Theorem: BabySNARK.is_sound
Proof state before:
case intro.intro.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
eqnH :
  (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
eqnV :
  (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
eqnB :
  (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
⊢ (((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:
suffices
    (((List.sum (List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)))
          + (List.sum (List.map (fun i => Polynomial.C (prover.fst Proof_Idx.B (SRS_Elements_Idx.βu i)) * u_wit i) (List.finRange n_wit)))) ^ 2
        - 1
      )
    =
    t * (List.sum (List.map (fun i => Polynomial.C (prover.fst Proof_Idx.H (SRS_Elements_Idx.βu i)) * u_wit i) (List.finRange n_wit))) by
  rw [this]
  apply Polynomial.mul_self_modByMonic
  assumption
Proof state after:
case intro.intro.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
eqnH :
  (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
eqnV :
  (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
eqnB :
  (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
⊢ ((List.map (fun i => Polynomial.C (stmt i) * u_stmt i) (List.finRange n_stmt)).sum +
          (List.map (fun i => Polynomial.C (prover.1 Proof_Idx.B (SRS_Elements_Idx.βu i)) * u_wit i)
              (List.finRange n_wit)).sum) ^
        2 -
      1 =
    t *
      (List.map (fun i => Polynomial.C (prover.1 Proof_Idx.H (SRS_Elements_Idx.βu i)) * u_wit i)
          (List.finRange n_wit)).sum