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 (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
Tactic:
simp_rw [BabySNARK] at eqnI eqnII eqnH eqnV eqnB
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
                        (match pairing with
                        | PairingsI_Idx.ht =>
                          match pf_elem with
                          | Proof_Idx.H => 1
                          | Proof_Idx.V => 0
                          | Proof_Idx.B => 0
                        | PairingsI_Idx.gg =>
                          match pf_elem with
                          | Proof_Idx.H => 0
                          | Proof_Idx.V => 0
                          | Proof_Idx.B => 0
                        | PairingsI_Idx.vv =>
                          match pf_elem with
                          | Proof_Idx.H => 0
                          | Proof_Idx.V => 1
                          | Proof_Idx.B => 0) *
                      (List.map
                          (fun SRS_elem =>
                            C (prover.1 pf_elem SRS_elem) *
                              match SRS_elem with
                              | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
                              | SRS_Elements_Idx.γ => X Vars_β
                              | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
                              | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
                          (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
                              [SRS_Elements_Idx.γβ] ++
                            List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum)
                  [Proof_Idx.H, Proof_Idx.V, Proof_Idx.B]).sum +
              (List.map
                  (fun SRS_elem =>
                    C
                        (match pairing with
                        | PairingsI_Idx.ht =>
                          match SRS_elem with
                          | SRS_Elements_Idx.τ_pow i => t.coeff ↑i
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.γβ => 0
                          | SRS_Elements_Idx.βu a => 0
                        | PairingsI_Idx.gg =>
                          match SRS_elem with
                          | SRS_Elements_Idx.τ_pow i => if ↑i = 0 then 1 else 0
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.γβ => 0
                          | SRS_Elements_Idx.βu a => 0
                        | PairingsI_Idx.vv =>
                          match SRS_elem with
                          | SRS_Elements_Idx.τ_pow i =>
                            (List.map (fun j => stmt j * (u_stmt j).coeff ↑i) (List.finRange n_stmt)).sum
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.γβ => 0
                          | SRS_Elements_Idx.βu a => 0) *
                      match SRS_elem with
                      | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
                      | SRS_Elements_Idx.γ => X Vars_β
                      | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
                      | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
                  (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
                      [SRS_Elements_Idx.γβ] ++
                    List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum) *
            ((List.map
                  (fun pf_elem =>
                    C
                        (match pairing with
                        | PairingsI_Idx.ht =>
                          match pf_elem with
                          | Proof_Idx.H => 0
                          | Proof_Idx.V => 0
                          | Proof_Idx.B => 0
                        | PairingsI_Idx.gg =>
                          match pf_elem with
                          | Proof_Idx.H => 0
                          | Proof_Idx.V => 0
                          | Proof_Idx.B => 0
                        | PairingsI_Idx.vv =>
                          match pf_elem with
                          | Proof_Idx.H => 0
                          | Proof_Idx.V => -1
                          | Proof_Idx.B => 0) *
                      (List.map
                          (fun SRS_elem =>
                            C (prover.2 pf_elem SRS_elem) *
                              match SRS_elem with
                              | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
                              | SRS_Elements_Idx.γ => X Vars_β
                              | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
                              | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
                          (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
                              [SRS_Elements_Idx.γβ] ++
                            List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum)
                  [Proof_Idx.H, Proof_Idx.V, Proof_Idx.B]).sum +
              (List.map
                  (fun SRS_elem =>
                    C
                        (match pairing with
                        | PairingsI_Idx.ht =>
                          match SRS_elem with
                          | SRS_Elements_Idx.τ_pow i => t.coeff ↑i
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.γβ => 0
                          | SRS_Elements_Idx.βu a => 0
                        | PairingsI_Idx.gg =>
                          match SRS_elem with
                          | SRS_Elements_Idx.τ_pow i => if ↑i = 0 then 1 else 0
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.γβ => 0
                          | SRS_Elements_Idx.βu a => 0
                        | PairingsI_Idx.vv =>
                          match SRS_elem with
                          | SRS_Elements_Idx.τ_pow i =>
                            -(List.map (fun j => stmt j * (u_stmt j).coeff ↑i) (List.finRange n_stmt)).sum
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.γβ => 0
                          | SRS_Elements_Idx.βu a => 0) *
                      match SRS_elem with
                      | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
                      | SRS_Elements_Idx.γ => X Vars_β
                      | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
                      | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
                  (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
                      [SRS_Elements_Idx.γβ] ++
                    List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum))
        [PairingsI_Idx.ht, PairingsI_Idx.gg, PairingsI_Idx.vv]).sum =
    0
eqnII :
  (List.map
        (fun pairing =>
          ((List.map
                  (fun pf_elem =>
                    C
                        (match pairing with
                        | PairingsII_Idx.bγ =>
                          match pf_elem with
                          | Proof_Idx.H => 0
                          | Proof_Idx.V => 0
                          | Proof_Idx.B => 1
                        | PairingsII_Idx.γβv =>
                          match pf_elem with
                          | Proof_Idx.H => 0
                          | Proof_Idx.V => 0
                          | Proof_Idx.B => 0) *
                      (List.map
                          (fun SRS_elem =>
                            C (prover.1 pf_elem SRS_elem) *
                              match SRS_elem with
                              | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
                              | SRS_Elements_Idx.γ => X Vars_β
                              | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
                              | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
                          (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
                              [SRS_Elements_Idx.γβ] ++
                            List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum)
                  [Proof_Idx.H, Proof_Idx.V, Proof_Idx.B]).sum +
              (List.map
                  (fun SRS_elem =>
                    C
                        (match pairing with
                        | PairingsII_Idx.bγ =>
                          match SRS_elem with
                          | SRS_Elements_Idx.τ_pow a => 0
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.γβ => 0
                          | SRS_Elements_Idx.βu a => 0
                        | PairingsII_Idx.γβv =>
                          match SRS_elem with
                          | SRS_Elements_Idx.τ_pow a => 0
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.γβ => 1
                          | SRS_Elements_Idx.βu a => 0) *
                      match SRS_elem with
                      | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
                      | SRS_Elements_Idx.γ => X Vars_β
                      | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
                      | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
                  (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
                      [SRS_Elements_Idx.γβ] ++
                    List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum) *
            ((List.map
                  (fun pf_elem =>
                    C
                        (match pairing with
                        | PairingsII_Idx.bγ =>
                          match pf_elem with
                          | Proof_Idx.H => 0
                          | Proof_Idx.V => 0
                          | Proof_Idx.B => 0
                        | PairingsII_Idx.γβv =>
                          match pf_elem with
                          | Proof_Idx.H => 0
                          | Proof_Idx.V => -1
                          | Proof_Idx.B => 0) *
                      (List.map
                          (fun SRS_elem =>
                            C (prover.2 pf_elem SRS_elem) *
                              match SRS_elem with
                              | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
                              | SRS_Elements_Idx.γ => X Vars_β
                              | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
                              | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
                          (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
                              [SRS_Elements_Idx.γβ] ++
                            List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum)
                  [Proof_Idx.H, Proof_Idx.V, Proof_Idx.B]).sum +
              (List.map
                  (fun SRS_elem =>
                    C
                        (match pairing with
                        | PairingsII_Idx.bγ =>
                          match SRS_elem with
                          | SRS_Elements_Idx.τ_pow a => 0
                          | SRS_Elements_Idx.γ => 1
                          | SRS_Elements_Idx.γβ => 0
                          | SRS_Elements_Idx.βu a => 0
                        | PairingsII_Idx.γβv =>
                          match SRS_elem with
                          | SRS_Elements_Idx.τ_pow a => 0
                          | SRS_Elements_Idx.γ => 0
                          | SRS_Elements_Idx.γβ => 0
                          | SRS_Elements_Idx.βu a => 0) *
                      match SRS_elem with
                      | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
                      | SRS_Elements_Idx.γ => X Vars_β
                      | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
                      | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
                  (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
                      [SRS_Elements_Idx.γβ] ++
                    List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum))
        [PairingsII_Idx.bγ, PairingsII_Idx.γβv]).sum =
    0
eqnH :
  (List.map
        (fun SRS_elem =>
          C (prover.1 Proof_Idx.H SRS_elem) *
            match SRS_elem with
            | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
            | SRS_Elements_Idx.γ => X Vars_β
            | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
            | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
        (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
            [SRS_Elements_Idx.γβ] ++
          List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum =
    (List.map
        (fun SRS_elem =>
          C (prover.2 Proof_Idx.H SRS_elem) *
            match SRS_elem with
            | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
            | SRS_Elements_Idx.γ => X Vars_β
            | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
            | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
        (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
            [SRS_Elements_Idx.γβ] ++
          List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum
eqnV :
  (List.map
        (fun SRS_elem =>
          C (prover.1 Proof_Idx.V SRS_elem) *
            match SRS_elem with
            | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
            | SRS_Elements_Idx.γ => X Vars_β
            | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
            | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
        (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
            [SRS_Elements_Idx.γβ] ++
          List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum =
    (List.map
        (fun SRS_elem =>
          C (prover.2 Proof_Idx.V SRS_elem) *
            match SRS_elem with
            | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
            | SRS_Elements_Idx.γ => X Vars_β
            | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
            | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
        (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
            [SRS_Elements_Idx.γβ] ++
          List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum
eqnB :
  (List.map
        (fun SRS_elem =>
          C (prover.1 Proof_Idx.B SRS_elem) *
            match SRS_elem with
            | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
            | SRS_Elements_Idx.γ => X Vars_β
            | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
            | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
        (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
            [SRS_Elements_Idx.γβ] ++
          List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).sum =
    (List.map
        (fun SRS_elem =>
          C (prover.2 Proof_Idx.B SRS_elem) *
            match SRS_elem with
            | SRS_Elements_Idx.τ_pow i => X Vars_τ ^ ↑i
            | SRS_Elements_Idx.γ => X Vars_β
            | SRS_Elements_Idx.γβ => X Vars_γ * X Vars_β
            | SRS_Elements_Idx.βu i => X Vars_β * (to_MvPolynomial_Option Vars) (u_wit i))
        (List.map (fun i => SRS_Elements_Idx.τ_pow i) (List.finRange n_var) ++ [SRS_Elements_Idx.γ] ++
            [SRS_Elements_Idx.γβ] ++
          List.map (fun i => SRS_Elements_Idx.βu i) (List.finRange n_wit))).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