clear eqns null
case intro
F : Type
inst✝ : Field F
n_stmt n_wit m d : ℕ
v_stmt : Fin n_stmt → Polynomial F
v_wit : Fin n_wit → Polynomial F
w_wit : Fin m → Polynomial F
v_0 w_0 : Polynomial F
r : Fin n_wit → F
stmt : GGPR.Stmt
prover : AGMProofSystemInstantiation.Prover F GGPR
eqns : ∀ (check_idx : GGPR.EqualityChecks), GGPR.check_poly prover stmt check_idx = 0
null :
  ∀ pfs ∈ GGPR.Identified_Proof_Elems,
    (List.map (fun SRS_elem => C (prover.1 pfs.1 SRS_elem) * GGPR.SRSElementValue_G1 SRS_elem)
          GGPR.ListSRSElements_G1).sum =
      (List.map (fun SRS_elem => C (prover.2 pfs.2 SRS_elem) * GGPR.SRSElementValue_G2 SRS_elem)
          GGPR.ListSRSElements_G2).sum
t : Polynomial F := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))
eqnI : GGPR.check_poly prover stmt ChecksIdx.CheckI = 0
eqnII : GGPR.check_poly prover stmt ChecksIdx.CheckII = 0
eqnIII : GGPR.check_poly prover stmt ChecksIdx.CheckIII = 0
eqnIV : GGPR.check_poly prover stmt ChecksIdx.CheckIV = 0
eqnV : GGPR.check_poly prover stmt ChecksIdx.CheckV = 0
⊢ ((v_0 + (List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)).sum +
            (List.map
                (fun i =>
                  Polynomial.C
                      (((fun prover =>
                              (fun i => prover.2 Proof_G2_Idx.H (SRS_Elements_Idx.EK_β_v i), fun i =>
                                prover.2 Proof_G2_Idx.H (SRS_Elements_Idx.EK_β_w i)))
                            prover).1
                        i) *
                    v_wit i)
                (List.finRange n_wit)).sum) *
          (w_0 +
            (List.map
                (fun i =>
                  Polynomial.C
                      (((fun prover =>
                              (fun i => prover.2 Proof_G2_Idx.H (SRS_Elements_Idx.EK_β_v i), fun i =>
                                prover.2 Proof_G2_Idx.H (SRS_Elements_Idx.EK_β_w i)))
                            prover).2
                        i) *
                    w_wit i)
                (List.finRange m)).sum)).modByMonic
      t =
    0
case intro
F : Type
inst✝ : Field F
n_stmt n_wit m d : ℕ
v_stmt : Fin n_stmt → Polynomial F
v_wit : Fin n_wit → Polynomial F
w_wit : Fin m → Polynomial F
v_0 w_0 : Polynomial F
r : Fin n_wit → F
stmt : GGPR.Stmt
prover : AGMProofSystemInstantiation.Prover F GGPR
t : Polynomial F := ∏ i : Fin n_wit, (Polynomial.X - Polynomial.C (r i))
eqnI : GGPR.check_poly prover stmt ChecksIdx.CheckI = 0
eqnII : GGPR.check_poly prover stmt ChecksIdx.CheckII = 0
eqnIII : GGPR.check_poly prover stmt ChecksIdx.CheckIII = 0
eqnIV : GGPR.check_poly prover stmt ChecksIdx.CheckIV = 0
eqnV : GGPR.check_poly prover stmt ChecksIdx.CheckV = 0
⊢ ((v_0 + (List.map (fun i => Polynomial.C (stmt i) * v_stmt i) (List.finRange n_stmt)).sum +
            (List.map
                (fun i =>
                  Polynomial.C
                      (((fun prover =>
                              (fun i => prover.2 Proof_G2_Idx.H (SRS_Elements_Idx.EK_β_v i), fun i =>
                                prover.2 Proof_G2_Idx.H (SRS_Elements_Idx.EK_β_w i)))
                            prover).1
                        i) *
                    v_wit i)
                (List.finRange n_wit)).sum) *
          (w_0 +
            (List.map
                (fun i =>
                  Polynomial.C
                      (((fun prover =>
                              (fun i => prover.2 Proof_G2_Idx.H (SRS_Elements_Idx.EK_β_v i), fun i =>
                                prover.2 Proof_G2_Idx.H (SRS_Elements_Idx.EK_β_w i)))
                            prover).2
                        i) *
                    w_wit i)
                (List.finRange m)).sum)).modByMonic
      t =
    0