Theorem: Groth16TypeIII.is_sound_quotient_version (FormalSnarksProject/Other/Groth16TypeIIIQuotientVersion.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
t : Polynomial F
⊢ soundness F Groth16TypeIII (Fin n_wit → F)
    (fun stmt wit =>
      (((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
t : Polynomial 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 =>
          (((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
t : Polynomial 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 =>
          (((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
t : Polynomial 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
⊢ (((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
t : Polynomial 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
⊢ (((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
t : Polynomial 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
⊢ (((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
t : Polynomial 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
⊢ (((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
t : Polynomial 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
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
t : Polynomial 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
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
t : Polynomial F
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
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
t : Polynomial F
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
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:
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
t : Polynomial F
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
eqn :
  (List.map
        (fun pairing =>
          ((List.map
                  (fun pf_elem =>
                    C
                        (match pairing with
                        | PairingsIdx.ab =>
                          match pf_elem with
                          | Proof_G1_Idx.A => 1
                          | Proof_G1_Idx.C => 0
                        | PairingsIdx.αβ =>
                          match pf_elem with
                          | Proof_G1_Idx.A => 0
                          | Proof_G1_Idx.C => 0
                        | PairingsIdx.stmtγ =>
                          match pf_elem with
                          | Proof_G1_Idx.A => 0
                          | Proof_G1_Idx.C => 0
                        | PairingsIdx.cδ =>
                          match pf_elem with
                          | Proof_G1_Idx.A => 0
                          | Proof_G1_Idx.C => 1) *
                      (List.map
                          (fun SRS_elem =>
                            C (prover.1 pf_elem SRS_elem) *
                              match SRS_elem with
                              | SRS_Elements_G1_Idx.α => X Vars_γ * X Vars_δ * X Vars_α
                              | SRS_Elements_G1_Idx.β => X Vars_γ * X Vars_δ * X Vars_β
                              | SRS_Elements_G1_Idx.δ => X Vars_γ * X Vars_δ * X Vars_δ
                              | SRS_Elements_G1_Idx.x_pow i => X Vars_γ * X Vars_δ * X Vars_x ^ ↑i
                              | SRS_Elements_G1_Idx.x_pow_times_t i =>
                                X Vars_γ * X Vars_x ^ ↑i * (to_MvPolynomial_Option Vars) t
                              | SRS_Elements_G1_Idx.y i =>
                                X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i) +
                                    X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i) +
                                  X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)
                              | SRS_Elements_G1_Idx.q i =>
                                X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i) +
                                    X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i) +
                                  X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i))
                          ([SRS_Elements_G1_Idx.α] ++ [SRS_Elements_G1_Idx.β] ++ [SRS_Elements_G1_Idx.δ] ++
                                  List.map (fun i => SRS_Elements_G1_Idx.x_pow i) (List.finRange n_var) ++
                                List.map (fun i => SRS_Elements_G1_Idx.x_pow_times_t i) (List.finRange (n_var - 1)) ++
                              List.map (fun i => SRS_Elements_G1_Idx.y i) (List.finRange n_stmt) ++
                            List.map (fun i => SRS_Elements_G1_Idx.q i) (List.finRange n_wit))).sum)
                  [Proof_G1_Idx.A, Proof_G1_Idx.C]).sum +
              (List.map
                  (fun SRS_elem =>
                    C
                        (match pairing with
                        | PairingsIdx.ab =>
                          match SRS_elem with
                          | SRS_Elements_G1_Idx.α => 0
                          | SRS_Elements_G1_Idx.β => 0
                          | SRS_Elements_G1_Idx.δ => 0
                          | SRS_Elements_G1_Idx.x_pow a => 0
                          | SRS_Elements_G1_Idx.x_pow_times_t a => 0
                          | SRS_Elements_G1_Idx.y a => 0
                          | SRS_Elements_G1_Idx.q a => 0
                        | PairingsIdx.αβ =>
                          match SRS_elem with
                          | SRS_Elements_G1_Idx.α => 1
                          | SRS_Elements_G1_Idx.β => 0
                          | SRS_Elements_G1_Idx.δ => 0
                          | SRS_Elements_G1_Idx.x_pow a => 0
                          | SRS_Elements_G1_Idx.x_pow_times_t a => 0
                          | SRS_Elements_G1_Idx.y a => 0
                          | SRS_Elements_G1_Idx.q a => 0
                        | PairingsIdx.stmtγ =>
                          match SRS_elem with
                          | SRS_Elements_G1_Idx.α => 0
                          | SRS_Elements_G1_Idx.β => 0
                          | SRS_Elements_G1_Idx.δ => 0
                          | SRS_Elements_G1_Idx.x_pow a => 0
                          | SRS_Elements_G1_Idx.x_pow_times_t a => 0
                          | SRS_Elements_G1_Idx.y i => stmt i
                          | SRS_Elements_G1_Idx.q a => 0
                        | PairingsIdx.cδ =>
                          match SRS_elem with
                          | SRS_Elements_G1_Idx.α => 0
                          | SRS_Elements_G1_Idx.β => 0
                          | SRS_Elements_G1_Idx.δ => 0
                          | SRS_Elements_G1_Idx.x_pow a => 0
                          | SRS_Elements_G1_Idx.x_pow_times_t a => 0
                          | SRS_Elements_G1_Idx.y a => 0
                          | SRS_Elements_G1_Idx.q a => 0) *
                      match SRS_elem with
                      | SRS_Elements_G1_Idx.α => X Vars_γ * X Vars_δ * X Vars_α
                      | SRS_Elements_G1_Idx.β => X Vars_γ * X Vars_δ * X Vars_β
                      | SRS_Elements_G1_Idx.δ => X Vars_γ * X Vars_δ * X Vars_δ
                      | SRS_Elements_G1_Idx.x_pow i => X Vars_γ * X Vars_δ * X Vars_x ^ ↑i
                      | SRS_Elements_G1_Idx.x_pow_times_t i =>
                        X Vars_γ * X Vars_x ^ ↑i * (to_MvPolynomial_Option Vars) t
                      | SRS_Elements_G1_Idx.y i =>
                        X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i) +
                            X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i) +
                          X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)
                      | SRS_Elements_G1_Idx.q i =>
                        X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i) +
                            X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i) +
                          X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i))
                  ([SRS_Elements_G1_Idx.α] ++ [SRS_Elements_G1_Idx.β] ++ [SRS_Elements_G1_Idx.δ] ++
                          List.map (fun i => SRS_Elements_G1_Idx.x_pow i) (List.finRange n_var) ++
                        List.map (fun i => SRS_Elements_G1_Idx.x_pow_times_t i) (List.finRange (n_var - 1)) ++
                      List.map (fun i => SRS_Elements_G1_Idx.y i) (List.finRange n_stmt) ++
                    List.map (fun i => SRS_Elements_G1_Idx.q i) (List.finRange n_wit))).sum) *
            ((List.map
                  (fun pf_elem =>
                    C
                        (match pairing with
                        | PairingsIdx.ab => -1
                        | PairingsIdx.αβ => 0
                        | PairingsIdx.stmtγ => 0
                        | PairingsIdx.cδ => 0) *
                      (List.map
                          (fun SRS_elem =>
                            C (prover.2 pf_elem SRS_elem) *
                              match SRS_elem with
                              | SRS_Elements_G2_Idx.β => X Vars_γ * X Vars_δ * X Vars_β
                              | SRS_Elements_G2_Idx.γ => X Vars_γ * X Vars_δ * X Vars_γ
                              | SRS_Elements_G2_Idx.δ => X Vars_γ * X Vars_δ * X Vars_δ
                              | SRS_Elements_G2_Idx.x_pow i => X Vars_γ * X Vars_δ * X Vars_x ^ ↑i)
                          ([SRS_Elements_G2_Idx.β] ++ [SRS_Elements_G2_Idx.γ] ++ [SRS_Elements_G2_Idx.δ] ++
                            List.map (fun i => SRS_Elements_G2_Idx.x_pow i) (List.finRange n_var))).sum)
                  [Proof_G2_Idx.B]).sum +
              (List.map
                  (fun SRS_elem =>
                    C
                        (match pairing with
                        | PairingsIdx.ab =>
                          match SRS_elem with
                          | SRS_Elements_G2_Idx.β => 0
                          | SRS_Elements_G2_Idx.γ => 0
                          | SRS_Elements_G2_Idx.δ => 0
                          | SRS_Elements_G2_Idx.x_pow a => 0
                        | PairingsIdx.αβ =>
                          match SRS_elem with
                          | SRS_Elements_G2_Idx.β => 1
                          | SRS_Elements_G2_Idx.γ => 0
                          | SRS_Elements_G2_Idx.δ => 0
                          | SRS_Elements_G2_Idx.x_pow a => 0
                        | PairingsIdx.stmtγ =>
                          match SRS_elem with
                          | SRS_Elements_G2_Idx.β => 0
                          | SRS_Elements_G2_Idx.γ => 1
                          | SRS_Elements_G2_Idx.δ => 0
                          | SRS_Elements_G2_Idx.x_pow a => 0
                        | PairingsIdx.cδ =>
                          match SRS_elem with
                          | SRS_Elements_G2_Idx.β => 0
                          | SRS_Elements_G2_Idx.γ => 0
                          | SRS_Elements_G2_Idx.δ => 1
                          | SRS_Elements_G2_Idx.x_pow a => 0) *
                      match SRS_elem with
                      | SRS_Elements_G2_Idx.β => X Vars_γ * X Vars_δ * X Vars_β
                      | SRS_Elements_G2_Idx.γ => X Vars_γ * X Vars_δ * X Vars_γ
                      | SRS_Elements_G2_Idx.δ => X Vars_γ * X Vars_δ * X Vars_δ
                      | SRS_Elements_G2_Idx.x_pow i => X Vars_γ * X Vars_δ * X Vars_x ^ ↑i)
                  ([SRS_Elements_G2_Idx.β] ++ [SRS_Elements_G2_Idx.γ] ++ [SRS_Elements_G2_Idx.δ] ++
                    List.map (fun i => SRS_Elements_G2_Idx.x_pow i) (List.finRange n_var))).sum))
        [PairingsIdx.ab, PairingsIdx.αβ, PairingsIdx.stmtγ, PairingsIdx.cδ]).sum =
    0
⊢ (((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
              (List.map
                  (fun i =>
                    Polynomial.C ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover i) *
                      u_wit i)
                  (List.finRange n_wit)).sum) *
            ((List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)).sum +
              (List.map
                  (fun i =>
                    Polynomial.C ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover i) *
                      v_wit i)
                  (List.finRange n_wit)).sum) -
          ((List.map (fun i => Polynomial.C (stmt i) * w_stmt i) (List.finRange n_stmt)).sum +
            (List.map
                (fun i =>
                  Polynomial.C ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover i) * w_wit i)
                (List.finRange n_wit)).sum)).modByMonic
      t =
    0

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

State before:
case intro
F : Type
inst✝ : Field F
n_stmt n_wit n_var : ℕ
u_stmt : Fin n_stmt → Polynomial F
u_wit : Fin n_wit → Polynomial F
v_stmt : Fin n_stmt → Polynomial F
v_wit : Fin n_wit → Polynomial F
w_stmt : Fin n_stmt → Polynomial F
w_wit : Fin n_wit → Polynomial F
t : Polynomial F
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
eqn :
  (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) +
          (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) +
            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) +
              ((List.map
                    (fun x =>
                      C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) * (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                    (List.finRange n_var)).sum +
                ((List.map
                      (fun x =>
                        C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                          (X Vars_γ * X Vars_x ^ ↑x * (to_MvPolynomial_Option Vars) t))
                      (List.finRange (n_var - 1))).sum +
                  ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y x)) *
                            (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt x) +
                                X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt x) +
                              X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt x)))
                        (List.finRange n_stmt)).sum +
                    (List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q x)) *
                            (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit x) +
                                X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit x) +
                              X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit x)))
                        (List.finRange n_wit)).sum)))))) *
        (-(List.map
                    (fun x =>
                      C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) * (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                    (List.finRange n_var)).sum +
              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β))) +
      (X Vars_γ * X Vars_δ * X Vars_α * (X Vars_γ * X Vars_δ * X Vars_β) +
        ((List.map
                (fun x =>
                  C (stmt x) *
                    (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt x) +
                        X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt x) +
                      X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt x)))
                (List.finRange n_stmt)).sum *
            (X Vars_γ * X Vars_δ * X Vars_γ) +
          (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) +
              (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) +
                (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) +
                  ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x)) *
                            (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                        (List.finRange n_var)).sum +
                    ((List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                              (X Vars_γ * X Vars_x ^ ↑x * (to_MvPolynomial_Option Vars) t))
                          (List.finRange (n_var - 1))).sum +
                      ((List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y x)) *
                                (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt x) +
                                    X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt x) +
                                  X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt x)))
                            (List.finRange n_stmt)).sum +
                        (List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q x)) *
                                (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit x) +
                                    X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit x) +
                                  X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit x)))
                            (List.finRange n_wit)).sum)))))) *
            (X Vars_γ * X Vars_δ * X Vars_δ))) =
    0
⊢ (((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
              (List.map
                  (fun i =>
                    Polynomial.C ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover i) *
                      u_wit i)
                  (List.finRange n_wit)).sum) *
            ((List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)).sum +
              (List.map
                  (fun i =>
                    Polynomial.C ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover i) *
                      v_wit i)
                  (List.finRange n_wit)).sum) -
          ((List.map (fun i => Polynomial.C (stmt i) * w_stmt i) (List.finRange n_stmt)).sum +
            (List.map
                (fun i =>
                  Polynomial.C ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover i) * w_wit i)
                (List.finRange n_wit)).sum)).modByMonic
      t =
    0
Tactic:
simp only [mul_add, add_mul, List.sum_map_add] 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
t : Polynomial F
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
eqn :
  C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) *
                -(List.map
                      (fun x =>
                        C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                          (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                      (List.finRange n_var)).sum +
              (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) *
                  -(List.map
                        (fun x =>
                          C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                            (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                        (List.finRange n_var)).sum +
                (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) *
                    -(List.map
                          (fun x =>
                            C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                              (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                          (List.finRange n_var)).sum +
                  ((List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                              (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                          (List.finRange n_var)).sum *
                      -(List.map
                            (fun x =>
                              C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                            (List.finRange n_var)).sum +
                    ((List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                (X Vars_γ * X Vars_x ^ ↑x * (to_MvPolynomial_Option Vars) t))
                            (List.finRange (n_var - 1))).sum *
                        -(List.map
                              (fun x =>
                                C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                  (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                              (List.finRange n_var)).sum +
                      ((List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                      (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i)))
                                  (List.finRange n_stmt)).sum *
                              -(List.map
                                    (fun x =>
                                      C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                        (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                                    (List.finRange n_var)).sum +
                            (List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                      (X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i)))
                                  (List.finRange n_stmt)).sum *
                              -(List.map
                                    (fun x =>
                                      C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                        (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                                    (List.finRange n_var)).sum +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                    (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                                (List.finRange n_stmt)).sum *
                            -(List.map
                                  (fun x =>
                                    C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                      (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                                  (List.finRange n_var)).sum +
                        ((List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                      (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i)))
                                  (List.finRange n_wit)).sum *
                              -(List.map
                                    (fun x =>
                                      C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                        (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                                    (List.finRange n_var)).sum +
                            (List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                      (X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i)))
                                  (List.finRange n_wit)).sum *
                              -(List.map
                                    (fun x =>
                                      C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                        (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                                    (List.finRange n_var)).sum +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                    (X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i)))
                                (List.finRange n_wit)).sum *
                            -(List.map
                                  (fun x =>
                                    C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                      (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                                  (List.finRange n_var)).sum)))))) +
            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) *
                -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
              (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) *
                  -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) *
                    -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                  ((List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                              (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                          (List.finRange n_var)).sum *
                      -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                    ((List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                (X Vars_γ * X Vars_x ^ ↑x * (to_MvPolynomial_Option Vars) t))
                            (List.finRange (n_var - 1))).sum *
                        -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                      ((List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                      (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i)))
                                  (List.finRange n_stmt)).sum *
                              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                            (List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                      (X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i)))
                                  (List.finRange n_stmt)).sum *
                              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                    (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                                (List.finRange n_stmt)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                        ((List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                      (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i)))
                                  (List.finRange n_wit)).sum *
                              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                            (List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                      (X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i)))
                                  (List.finRange n_wit)).sum *
                              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                    (X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i)))
                                (List.finRange n_wit)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                                (X Vars_γ * X Vars_δ * X Vars_δ))))))))) +
          (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) *
              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) *
                -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
              (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) *
                  -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                            (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                        (List.finRange n_var)).sum *
                    -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                  ((List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                              (X Vars_γ * X Vars_x ^ ↑x * (to_MvPolynomial_Option Vars) t))
                          (List.finRange (n_var - 1))).sum *
                      -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                    ((List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                    (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i)))
                                (List.finRange n_stmt)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                    (X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i)))
                                (List.finRange n_stmt)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                  (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                              (List.finRange n_stmt)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                      ((List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                    (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i)))
                                (List.finRange n_wit)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                    (X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i)))
                                (List.finRange n_wit)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                  (X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i)))
                              (List.finRange n_wit)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                              (X Vars_γ * X Vars_δ * X Vars_γ))))))))) +
        (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) *
            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
          (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) *
              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) *
                -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
              ((List.map
                      (fun x =>
                        C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                          (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                      (List.finRange n_var)).sum *
                  -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                            (X Vars_γ * X Vars_x ^ ↑x * (to_MvPolynomial_Option Vars) t))
                        (List.finRange (n_var - 1))).sum *
                    -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                  ((List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                  (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i)))
                              (List.finRange n_stmt)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                  (X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i)))
                              (List.finRange n_stmt)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                      (List.map
                            (fun i =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                            (List.finRange n_stmt)).sum *
                        -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                    ((List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                  (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i)))
                              (List.finRange n_wit)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                  (X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i)))
                              (List.finRange n_wit)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                      (List.map
                            (fun i =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                (X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i)))
                            (List.finRange n_wit)).sum *
                        -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β))))))))) +
      (X Vars_γ * X Vars_δ * X Vars_α * (X Vars_γ * X Vars_δ * X Vars_β) +
        ((List.map (fun i => C (stmt i) * (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i)))
                    (List.finRange n_stmt)).sum *
                (X Vars_γ * X Vars_δ * X Vars_γ) +
              (List.map (fun i => C (stmt i) * (X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i)))
                    (List.finRange n_stmt)).sum *
                (X Vars_γ * X Vars_δ * X Vars_γ) +
            (List.map (fun i => C (stmt i) * (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                  (List.finRange n_stmt)).sum *
              (X Vars_γ * X Vars_δ * X Vars_γ) +
          (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) *
              (X Vars_γ * X Vars_δ * X Vars_δ) +
            (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) *
                (X Vars_γ * X Vars_δ * X Vars_δ) +
              (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) *
                  (X Vars_γ * X Vars_δ * X Vars_δ) +
                ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x)) *
                            (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                        (List.finRange n_var)).sum *
                    (X Vars_γ * X Vars_δ * X Vars_δ) +
                  ((List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                              (X Vars_γ * X Vars_x ^ ↑x * (to_MvPolynomial_Option Vars) t))
                          (List.finRange (n_var - 1))).sum *
                      (X Vars_γ * X Vars_δ * X Vars_δ) +
                    ((List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y i)) *
                                    (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i)))
                                (List.finRange n_stmt)).sum *
                            (X Vars_γ * X Vars_δ * X Vars_δ) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y i)) *
                                    (X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i)))
                                (List.finRange n_stmt)).sum *
                            (X Vars_γ * X Vars_δ * X Vars_δ) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y i)) *
                                  (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                              (List.finRange n_stmt)).sum *
                          (X Vars_γ * X Vars_δ * X Vars_δ) +
                      ((List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) *
                                    (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i)))
                                (List.finRange n_wit)).sum *
                            (X Vars_γ * X Vars_δ * X Vars_δ) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) *
                                    (X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i)))
                                (List.finRange n_wit)).sum *
                            (X Vars_γ * X Vars_δ * X Vars_δ) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) *
                                  (X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i)))
                              (List.finRange n_wit)).sum *
                          (X Vars_γ * X Vars_δ * X Vars_δ)))))))))) =
    0
⊢ (((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
              (List.map
                  (fun i =>
                    Polynomial.C ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover i) *
                      u_wit i)
                  (List.finRange n_wit)).sum) *
            ((List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)).sum +
              (List.map
                  (fun i =>
                    Polynomial.C ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover i) *
                      v_wit i)
                  (List.finRange n_wit)).sum) -
          ((List.map (fun i => Polynomial.C (stmt i) * w_stmt i) (List.finRange n_stmt)).sum +
            (List.map
                (fun i =>
                  Polynomial.C ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover i) * w_wit i)
                (List.finRange n_wit)).sum)).modByMonic
      t =
    0

State before:
case intro
F : Type
inst✝ : Field F
n_stmt n_wit n_var : ℕ
u_stmt : Fin n_stmt → Polynomial F
u_wit : Fin n_wit → Polynomial F
v_stmt : Fin n_stmt → Polynomial F
v_wit : Fin n_wit → Polynomial F
w_stmt : Fin n_stmt → Polynomial F
w_wit : Fin n_wit → Polynomial F
t : Polynomial F
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
eqn :
  C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) *
                -(List.map
                      (fun x =>
                        C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                          (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                      (List.finRange n_var)).sum +
              (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) *
                  -(List.map
                        (fun x =>
                          C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                            (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                        (List.finRange n_var)).sum +
                (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) *
                    -(List.map
                          (fun x =>
                            C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                              (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                          (List.finRange n_var)).sum +
                  ((List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                              (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                          (List.finRange n_var)).sum *
                      -(List.map
                            (fun x =>
                              C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                            (List.finRange n_var)).sum +
                    ((List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                (X Vars_γ * X Vars_x ^ ↑x * (to_MvPolynomial_Option Vars) t))
                            (List.finRange (n_var - 1))).sum *
                        -(List.map
                              (fun x =>
                                C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                  (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                              (List.finRange n_var)).sum +
                      ((List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                      (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i)))
                                  (List.finRange n_stmt)).sum *
                              -(List.map
                                    (fun x =>
                                      C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                        (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                                    (List.finRange n_var)).sum +
                            (List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                      (X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i)))
                                  (List.finRange n_stmt)).sum *
                              -(List.map
                                    (fun x =>
                                      C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                        (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                                    (List.finRange n_var)).sum +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                    (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                                (List.finRange n_stmt)).sum *
                            -(List.map
                                  (fun x =>
                                    C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                      (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                                  (List.finRange n_var)).sum +
                        ((List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                      (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i)))
                                  (List.finRange n_wit)).sum *
                              -(List.map
                                    (fun x =>
                                      C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                        (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                                    (List.finRange n_var)).sum +
                            (List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                      (X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i)))
                                  (List.finRange n_wit)).sum *
                              -(List.map
                                    (fun x =>
                                      C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                        (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                                    (List.finRange n_var)).sum +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                    (X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i)))
                                (List.finRange n_wit)).sum *
                            -(List.map
                                  (fun x =>
                                    C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow x)) *
                                      (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                                  (List.finRange n_var)).sum)))))) +
            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) *
                -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
              (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) *
                  -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) *
                    -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                  ((List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                              (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                          (List.finRange n_var)).sum *
                      -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                    ((List.map
                            (fun x =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                                (X Vars_γ * X Vars_x ^ ↑x * (to_MvPolynomial_Option Vars) t))
                            (List.finRange (n_var - 1))).sum *
                        -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                      ((List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                      (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i)))
                                  (List.finRange n_stmt)).sum *
                              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                            (List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                      (X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i)))
                                  (List.finRange n_stmt)).sum *
                              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                    (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                                (List.finRange n_stmt)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                        ((List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                      (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i)))
                                  (List.finRange n_wit)).sum *
                              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                            (List.map
                                  (fun i =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                      (X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i)))
                                  (List.finRange n_wit)).sum *
                              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ)) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                    (X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i)))
                                (List.finRange n_wit)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ) *
                                (X Vars_γ * X Vars_δ * X Vars_δ))))))))) +
          (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) *
              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) *
                -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
              (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) *
                  -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                            (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                        (List.finRange n_var)).sum *
                    -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                  ((List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                              (X Vars_γ * X Vars_x ^ ↑x * (to_MvPolynomial_Option Vars) t))
                          (List.finRange (n_var - 1))).sum *
                      -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                    ((List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                    (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i)))
                                (List.finRange n_stmt)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                    (X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i)))
                                (List.finRange n_stmt)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                  (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                              (List.finRange n_stmt)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                      ((List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                    (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i)))
                                (List.finRange n_wit)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                    (X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i)))
                                (List.finRange n_wit)).sum *
                            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) * (X Vars_γ * X Vars_δ * X Vars_γ)) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                  (X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i)))
                              (List.finRange n_wit)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ) *
                              (X Vars_γ * X Vars_δ * X Vars_γ))))))))) +
        (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) *
            -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
          (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) *
              -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) *
                -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
              ((List.map
                      (fun x =>
                        C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow x)) *
                          (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                      (List.finRange n_var)).sum *
                  -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                            (X Vars_γ * X Vars_x ^ ↑x * (to_MvPolynomial_Option Vars) t))
                        (List.finRange (n_var - 1))).sum *
                    -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                  ((List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                  (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i)))
                              (List.finRange n_stmt)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                  (X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i)))
                              (List.finRange n_stmt)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                      (List.map
                            (fun i =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y i)) *
                                (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                            (List.finRange n_stmt)).sum *
                        -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                    ((List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                  (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i)))
                              (List.finRange n_wit)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                  (X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i)))
                              (List.finRange n_wit)).sum *
                          -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β)) +
                      (List.map
                            (fun i =>
                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q i)) *
                                (X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i)))
                            (List.finRange n_wit)).sum *
                        -(C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β))))))))) +
      (X Vars_γ * X Vars_δ * X Vars_α * (X Vars_γ * X Vars_δ * X Vars_β) +
        ((List.map (fun i => C (stmt i) * (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i)))
                    (List.finRange n_stmt)).sum *
                (X Vars_γ * X Vars_δ * X Vars_γ) +
              (List.map (fun i => C (stmt i) * (X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i)))
                    (List.finRange n_stmt)).sum *
                (X Vars_γ * X Vars_δ * X Vars_γ) +
            (List.map (fun i => C (stmt i) * (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                  (List.finRange n_stmt)).sum *
              (X Vars_γ * X Vars_δ * X Vars_γ) +
          (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α) * (X Vars_γ * X Vars_δ * X Vars_α) *
              (X Vars_γ * X Vars_δ * X Vars_δ) +
            (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β) * (X Vars_γ * X Vars_δ * X Vars_β) *
                (X Vars_γ * X Vars_δ * X Vars_δ) +
              (C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ) * (X Vars_γ * X Vars_δ * X Vars_δ) *
                  (X Vars_γ * X Vars_δ * X Vars_δ) +
                ((List.map
                        (fun x =>
                          C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow x)) *
                            (X Vars_γ * X Vars_δ * X Vars_x ^ ↑x))
                        (List.finRange n_var)).sum *
                    (X Vars_γ * X Vars_δ * X Vars_δ) +
                  ((List.map
                          (fun x =>
                            C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t x)) *
                              (X Vars_γ * X Vars_x ^ ↑x * (to_MvPolynomial_Option Vars) t))
                          (List.finRange (n_var - 1))).sum *
                      (X Vars_γ * X Vars_δ * X Vars_δ) +
                    ((List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y i)) *
                                    (X Vars_β * X Vars_δ * (to_MvPolynomial_Option Vars) (u_stmt i)))
                                (List.finRange n_stmt)).sum *
                            (X Vars_γ * X Vars_δ * X Vars_δ) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y i)) *
                                    (X Vars_α * X Vars_δ * (to_MvPolynomial_Option Vars) (v_stmt i)))
                                (List.finRange n_stmt)).sum *
                            (X Vars_γ * X Vars_δ * X Vars_δ) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y i)) *
                                  (X Vars_δ * (to_MvPolynomial_Option Vars) (w_stmt i)))
                              (List.finRange n_stmt)).sum *
                          (X Vars_γ * X Vars_δ * X Vars_δ) +
                      ((List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) *
                                    (X Vars_β * X Vars_γ * (to_MvPolynomial_Option Vars) (u_wit i)))
                                (List.finRange n_wit)).sum *
                            (X Vars_γ * X Vars_δ * X Vars_δ) +
                          (List.map
                                (fun i =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) *
                                    (X Vars_α * X Vars_γ * (to_MvPolynomial_Option Vars) (v_wit i)))
                                (List.finRange n_wit)).sum *
                            (X Vars_γ * X Vars_δ * X Vars_δ) +
                        (List.map
                              (fun i =>
                                C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) *
                                  (X Vars_γ * (to_MvPolynomial_Option Vars) (w_wit i)))
                              (List.finRange n_wit)).sum *
                          (X Vars_γ * X Vars_δ * X Vars_δ)))))))))) =
    0
⊢ (((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
              (List.map
                  (fun i =>
                    Polynomial.C ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover i) *
                      u_wit i)
                  (List.finRange n_wit)).sum) *
            ((List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)).sum +
              (List.map
                  (fun i =>
                    Polynomial.C ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover i) *
                      v_wit i)
                  (List.finRange n_wit)).sum) -
          ((List.map (fun i => Polynomial.C (stmt i) * w_stmt i) (List.finRange n_stmt)).sum +
            (List.map
                (fun i =>
                  Polynomial.C ((fun prover i => prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q i)) prover i) * w_wit i)
                (List.finRange n_wit)).sum)).modByMonic
      t =
    0
Tactic:
simp only [
  mul_assoc,
  mul_left_comm (C _) (X (some _)) _, mul_left_comm (List.sum _) (X (some _)) _,
  mul_comm (C _) (X (some _)), mul_comm (List.sum _) (X (some _)),
  neg_mul, mul_neg,
  List.sum_map_mul_right, List.sum_map_mul_left] 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
t : Polynomial F
stmt : Groth16TypeIII.Stmt
prover : Prover F Groth16TypeIII
eqn :
  -(X Vars_γ *
                  (X Vars_δ *
                    (X Vars_α *
                      (X Vars_γ *
                        (X Vars_δ *
                          (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                            (List.map
                                (fun b => C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                (List.finRange n_var)).sum)))))) +
              (-(X Vars_γ *
                    (X Vars_δ *
                      (X Vars_β *
                        (X Vars_γ *
                          (X Vars_δ *
                            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                              (List.map
                                  (fun b => C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                  (List.finRange n_var)).sum)))))) +
                (-(X Vars_γ *
                      (X Vars_δ *
                        (X Vars_δ *
                          (X Vars_γ *
                            (X Vars_δ *
                              (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                                (List.map
                                    (fun b => C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                    (List.finRange n_var)).sum)))))) +
                  (-(X Vars_γ *
                        (X Vars_δ *
                          (X Vars_γ *
                            (X Vars_δ *
                              ((List.map
                                    (fun b => C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                    (List.finRange n_var)).sum *
                                (List.map
                                    (fun b => C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                    (List.finRange n_var)).sum))))) +
                    (-(X Vars_γ *
                          (X Vars_γ *
                            (X Vars_δ *
                              ((List.map
                                    (fun b =>
                                      C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t b)) *
                                        (X Vars_x ^ ↑b * (to_MvPolynomial_Option Vars) t))
                                    (List.finRange (n_var - 1))).sum *
                                (List.map
                                    (fun b => C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                    (List.finRange n_var)).sum)))) +
                      (-(X Vars_β *
                                (X Vars_δ *
                                  (X Vars_γ *
                                    (X Vars_δ *
                                      ((List.map
                                            (fun b =>
                                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y b)) *
                                                (to_MvPolynomial_Option Vars) (u_stmt b))
                                            (List.finRange n_stmt)).sum *
                                        (List.map
                                            (fun b =>
                                              C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                            (List.finRange n_var)).sum))))) +
                            -(X Vars_α *
                                (X Vars_δ *
                                  (X Vars_γ *
                                    (X Vars_δ *
                                      ((List.map
                                            (fun b =>
                                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y b)) *
                                                (to_MvPolynomial_Option Vars) (v_stmt b))
                                            (List.finRange n_stmt)).sum *
                                        (List.map
                                            (fun b =>
                                              C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                            (List.finRange n_var)).sum))))) +
                          -(X Vars_δ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  ((List.map
                                        (fun b =>
                                          C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y b)) *
                                            (to_MvPolynomial_Option Vars) (w_stmt b))
                                        (List.finRange n_stmt)).sum *
                                    (List.map
                                        (fun b =>
                                          C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                        (List.finRange n_var)).sum)))) +
                        (-(X Vars_β *
                                (X Vars_γ *
                                  (X Vars_γ *
                                    (X Vars_δ *
                                      ((List.map
                                            (fun b =>
                                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q b)) *
                                                (to_MvPolynomial_Option Vars) (u_wit b))
                                            (List.finRange n_wit)).sum *
                                        (List.map
                                            (fun b =>
                                              C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                            (List.finRange n_var)).sum))))) +
                            -(X Vars_α *
                                (X Vars_γ *
                                  (X Vars_γ *
                                    (X Vars_δ *
                                      ((List.map
                                            (fun b =>
                                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q b)) *
                                                (to_MvPolynomial_Option Vars) (v_wit b))
                                            (List.finRange n_wit)).sum *
                                        (List.map
                                            (fun b =>
                                              C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                            (List.finRange n_var)).sum))))) +
                          -(X Vars_γ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  ((List.map
                                        (fun b =>
                                          C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q b)) *
                                            (to_MvPolynomial_Option Vars) (w_wit b))
                                        (List.finRange n_wit)).sum *
                                    (List.map
                                        (fun b =>
                                          C (prover.2 Proof_G2_Idx.B (SRS_Elements_G2_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                        (List.finRange n_var)).sum)))))))))) +
            (-(X Vars_γ *
                  (X Vars_δ *
                    (X Vars_α *
                      (X Vars_γ *
                        (X Vars_δ *
                          (X Vars_δ *
                            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                              C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))))))) +
              (-(X Vars_γ *
                    (X Vars_δ *
                      (X Vars_β *
                        (X Vars_γ *
                          (X Vars_δ *
                            (X Vars_δ *
                              (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                                C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))))))) +
                (-(X Vars_γ *
                      (X Vars_δ *
                        (X Vars_δ *
                          (X Vars_γ *
                            (X Vars_δ *
                              (X Vars_δ *
                                (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                                  C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))))))) +
                  (-(X Vars_γ *
                        (X Vars_δ *
                          (X Vars_γ *
                            (X Vars_δ *
                              (X Vars_δ *
                                ((List.map
                                      (fun b =>
                                        C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                      (List.finRange n_var)).sum *
                                  C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))) +
                    (-(X Vars_γ *
                          (X Vars_γ *
                            (X Vars_δ *
                              (X Vars_δ *
                                ((List.map
                                      (fun b =>
                                        C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t b)) *
                                          (X Vars_x ^ ↑b * (to_MvPolynomial_Option Vars) t))
                                      (List.finRange (n_var - 1))).sum *
                                  C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))))) +
                      (-(X Vars_β *
                                (X Vars_δ *
                                  (X Vars_γ *
                                    (X Vars_δ *
                                      (X Vars_δ *
                                        ((List.map
                                              (fun b =>
                                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y b)) *
                                                  (to_MvPolynomial_Option Vars) (u_stmt b))
                                              (List.finRange n_stmt)).sum *
                                          C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))) +
                            -(X Vars_α *
                                (X Vars_δ *
                                  (X Vars_γ *
                                    (X Vars_δ *
                                      (X Vars_δ *
                                        ((List.map
                                              (fun b =>
                                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y b)) *
                                                  (to_MvPolynomial_Option Vars) (v_stmt b))
                                              (List.finRange n_stmt)).sum *
                                          C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))) +
                          -(X Vars_δ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  (X Vars_δ *
                                    ((List.map
                                          (fun b =>
                                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y b)) *
                                              (to_MvPolynomial_Option Vars) (w_stmt b))
                                          (List.finRange n_stmt)).sum *
                                      C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ)))))) +
                        (-(X Vars_β *
                                (X Vars_γ *
                                  (X Vars_γ *
                                    (X Vars_δ *
                                      (X Vars_δ *
                                        ((List.map
                                              (fun b =>
                                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q b)) *
                                                  (to_MvPolynomial_Option Vars) (u_wit b))
                                              (List.finRange n_wit)).sum *
                                          C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))) +
                            -(X Vars_α *
                                (X Vars_γ *
                                  (X Vars_γ *
                                    (X Vars_δ *
                                      (X Vars_δ *
                                        ((List.map
                                              (fun b =>
                                                C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q b)) *
                                                  (to_MvPolynomial_Option Vars) (v_wit b))
                                              (List.finRange n_wit)).sum *
                                          C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))) +
                          -(X Vars_γ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  (X Vars_δ *
                                    ((List.map
                                          (fun b =>
                                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q b)) *
                                              (to_MvPolynomial_Option Vars) (w_wit b))
                                          (List.finRange n_wit)).sum *
                                      C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.δ))))))))))))) +
          (-(X Vars_γ *
                (X Vars_δ *
                  (X Vars_α *
                    (X Vars_γ *
                      (X Vars_δ *
                        (X Vars_γ *
                          (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                            C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))))))) +
            (-(X Vars_γ *
                  (X Vars_δ *
                    (X Vars_β *
                      (X Vars_γ *
                        (X Vars_δ *
                          (X Vars_γ *
                            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                              C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))))))) +
              (-(X Vars_γ *
                    (X Vars_δ *
                      (X Vars_δ *
                        (X Vars_γ *
                          (X Vars_δ *
                            (X Vars_γ *
                              (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                                C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))))))) +
                (-(X Vars_γ *
                      (X Vars_δ *
                        (X Vars_γ *
                          (X Vars_δ *
                            (X Vars_γ *
                              ((List.map
                                    (fun b => C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                    (List.finRange n_var)).sum *
                                C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))) +
                  (-(X Vars_γ *
                        (X Vars_γ *
                          (X Vars_δ *
                            (X Vars_γ *
                              ((List.map
                                    (fun b =>
                                      C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t b)) *
                                        (X Vars_x ^ ↑b * (to_MvPolynomial_Option Vars) t))
                                    (List.finRange (n_var - 1))).sum *
                                C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))))) +
                    (-(X Vars_β *
                              (X Vars_δ *
                                (X Vars_γ *
                                  (X Vars_δ *
                                    (X Vars_γ *
                                      ((List.map
                                            (fun b =>
                                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y b)) *
                                                (to_MvPolynomial_Option Vars) (u_stmt b))
                                            (List.finRange n_stmt)).sum *
                                        C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))) +
                          -(X Vars_α *
                              (X Vars_δ *
                                (X Vars_γ *
                                  (X Vars_δ *
                                    (X Vars_γ *
                                      ((List.map
                                            (fun b =>
                                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y b)) *
                                                (to_MvPolynomial_Option Vars) (v_stmt b))
                                            (List.finRange n_stmt)).sum *
                                        C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))) +
                        -(X Vars_δ *
                            (X Vars_γ *
                              (X Vars_δ *
                                (X Vars_γ *
                                  ((List.map
                                        (fun b =>
                                          C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y b)) *
                                            (to_MvPolynomial_Option Vars) (w_stmt b))
                                        (List.finRange n_stmt)).sum *
                                    C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ)))))) +
                      (-(X Vars_β *
                              (X Vars_γ *
                                (X Vars_γ *
                                  (X Vars_δ *
                                    (X Vars_γ *
                                      ((List.map
                                            (fun b =>
                                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q b)) *
                                                (to_MvPolynomial_Option Vars) (u_wit b))
                                            (List.finRange n_wit)).sum *
                                        C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))) +
                          -(X Vars_α *
                              (X Vars_γ *
                                (X Vars_γ *
                                  (X Vars_δ *
                                    (X Vars_γ *
                                      ((List.map
                                            (fun b =>
                                              C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q b)) *
                                                (to_MvPolynomial_Option Vars) (v_wit b))
                                            (List.finRange n_wit)).sum *
                                        C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))) +
                        -(X Vars_γ *
                            (X Vars_γ *
                              (X Vars_δ *
                                (X Vars_γ *
                                  ((List.map
                                        (fun b =>
                                          C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q b)) *
                                            (to_MvPolynomial_Option Vars) (w_wit b))
                                        (List.finRange n_wit)).sum *
                                    C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.γ))))))))))))) +
        (-(X Vars_γ *
              (X Vars_δ *
                (X Vars_α *
                  (X Vars_γ *
                    (X Vars_δ *
                      (X Vars_β *
                        (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.α) *
                          C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))))) +
          (-(X Vars_γ *
                (X Vars_δ *
                  (X Vars_β *
                    (X Vars_γ *
                      (X Vars_δ *
                        (X Vars_β *
                          (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.β) *
                            C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))))) +
            (-(X Vars_γ *
                  (X Vars_δ *
                    (X Vars_δ *
                      (X Vars_γ *
                        (X Vars_δ *
                          (X Vars_β *
                            (C (prover.1 Proof_G1_Idx.A SRS_Elements_G1_Idx.δ) *
                              C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))))) +
              (-(X Vars_γ *
                    (X Vars_δ *
                      (X Vars_γ *
                        (X Vars_δ *
                          (X Vars_β *
                            ((List.map
                                  (fun b => C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                  (List.finRange n_var)).sum *
                              C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))) +
                (-(X Vars_γ *
                      (X Vars_γ *
                        (X Vars_δ *
                          (X Vars_β *
                            ((List.map
                                  (fun b =>
                                    C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.x_pow_times_t b)) *
                                      (X Vars_x ^ ↑b * (to_MvPolynomial_Option Vars) t))
                                  (List.finRange (n_var - 1))).sum *
                              C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))) +
                  (-(X Vars_β *
                            (X Vars_δ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  (X Vars_β *
                                    ((List.map
                                          (fun b =>
                                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y b)) *
                                              (to_MvPolynomial_Option Vars) (u_stmt b))
                                          (List.finRange n_stmt)).sum *
                                      C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))) +
                        -(X Vars_α *
                            (X Vars_δ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  (X Vars_β *
                                    ((List.map
                                          (fun b =>
                                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y b)) *
                                              (to_MvPolynomial_Option Vars) (v_stmt b))
                                          (List.finRange n_stmt)).sum *
                                      C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))) +
                      -(X Vars_δ *
                          (X Vars_γ *
                            (X Vars_δ *
                              (X Vars_β *
                                ((List.map
                                      (fun b =>
                                        C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.y b)) *
                                          (to_MvPolynomial_Option Vars) (w_stmt b))
                                      (List.finRange n_stmt)).sum *
                                  C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β)))))) +
                    (-(X Vars_β *
                            (X Vars_γ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  (X Vars_β *
                                    ((List.map
                                          (fun b =>
                                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q b)) *
                                              (to_MvPolynomial_Option Vars) (u_wit b))
                                          (List.finRange n_wit)).sum *
                                      C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))) +
                        -(X Vars_α *
                            (X Vars_γ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  (X Vars_β *
                                    ((List.map
                                          (fun b =>
                                            C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q b)) *
                                              (to_MvPolynomial_Option Vars) (v_wit b))
                                          (List.finRange n_wit)).sum *
                                      C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))) +
                      -(X Vars_γ *
                          (X Vars_γ *
                            (X Vars_δ *
                              (X Vars_β *
                                ((List.map
                                      (fun b =>
                                        C (prover.1 Proof_G1_Idx.A (SRS_Elements_G1_Idx.q b)) *
                                          (to_MvPolynomial_Option Vars) (w_wit b))
                                      (List.finRange n_wit)).sum *
                                  C (prover.2 Proof_G2_Idx.B SRS_Elements_G2_Idx.β))))))))))))) +
      (X Vars_γ * (X Vars_δ * (X Vars_α * (X Vars_γ * (X Vars_δ * X Vars_β)))) +
        (X Vars_β *
                (X Vars_δ *
                  (X Vars_γ *
                    (X Vars_δ *
                      (X Vars_γ *
                        (List.map (fun b => C (stmt b) * (to_MvPolynomial_Option Vars) (u_stmt b))
                            (List.finRange n_stmt)).sum)))) +
              X Vars_α *
                (X Vars_δ *
                  (X Vars_γ *
                    (X Vars_δ *
                      (X Vars_γ *
                        (List.map (fun b => C (stmt b) * (to_MvPolynomial_Option Vars) (v_stmt b))
                            (List.finRange n_stmt)).sum)))) +
            X Vars_δ *
              (X Vars_γ *
                (X Vars_δ *
                  (X Vars_γ *
                    (List.map (fun b => C (stmt b) * (to_MvPolynomial_Option Vars) (w_stmt b))
                        (List.finRange n_stmt)).sum))) +
          (X Vars_γ *
              (X Vars_δ *
                (X Vars_α * (X Vars_γ * (X Vars_δ * (X Vars_δ * C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.α)))))) +
            (X Vars_γ *
                (X Vars_δ *
                  (X Vars_β *
                    (X Vars_γ * (X Vars_δ * (X Vars_δ * C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.β)))))) +
              (X Vars_γ *
                  (X Vars_δ *
                    (X Vars_δ *
                      (X Vars_γ * (X Vars_δ * (X Vars_δ * C (prover.1 Proof_G1_Idx.C SRS_Elements_G1_Idx.δ)))))) +
                (X Vars_γ *
                    (X Vars_δ *
                      (X Vars_γ *
                        (X Vars_δ *
                          (X Vars_δ *
                            (List.map
                                (fun b => C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow b)) * X Vars_x ^ ↑b)
                                (List.finRange n_var)).sum)))) +
                  (X Vars_γ *
                      (X Vars_γ *
                        (X Vars_δ *
                          (X Vars_δ *
                            (List.map
                                (fun b =>
                                  C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.x_pow_times_t b)) *
                                    (X Vars_x ^ ↑b * (to_MvPolynomial_Option Vars) t))
                                (List.finRange (n_var - 1))).sum))) +
                    (X Vars_β *
                            (X Vars_δ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  (X Vars_δ *
                                    (List.map
                                        (fun b =>
                                          C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y b)) *
                                            (to_MvPolynomial_Option Vars) (u_stmt b))
                                        (List.finRange n_stmt)).sum)))) +
                          X Vars_α *
                            (X Vars_δ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  (X Vars_δ *
                                    (List.map
                                        (fun b =>
                                          C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y b)) *
                                            (to_MvPolynomial_Option Vars) (v_stmt b))
                                        (List.finRange n_stmt)).sum)))) +
                        X Vars_δ *
                          (X Vars_γ *
                            (X Vars_δ *
                              (X Vars_δ *
                                (List.map
                                    (fun b =>
                                      C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.y b)) *
                                        (to_MvPolynomial_Option Vars) (w_stmt b))
                                    (List.finRange n_stmt)).sum))) +
                      (X Vars_β *
                            (X Vars_γ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  (X Vars_δ *
                                    (List.map
                                        (fun b =>
                                          C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q b)) *
                                            (to_MvPolynomial_Option Vars) (u_wit b))
                                        (List.finRange n_wit)).sum)))) +
                          X Vars_α *
                            (X Vars_γ *
                              (X Vars_γ *
                                (X Vars_δ *
                                  (X Vars_δ *
                                    (List.map
                                        (fun b =>
                                          C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q b)) *
                                            (to_MvPolynomial_Option Vars) (v_wit b))
                                        (List.finRange n_wit)).sum)))) +
                        X Vars_γ *
                          (X Vars_γ *
                            (X Vars_δ *
                              (X Vars_δ *
                                (List.map
                                    (fun b =>
                                      C (prover.1 Proof_G1_Idx.C (SRS_Elements_G1_Idx.q b)) *
                                        (to_MvPolynomial_Option Vars) (w_wit b))
                                    (List.finRange n_wit)).sum)))))))))))) =
    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