Theorem: changeExponent_soundness
Proof state before:
case intro.a
F : Type
inst✝ : Field F
𝓟 : AGMProofSystemInstantiation F
sample : 𝓟.Sample
d : ℕ
hTypeIII : 𝓟.Identified_Proof_Elems = []
Wit : Type
relation : 𝓟.Stmt → Wit → Prop
extractor : AGMProofSystemInstantiation.Prover F 𝓟 → Wit
stmt : (changeExponent_G1 𝓟 sample d).Stmt
agm : AGMProofSystemInstantiation.Prover F (changeExponent_G1 𝓟 sample d)
poly_checks_pass :
  ∀ (check_idx : (changeExponent_G1 𝓟 sample d).EqualityChecks),
    (List.map
          (fun pairing =>
            ((List.map
                    (fun pf_elem =>
                      MvPolynomial.C
                          ((changeExponent_G1 𝓟 sample d).verificationPairingProof_G1 stmt check_idx pairing pf_elem) *
                        (List.map
                            (fun SRS_elem =>
                              MvPolynomial.C (agm.1 pf_elem SRS_elem) *
                                (changeExponent_G1 𝓟 sample d).SRSElementValue_G1 SRS_elem)
                            (changeExponent_G1 𝓟 sample d).ListSRSElements_G1).sum)
                    (changeExponent_G1 𝓟 sample d).ListProof_G1).sum +
                (List.map
                    (fun SRS_elem =>
                      MvPolynomial.C
                          ((changeExponent_G1 𝓟 sample d).verificationPairingSRS_G1 stmt check_idx pairing SRS_elem) *
                        (changeExponent_G1 𝓟 sample d).SRSElementValue_G1 SRS_elem)
                    (changeExponent_G1 𝓟 sample d).ListSRSElements_G1).sum) *
              ((List.map
                    (fun pf_elem =>
                      MvPolynomial.C
                          ((changeExponent_G1 𝓟 sample d).verificationPairingProof_G2 stmt check_idx pairing pf_elem) *
                        (List.map
                            (fun SRS_elem =>
                              MvPolynomial.C (agm.2 pf_elem SRS_elem) *
                                (changeExponent_G1 𝓟 sample d).SRSElementValue_G2 SRS_elem)
                            (changeExponent_G1 𝓟 sample d).ListSRSElements_G2).sum)
                    (changeExponent_G1 𝓟 sample d).ListProof_G2).sum +
                (List.map
                    (fun SRS_elem =>
                      MvPolynomial.C
                          ((changeExponent_G1 𝓟 sample d).verificationPairingSRS_G2 stmt check_idx pairing SRS_elem) *
                        (changeExponent_G1 𝓟 sample d).SRSElementValue_G2 SRS_elem)
                    (changeExponent_G1 𝓟 sample d).ListSRSElements_G2).sum))
          ((changeExponent_G1 𝓟 sample d).ListPairings check_idx)).sum =
      0
right✝ :
  ∀ pfs ∈ (changeExponent_G1 𝓟 sample d).Identified_Proof_Elems,
    (List.map
          (fun SRS_elem =>
            MvPolynomial.C (agm.1 pfs.1 SRS_elem) * (changeExponent_G1 𝓟 sample d).SRSElementValue_G1 SRS_elem)
          (changeExponent_G1 𝓟 sample d).ListSRSElements_G1).sum =
      (List.map
          (fun SRS_elem =>
            MvPolynomial.C (agm.2 pfs.2 SRS_elem) * (changeExponent_G1 𝓟 sample d).SRSElementValue_G2 SRS_elem)
          (changeExponent_G1 𝓟 sample d).ListSRSElements_G2).sum
⊢ (∀ (check_idx : 𝓟.EqualityChecks),
      (List.map
            (fun pairing =>
              ((List.map
                      (fun pf_elem =>
                        MvPolynomial.C (𝓟.verificationPairingProof_G1 stmt check_idx pairing pf_elem) *
                          (List.map
                              (fun SRS_elem => MvPolynomial.C (agm.1 pf_elem SRS_elem) * 𝓟.SRSElementValue_G1 SRS_elem)
                              𝓟.ListSRSElements_G1).sum)
                      𝓟.ListProof_G1).sum +
                  (List.map
                      (fun SRS_elem =>
                        MvPolynomial.C (𝓟.verificationPairingSRS_G1 stmt check_idx pairing SRS_elem) *
                          𝓟.SRSElementValue_G1 SRS_elem)
                      𝓟.ListSRSElements_G1).sum) *
                ((List.map
                      (fun pf_elem =>
                        MvPolynomial.C (𝓟.verificationPairingProof_G2 stmt check_idx pairing pf_elem) *
                          (List.map
                              (fun SRS_elem => MvPolynomial.C (agm.2 pf_elem SRS_elem) * 𝓟.SRSElementValue_G2 SRS_elem)
                              𝓟.ListSRSElements_G2).sum)
                      𝓟.ListProof_G2).sum +
                  (List.map
                      (fun SRS_elem =>
                        MvPolynomial.C (𝓟.verificationPairingSRS_G2 stmt check_idx pairing SRS_elem) *
                          𝓟.SRSElementValue_G2 SRS_elem)
                      𝓟.ListSRSElements_G2).sum))
            (𝓟.ListPairings check_idx)).sum =
        0) ∧
    ∀ pfs ∈ 𝓟.Identified_Proof_Elems,
      (List.map (fun SRS_elem => MvPolynomial.C (agm.1 pfs.1 SRS_elem) * 𝓟.SRSElementValue_G1 SRS_elem)
            𝓟.ListSRSElements_G1).sum =
        (List.map (fun SRS_elem => MvPolynomial.C (agm.2 pfs.2 SRS_elem) * 𝓟.SRSElementValue_G2 SRS_elem)
            𝓟.ListSRSElements_G2).sum
Tactic:
constructor
Proof state after:
case intro.a.left
F : Type
inst✝ : Field F
𝓟 : AGMProofSystemInstantiation F
sample : 𝓟.Sample
d : ℕ
hTypeIII : 𝓟.Identified_Proof_Elems = []
Wit : Type
relation : 𝓟.Stmt → Wit → Prop
extractor : AGMProofSystemInstantiation.Prover F 𝓟 → Wit
stmt : (changeExponent_G1 𝓟 sample d).Stmt
agm : AGMProofSystemInstantiation.Prover F (changeExponent_G1 𝓟 sample d)
poly_checks_pass :
  ∀ (check_idx : (changeExponent_G1 𝓟 sample d).EqualityChecks),
    (List.map
          (fun pairing =>
            ((List.map
                    (fun pf_elem =>
                      MvPolynomial.C
                          ((changeExponent_G1 𝓟 sample d).verificationPairingProof_G1 stmt check_idx pairing pf_elem) *
                        (List.map
                            (fun SRS_elem =>
                              MvPolynomial.C (agm.1 pf_elem SRS_elem) *
                                (changeExponent_G1 𝓟 sample d).SRSElementValue_G1 SRS_elem)
                            (changeExponent_G1 𝓟 sample d).ListSRSElements_G1).sum)
                    (changeExponent_G1 𝓟 sample d).ListProof_G1).sum +
                (List.map
                    (fun SRS_elem =>
                      MvPolynomial.C
                          ((changeExponent_G1 𝓟 sample d).verificationPairingSRS_G1 stmt check_idx pairing SRS_elem) *
                        (changeExponent_G1 𝓟 sample d).SRSElementValue_G1 SRS_elem)
                    (changeExponent_G1 𝓟 sample d).ListSRSElements_G1).sum) *
              ((List.map
                    (fun pf_elem =>
                      MvPolynomial.C
                          ((changeExponent_G1 𝓟 sample d).verificationPairingProof_G2 stmt check_idx pairing pf_elem) *
                        (List.map
                            (fun SRS_elem =>
                              MvPolynomial.C (agm.2 pf_elem SRS_elem) *
                                (changeExponent_G1 𝓟 sample d).SRSElementValue_G2 SRS_elem)
                            (changeExponent_G1 𝓟 sample d).ListSRSElements_G2).sum)
                    (changeExponent_G1 𝓟 sample d).ListProof_G2).sum +
                (List.map
                    (fun SRS_elem =>
                      MvPolynomial.C
                          ((changeExponent_G1 𝓟 sample d).verificationPairingSRS_G2 stmt check_idx pairing SRS_elem) *
                        (changeExponent_G1 𝓟 sample d).SRSElementValue_G2 SRS_elem)
                    (changeExponent_G1 𝓟 sample d).ListSRSElements_G2).sum))
          ((changeExponent_G1 𝓟 sample d).ListPairings check_idx)).sum =
      0
right✝ :
  ∀ pfs ∈ (changeExponent_G1 𝓟 sample d).Identified_Proof_Elems,
    (List.map
          (fun SRS_elem =>
            MvPolynomial.C (agm.1 pfs.1 SRS_elem) * (changeExponent_G1 𝓟 sample d).SRSElementValue_G1 SRS_elem)
          (changeExponent_G1 𝓟 sample d).ListSRSElements_G1).sum =
      (List.map
          (fun SRS_elem =>
            MvPolynomial.C (agm.2 pfs.2 SRS_elem) * (changeExponent_G1 𝓟 sample d).SRSElementValue_G2 SRS_elem)
          (changeExponent_G1 𝓟 sample d).ListSRSElements_G2).sum
⊢ ∀ (check_idx : 𝓟.EqualityChecks),
    (List.map
          (fun pairing =>
            ((List.map
                    (fun pf_elem =>
                      MvPolynomial.C (𝓟.verificationPairingProof_G1 stmt check_idx pairing pf_elem) *
                        (List.map
                            (fun SRS_elem => MvPolynomial.C (agm.1 pf_elem SRS_elem) * 𝓟.SRSElementValue_G1 SRS_elem)
                            𝓟.ListSRSElements_G1).sum)
                    𝓟.ListProof_G1).sum +
                (List.map
                    (fun SRS_elem =>
                      MvPolynomial.C (𝓟.verificationPairingSRS_G1 stmt check_idx pairing SRS_elem) *
                        𝓟.SRSElementValue_G1 SRS_elem)
                    𝓟.ListSRSElements_G1).sum) *
              ((List.map
                    (fun pf_elem =>
                      MvPolynomial.C (𝓟.verificationPairingProof_G2 stmt check_idx pairing pf_elem) *
                        (List.map
                            (fun SRS_elem => MvPolynomial.C (agm.2 pf_elem SRS_elem) * 𝓟.SRSElementValue_G2 SRS_elem)
                            𝓟.ListSRSElements_G2).sum)
                    𝓟.ListProof_G2).sum +
                (List.map
                    (fun SRS_elem =>
                      MvPolynomial.C (𝓟.verificationPairingSRS_G2 stmt check_idx pairing SRS_elem) *
                        𝓟.SRSElementValue_G2 SRS_elem)
                    𝓟.ListSRSElements_G2).sum))
          (𝓟.ListPairings check_idx)).sum =
      0

case intro.a.right
F : Type
inst✝ : Field F
𝓟 : AGMProofSystemInstantiation F
sample : 𝓟.Sample
d : ℕ
hTypeIII : 𝓟.Identified_Proof_Elems = []
Wit : Type
relation : 𝓟.Stmt → Wit → Prop
extractor : AGMProofSystemInstantiation.Prover F 𝓟 → Wit
stmt : (changeExponent_G1 𝓟 sample d).Stmt
agm : AGMProofSystemInstantiation.Prover F (changeExponent_G1 𝓟 sample d)
poly_checks_pass :
  ∀ (check_idx : (changeExponent_G1 𝓟 sample d).EqualityChecks),
    (List.map
          (fun pairing =>
            ((List.map
                    (fun pf_elem =>
                      MvPolynomial.C
                          ((changeExponent_G1 𝓟 sample d).verificationPairingProof_G1 stmt check_idx pairing pf_elem) *
                        (List.map
                            (fun SRS_elem =>
                              MvPolynomial.C (agm.1 pf_elem SRS_elem) *
                                (changeExponent_G1 𝓟 sample d).SRSElementValue_G1 SRS_elem)
                            (changeExponent_G1 𝓟 sample d).ListSRSElements_G1).sum)
                    (changeExponent_G1 𝓟 sample d).ListProof_G1).sum +
                (List.map
                    (fun SRS_elem =>
                      MvPolynomial.C
                          ((changeExponent_G1 𝓟 sample d).verificationPairingSRS_G1 stmt check_idx pairing SRS_elem) *
                        (changeExponent_G1 𝓟 sample d).SRSElementValue_G1 SRS_elem)
                    (changeExponent_G1 𝓟 sample d).ListSRSElements_G1).sum) *
              ((List.map
                    (fun pf_elem =>
                      MvPolynomial.C
                          ((changeExponent_G1 𝓟 sample d).verificationPairingProof_G2 stmt check_idx pairing pf_elem) *
                        (List.map
                            (fun SRS_elem =>
                              MvPolynomial.C (agm.2 pf_elem SRS_elem) *
                                (changeExponent_G1 𝓟 sample d).SRSElementValue_G2 SRS_elem)
                            (changeExponent_G1 𝓟 sample d).ListSRSElements_G2).sum)
                    (changeExponent_G1 𝓟 sample d).ListProof_G2).sum +
                (List.map
                    (fun SRS_elem =>
                      MvPolynomial.C
                          ((changeExponent_G1 𝓟 sample d).verificationPairingSRS_G2 stmt check_idx pairing SRS_elem) *
                        (changeExponent_G1 𝓟 sample d).SRSElementValue_G2 SRS_elem)
                    (changeExponent_G1 𝓟 sample d).ListSRSElements_G2).sum))
          ((changeExponent_G1 𝓟 sample d).ListPairings check_idx)).sum =
      0
right✝ :
  ∀ pfs ∈ (changeExponent_G1 𝓟 sample d).Identified_Proof_Elems,
    (List.map
          (fun SRS_elem =>
            MvPolynomial.C (agm.1 pfs.1 SRS_elem) * (changeExponent_G1 𝓟 sample d).SRSElementValue_G1 SRS_elem)
          (changeExponent_G1 𝓟 sample d).ListSRSElements_G1).sum =
      (List.map
          (fun SRS_elem =>
            MvPolynomial.C (agm.2 pfs.2 SRS_elem) * (changeExponent_G1 𝓟 sample d).SRSElementValue_G2 SRS_elem)
          (changeExponent_G1 𝓟 sample d).ListSRSElements_G2).sum
⊢ ∀ pfs ∈ 𝓟.Identified_Proof_Elems,
    (List.map (fun SRS_elem => MvPolynomial.C (agm.1 pfs.1 SRS_elem) * 𝓟.SRSElementValue_G1 SRS_elem)
          𝓟.ListSRSElements_G1).sum =
      (List.map (fun SRS_elem => MvPolynomial.C (agm.2 pfs.2 SRS_elem) * 𝓟.SRSElementValue_G2 SRS_elem)
          𝓟.ListSRSElements_G2).sum