange n_stmt) ++
                  List.map
                    (fun x =>
                      C 0 * (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                          C 0 * (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                        C 0 * (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                    (List.finRange n_wit)).sum *
              ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                      [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                    [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                  List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                    (List.finRange n_var)).sum),
        [C 0 *
                    ([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) ++
                            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 *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum,
                  C 0 *
                    ([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) ++
                            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 *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum].sum *
              [C 0 *
                    ([C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                            [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.δ) *
                              (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        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].sum +
            ([C 1 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                            [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                          [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var) ++
                      List.map
                        (fun x =>
                          C 0 *
                            (X (some Vars.γ) * X none ^ ↑x *
                              (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                        (List.finRange (n_var - 1)) ++
                    List.map
                      (fun x =>
                        C 0 * (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                            C 0 * (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                          C 0 * (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                      (List.finRange n_stmt) ++
                  List.map
                    (fun x =>
                      C 0 * (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                          C 0 * (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                        C 0 * (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                    (List.finRange n_wit)).sum *
              [C 0 *
                    ([C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                            [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.δ) *
                              (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        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].sum +
          ([C 0 *
                    ([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) ++
                            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 *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum,
                  C 0 *
                    ([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) ++
                            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 *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum].sum *
              ([C 1 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                      [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                    [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                  List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                    (List.finRange n_var)).sum +
            ([C 1 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                            [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                          [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var) ++
                      List.map
                        (fun x =>
                          C 0 *
                            (X (some Vars.γ) * X none ^ ↑x *
                              (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                        (List.finRange (n_var - 1)) ++
                    List.map
                      (fun x =>
                        C 0 * (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                            C 0 * (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                          C 0 * (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                      (List.finRange n_stmt) ++
                  List.map
                    (fun x =>
                      C 0 * (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                          C 0 * (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                        C 0 * (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                    (List.finRange n_wit)).sum *
              ([C 1 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                      [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                    [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                  List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                    (List.finRange n_var)).sum),
        [C 0 *
                    ([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) ++
                            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 *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum,
                  C 0 *
                    ([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) ++
                            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 *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum].sum *
              [C 0 *
                    ([C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                            [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.δ) *
                              (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        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].sum +
            ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                            [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                          [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var) ++
                      List.map
                        (fun x =>
                          C 0 *
                            (X (some Vars.γ) * X none ^ ↑x *
                              (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                        (List.finRange (n_var - 1)) ++
                    List.map
                      (fun x =>
                        C (stmt x) * (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                            C (stmt x) *
                              (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                          C (stmt x) * (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                      (List.finRange n_stmt) ++
                  List.map
                    (fun x =>
                      C 0 * (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                          C 0 * (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                        C 0 * (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                    (List.finRange n_wit)).sum *
              [C 0 *
                    ([C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                            [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.δ) *
                              (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        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].sum +
          ([C 0 *
                    ([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) ++
                            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 *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum,
                  C 0 *
                    ([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) ++
                            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 *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum].sum *
              ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                      [C 1 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                    [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                  List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                    (List.finRange n_var)).sum +
            ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                            [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                          [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var) ++
                      List.map
                        (fun x =>
                          C 0 *
                            (X (some Vars.γ) * X none ^ ↑x *
                              (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                        (List.finRange (n_var - 1)) ++
                    List.map
                      (fun x =>
                        C (stmt x) * (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                            C (stmt x) *
                              (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                          C (stmt x) * (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                      (List.finRange n_stmt) ++
                  List.map
                    (fun x =>
                      C 0 * (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                          C 0 * (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                        C 0 * (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                    (List.finRange n_wit)).sum *
              ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                      [C 1 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                    [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                  List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                    (List.finRange n_var)).sum),
        [C 0 *
                    ([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) ++
                            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 *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum,
                  C 1 *
                    ([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) ++
                            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 *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum].sum *
              [C 0 *
                    ([C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                            [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.δ) *
                              (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        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].sum +
            ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                            [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                          [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var) ++
                      List.map
                        (fun x =>
                          C 0 *
                            (X (some Vars.γ) * X none ^ ↑x *
                              (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                        (List.finRange (n_var - 1)) ++
                    List.map
                      (fun x =>
                        C 0 * (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                            C 0 * (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                          C 0 * (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                      (List.finRange n_stmt) ++
                  List.map
                    (fun x =>
                      C 0 * (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                          C 0 * (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                        C 0 * (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                    (List.finRange n_wit)).sum *
              [C 0 *
                    ([C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) *
                                (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                            [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.δ) *
                              (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        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].sum +
          ([C 0 *
                    ([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) ++
                            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 *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum,
                  C 1 *
                    ([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) ++
                            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 *
                                    (to_MvPolynomial_Option Vars)
                                      (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                              (List.finRange (n_var - 1)) ++
                          List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                    (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                  (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt) ++
                        List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                  (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                          (List.finRange n_wit)).sum].sum *
              ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                      [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                    [C 1 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                  List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                    (List.finRange n_var)).sum +
            ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.α))] ++
                            [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                          [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                        List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                          (List.finRange n_var) ++
                      List.map
                        (fun x =>
                          C 0 *
                            (X (some Vars.γ) * X none ^ ↑x *
                              (to_MvPolynomial_Option Vars) (∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i)))))
                        (List.finRange (n_var - 1)) ++
                    List.map
                      (fun x =>
                        C 0 * (X (some Vars.β) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (u_stmt x)) +
                            C 0 * (X (some Vars.α) * X (some Vars.δ) * (to_MvPolynomial_Option Vars) (v_stmt x)) +
                          C 0 * (X (some Vars.δ) * (to_MvPolynomial_Option Vars) (w_stmt x)))
                      (List.finRange n_stmt) ++
                  List.map
                    (fun x =>
                      C 0 * (X (some Vars.β) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (u_wit x)) +
                          C 0 * (X (some Vars.α) * X (some Vars.γ) * (to_MvPolynomial_Option Vars) (v_wit x)) +
                        C 0 * (X (some Vars.γ) * (to_MvPolynomial_Option Vars) (w_wit x)))
                    (List.finRange n_wit)).sum *
              ([C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.β))] ++
                      [C 0 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.γ))] ++
                    [C 1 * (X (some Vars.γ) * X (some Vars.δ) * X (some Vars.δ))] ++
                  List.map (fun x => C 0 * (X (some Vars.γ) * X (some Vars.δ) * X none ^ ↑x))
                    (List.finRange n_var)).sum)].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