Theorem: BabySNARK.is_sound
Proof state before:
case intro
F : Type
inst✝ : Field F
n_stmt n_wit n_var : ℕ
u_stmt : Fin n_stmt → Polynomial F
u_wit : Fin n_wit → Polynomial F
v_stmt : Fin n_stmt → Polynomial F
v_wit : Fin n_wit → Polynomial F
w_stmt : Fin n_stmt → Polynomial F
w_wit : Fin n_wit → Polynomial F
t : Polynomial F
ht : (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum = t
ht0 : t.Monic
stmt : BabySNARK.Stmt
prover : Prover F BabySNARK
typeI_identification :
  ∀ pfs ∈ BabySNARK.Identified_Proof_Elems,
    (List.map (fun SRS_elem => C (prover.1 pfs.1 SRS_elem) * BabySNARK.SRSElementValue_G1 SRS_elem)
          BabySNARK.ListSRSElements_G1).sum =
      (List.map (fun SRS_elem => C (prover.2 pfs.2 SRS_elem) * BabySNARK.SRSElementValue_G2 SRS_elem)
          BabySNARK.ListSRSElements_G2).sum
eqnI :
  (List.map
        (fun pairing =>
          ((List.map
                  (fun pf_elem =>
                    C (BabySNARK.verificationPairingProof_G1 stmt ChecksIdx.CheckI pairing pf_elem) *
                      (List.map (fun SRS_elem => C (prover.1 pf_elem SRS_elem) * BabySNARK.SRSElementValue_G1 SRS_elem)
                          BabySNARK.ListSRSElements_G1).sum)
                  BabySNARK.ListProof_G1).sum +
              (List.map
                  (fun SRS_elem =>
                    C (BabySNARK.verificationPairingSRS_G1 stmt ChecksIdx.CheckI pairing SRS_elem) *
                      BabySNARK.SRSElementValue_G1 SRS_elem)
                  BabySNARK.ListSRSElements_G1).sum) *
            ((List.map
                  (fun pf_elem =>
                    C (BabySNARK.verificationPairingProof_G2 stmt ChecksIdx.CheckI pairing pf_elem) *
                      (List.map (fun SRS_elem => C (prover.2 pf_elem SRS_elem) * BabySNARK.SRSElementValue_G2 SRS_elem)
                          BabySNARK.ListSRSElements_G2).sum)
                  BabySNARK.ListProof_G2).sum +
              (List.map
                  (fun SRS_elem =>
                    C (BabySNARK.verificationPairingSRS_G2 stmt ChecksIdx.CheckI pairing SRS_elem) *
                      BabySNARK.SRSElementValue_G2 SRS_elem)
                  BabySNARK.ListSRSElements_G2).sum))
        (BabySNARK.ListPairings ChecksIdx.CheckI)).sum =
    0
eqnII :
  (List.map
        (fun pairing =>
          ((List.map
                  (fun pf_elem =>
                    C (BabySNARK.verificationPairingProof_G1 stmt ChecksIdx.CheckII pairing pf_elem) *
                      (List.map (fun SRS_elem => C (prover.1 pf_elem SRS_elem) * BabySNARK.SRSElementValue_G1 SRS_elem)
                          BabySNARK.ListSRSElements_G1).sum)
                  BabySNARK.ListProof_G1).sum +
              (List.map
                  (fun SRS_elem =>
                    C (BabySNARK.verificationPairingSRS_G1 stmt ChecksIdx.CheckII pairing SRS_elem) *
                      BabySNARK.SRSElementValue_G1 SRS_elem)
                  BabySNARK.ListSRSElements_G1).sum) *
            ((List.map
                  (fun pf_elem =>
                    C (BabySNARK.verificationPairingProof_G2 stmt ChecksIdx.CheckII pairing pf_elem) *
                      (List.map (fun SRS_elem => C (prover.2 pf_elem SRS_elem) * BabySNARK.SRSElementValue_G2 SRS_elem)
                          BabySNARK.ListSRSElements_G2).sum)
                  BabySNARK.ListProof_G2).sum +
              (List.map
                  (fun SRS_elem =>
                    C (BabySNARK.verificationPairingSRS_G2 stmt ChecksIdx.CheckII pairing SRS_elem) *
                      BabySNARK.SRSElementValue_G2 SRS_elem)
                  BabySNARK.ListSRSElements_G2).sum))
        (BabySNARK.ListPairings ChecksIdx.CheckII)).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_Idx.B (SRS_Elements_Idx.βu i)) prover i) * u_wit i)
                  (List.finRange n_wit)).sum) ^
            2 -
          1).modByMonic
      t =
    0
Tactic:
simp only [identified_proof_elems_def, Bool.not_eq_true, List.mem_cons, List.mem_singleton,
  forall_eq_or_imp, forall_eq] at typeI_identification
Proof state after:
case intro
F : Type
inst✝ : Field F
n_stmt n_wit n_var : ℕ
u_stmt : Fin n_stmt → Polynomial F
u_wit : Fin n_wit → Polynomial F
v_stmt : Fin n_stmt → Polynomial F
v_wit : Fin n_wit → Polynomial F
w_stmt : Fin n_stmt → Polynomial F
w_wit : Fin n_wit → Polynomial F
t : Polynomial F
ht : (List.map (fun x => Polynomial.C (t.coeff ↑x) * Polynomial.X ^ ↑x) (List.finRange n_var)).sum = t
ht0 : t.Monic
stmt : BabySNARK.Stmt
prover : Prover F BabySNARK
eqnI :
  (List.map
        (fun pairing =>
          ((List.map
                  (fun pf_elem =>
                    C (BabySNARK.verificationPairingProof_G1 stmt ChecksIdx.CheckI pairing pf_elem) *
                      (List.map (fun SRS_elem => C (prover.1 pf_elem SRS_elem) * BabySNARK.SRSElementValue_G1 SRS_elem)
                          BabySNARK.ListSRSElements_G1).sum)
                  BabySNARK.ListProof_G1).sum +
              (List.map
                  (fun SRS_elem =>
                    C (BabySNARK.verificationPairingSRS_G1 stmt ChecksIdx.CheckI pairing SRS_elem) *
                      BabySNARK.SRSElementValue_G1 SRS_elem)
                  BabySNARK.ListSRSElements_G1).sum) *
            ((List.map
                  (fun pf_elem =>
                    C (BabySNARK.verificationPairingProof_G2 stmt ChecksIdx.CheckI pairing pf_elem) *
                      (List.map (fun SRS_elem => C (prover.2 pf_elem SRS_elem) * BabySNARK.SRSElementValue_G2 SRS_elem)
                          BabySNARK.ListSRSElements_G2).sum)
                  BabySNARK.ListProof_G2).sum +
              (List.map
                  (fun SRS_elem =>
                    C (BabySNARK.verificationPairingSRS_G2 stmt ChecksIdx.CheckI pairing SRS_elem) *
                      BabySNARK.SRSElementValue_G2 SRS_elem)
                  BabySNARK.ListSRSElements_G2).sum))
        (BabySNARK.ListPairings ChecksIdx.CheckI)).sum =
    0
eqnII :
  (List.map
        (fun pairing =>
          ((List.map
                  (fun pf_elem =>
                    C (BabySNARK.verificationPairingProof_G1 stmt ChecksIdx.CheckII pairing pf_elem) *
                      (List.map (fun SRS_elem => C (prover.1 pf_elem SRS_elem) * BabySNARK.SRSElementValue_G1 SRS_elem)
                          BabySNARK.ListSRSElements_G1).sum)
                  BabySNARK.ListProof_G1).sum +
              (List.map
                  (fun SRS_elem =>
                    C (BabySNARK.verificationPairingSRS_G1 stmt ChecksIdx.CheckII pairing SRS_elem) *
                      BabySNARK.SRSElementValue_G1 SRS_elem)
                  BabySNARK.ListSRSElements_G1).sum) *
            ((List.map
                  (fun pf_elem =>
                    C (BabySNARK.verificationPairingProof_G2 stmt ChecksIdx.CheckII pairing pf_elem) *
                      (List.map (fun SRS_elem => C (prover.2 pf_elem SRS_elem) * BabySNARK.SRSElementValue_G2 SRS_elem)
                          BabySNARK.ListSRSElements_G2).sum)
                  BabySNARK.ListProof_G2).sum +
              (List.map
                  (fun SRS_elem =>
                    C (BabySNARK.verificationPairingSRS_G2 stmt ChecksIdx.CheckII pairing SRS_elem) *
                      BabySNARK.SRSElementValue_G2 SRS_elem)
                  BabySNARK.ListSRSElements_G2).sum))
        (BabySNARK.ListPairings ChecksIdx.CheckII)).sum =
    0
typeI_identification :
  (List.map (fun SRS_elem => C (prover.1 Proof_Idx.H SRS_elem) * BabySNARK.SRSElementValue_G1 SRS_elem)
          BabySNARK.ListSRSElements_G1).sum =
      (List.map (fun SRS_elem => C (prover.2 Proof_Idx.H SRS_elem) * BabySNARK.SRSElementValue_G2 SRS_elem)
          BabySNARK.ListSRSElements_G2).sum ∧
    (List.map (fun SRS_elem => C (prover.1 Proof_Idx.V SRS_elem) * BabySNARK.SRSElementValue_G1 SRS_elem)
            BabySNARK.ListSRSElements_G1).sum =
        (List.map (fun SRS_elem => C (prover.2 Proof_Idx.V SRS_elem) * BabySNARK.SRSElementValue_G2 SRS_elem)
            BabySNARK.ListSRSElements_G2).sum ∧
      (List.map (fun SRS_elem => C (prover.1 Proof_Idx.B SRS_elem) * BabySNARK.SRSElementValue_G1 SRS_elem)
              BabySNARK.ListSRSElements_G1).sum =
          (List.map (fun SRS_elem => C (prover.2 Proof_Idx.B SRS_elem) * BabySNARK.SRSElementValue_G2 SRS_elem)
              BabySNARK.ListSRSElements_G2).sum ∧
        ∀ a ∈ [],
          (List.map (fun SRS_elem => C (prover.1 a.1 SRS_elem) * BabySNARK.SRSElementValue_G1 SRS_elem)
                BabySNARK.ListSRSElements_G1).sum =
            (List.map (fun SRS_elem => C (prover.2 a.2 SRS_elem) * BabySNARK.SRSElementValue_G2 SRS_elem)
                BabySNARK.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_Idx.B (SRS_Elements_Idx.βu i)) prover i) * u_wit i)
                  (List.finRange n_wit)).sum) ^
            2 -
          1).modByMonic
      t =
    0