Theorem: collapseSRSElement_G1_soundness
Proof state before:
case intro.a.left
F : Type
inst✝¹ : Field F
𝓟 : AGMProofSystemInstantiation F
hTypeIII : 𝓟.Identified_Proof_Elems = []
inst✝ : DecidableEq 𝓟.SRSElements_G1
twin1 twin2 : 𝓟.SRSElements_G1
not_same : twin1 ≠ twin2
hcount1 : List.count twin1 𝓟.ListSRSElements_G1 = 1
hcount2 : List.count twin2 𝓟.ListSRSElements_G1 = 1
interchangeable : ∀ (idx : 𝓟.Proof_G1) (agm : 𝓟.Proof_G1 → 𝓟.SRSElements_G1 → F), agm idx twin1 = agm idx twin2
interchangeable' :
  ∀ (stmt : 𝓟.Stmt) (check_idx : 𝓟.EqualityChecks) (pairing : 𝓟.Pairings check_idx),
    𝓟.verificationPairingSRS_G1 stmt check_idx pairing twin1 = 𝓟.verificationPairingSRS_G1 stmt check_idx pairing twin2
Wit : Type
relation : 𝓟.Stmt → Wit → Prop
extractor : AGMProofSystemInstantiation.Prover F 𝓟 → Wit
stmt : (collapseSRSElement_G1 𝓟 twin1 twin2).Stmt
agm : AGMProofSystemInstantiation.Prover F (collapseSRSElement_G1 𝓟 twin1 twin2)
check_idx : 𝓟.EqualityChecks
poly_checks_pass :
  (List.map
        (fun pairing =>
          ((List.map
                  (fun pf_elem =>
                    MvPolynomial.C
                        ((collapseSRSElement_G1 𝓟 twin1 twin2).verificationPairingProof_G1 stmt check_idx pairing
                          pf_elem) *
                      (List.map
                          (fun SRS_elem =>
                            MvPolynomial.C (agm.1 pf_elem SRS_elem) *
                              (collapseSRSElement_G1 𝓟 twin1 twin2).SRSElementValue_G1 SRS_elem)
                          (collapseSRSElement_G1 𝓟 twin1 twin2).ListSRSElements_G1).sum)
                  (collapseSRSElement_G1 𝓟 twin1 twin2).ListProof_G1).sum +
              (List.map
                  (fun SRS_elem =>
                    MvPolynomial.C
                        ((collapseSRSElement_G1 𝓟 twin1 twin2).verificationPairingSRS_G1 stmt check_idx pairing
                          SRS_elem) *
                      (collapseSRSElement_G1 𝓟 twin1 twin2).SRSElementValue_G1 SRS_elem)
                  (collapseSRSElement_G1 𝓟 twin1 twin2).ListSRSElements_G1).sum) *
            ((List.map
                  (fun pf_elem =>
                    MvPolynomial.C
                        ((collapseSRSElement_G1 𝓟 twin1 twin2).verificationPairingProof_G2 stmt check_idx pairing
                          pf_elem) *
                      (List.map
                          (fun SRS_elem =>
                            MvPolynomial.C (agm.2 pf_elem SRS_elem) *
                              (collapseSRSElement_G1 𝓟 twin1 twin2).SRSElementValue_G2 SRS_elem)
                          (collapseSRSElement_G1 𝓟 twin1 twin2).ListSRSElements_G2).sum)
                  (collapseSRSElement_G1 𝓟 twin1 twin2).ListProof_G2).sum +
              (List.map
                  (fun SRS_elem =>
                    MvPolynomial.C
                        ((collapseSRSElement_G1 𝓟 twin1 twin2).verificationPairingSRS_G2 stmt check_idx pairing
                          SRS_elem) *
                      (collapseSRSElement_G1 𝓟 twin1 twin2).SRSElementValue_G2 SRS_elem)
                  (collapseSRSElement_G1 𝓟 twin1 twin2).ListSRSElements_G2).sum))
        ((collapseSRSElement_G1 𝓟 twin1 twin2).ListPairings check_idx)).sum =
    0
⊢ (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
Tactic:
unfold collapseSRSElement_G1 at poly_checks_pass
Proof state after:
case intro.a.left
F : Type
inst✝¹ : Field F
𝓟 : AGMProofSystemInstantiation F
hTypeIII : 𝓟.Identified_Proof_Elems = []
inst✝ : DecidableEq 𝓟.SRSElements_G1
twin1 twin2 : 𝓟.SRSElements_G1
not_same : twin1 ≠ twin2
hcount1 : List.count twin1 𝓟.ListSRSElements_G1 = 1
hcount2 : List.count twin2 𝓟.ListSRSElements_G1 = 1
interchangeable : ∀ (idx : 𝓟.Proof_G1) (agm : 𝓟.Proof_G1 → 𝓟.SRSElements_G1 → F), agm idx twin1 = agm idx twin2
interchangeable' :
  ∀ (stmt : 𝓟.Stmt) (check_idx : 𝓟.EqualityChecks) (pairing : 𝓟.Pairings check_idx),
    𝓟.verificationPairingSRS_G1 stmt check_idx pairing twin1 = 𝓟.verificationPairingSRS_G1 stmt check_idx pairing twin2
Wit : Type
relation : 𝓟.Stmt → Wit → Prop
extractor : AGMProofSystemInstantiation.Prover F 𝓟 → Wit
stmt : (collapseSRSElement_G1 𝓟 twin1 twin2).Stmt
agm : AGMProofSystemInstantiation.Prover F (collapseSRSElement_G1 𝓟 twin1 twin2)
check_idx : 𝓟.EqualityChecks
poly_checks_pass :
  (List.map
        (fun pairing =>
          ((List.map
                  (fun pf_elem =>
                    MvPolynomial.C
                        ({ Stmt := 𝓟.Stmt, Sample := 𝓟.Sample, SRSElements_G1 := 𝓟.SRSElements_G1,
                              ListSRSElements_G1 := 𝓟.ListSRSElements_G1, SRSElements_G2 := 𝓟.SRSElements_G2,
                              ListSRSElements_G2 := 𝓟.ListSRSElements_G2,
                              SRSElementValue_G1 := fun srs =>
                                if srs = twin1 then 𝓟.SRSElementValue_G1 twin1 + 𝓟.SRSElementValue_G1 twin2
                                else if srs = twin2 then 0 else 𝓟.SRSElementValue_G1 srs,
                              SRSElementValue_G2 := 𝓟.SRSElementValue_G2, Proof_G1 := 𝓟.Proof_G1,
                              ListProof_G1 := 𝓟.ListProof_G1, Proof_G2 := 𝓟.Proof_G2, ListProof_G2 := 𝓟.ListProof_G2,
                              EqualityChecks := 𝓟.EqualityChecks, Pairings := 𝓟.Pairings,
                              ListPairings := 𝓟.ListPairings, verificationPairingSRS_G1 := 𝓟.verificationPairingSRS_G1,
                              verificationPairingSRS_G2 := 𝓟.verificationPairingSRS_G2,
                              verificationPairingProof_G1 := 𝓟.verificationPairingProof_G1,
                              verificationPairingProof_G2 := 𝓟.verificationPairingProof_G2,
                              Identified_Proof_Elems := 𝓟.Identified_Proof_Elems }.verificationPairingProof_G1
                          stmt check_idx pairing pf_elem) *
                      (List.map
                          (fun SRS_elem =>
                            MvPolynomial.C (agm.1 pf_elem SRS_elem) *
                              { Stmt := 𝓟.Stmt, Sample := 𝓟.Sample, SRSElements_G1 := 𝓟.SRSElements_G1,
                                    ListSRSElements_G1 := 𝓟.ListSRSElements_G1, SRSElements_G2 := 𝓟.SRSElements_G2,
                                    ListSRSElements_G2 := 𝓟.ListSRSElements_G2,
                                    SRSElementValue_G1 := fun srs =>
                                      if srs = twin1 then 𝓟.SRSElementValue_G1 twin1 + 𝓟.SRSElementValue_G1 twin2
                                      else if srs = twin2 then 0 else 𝓟.SRSElementValue_G1 srs,
                                    SRSElementValue_G2 := 𝓟.SRSElementValue_G2, Proof_G1 := 𝓟.Proof_G1,
                                    ListProof_G1 := 𝓟.ListProof_G1, Proof_G2 := 𝓟.Proof_G2,
                                    ListProof_G2 := 𝓟.ListProof_G2, EqualityChecks := 𝓟.EqualityChecks,
                                    Pairings := 𝓟.Pairings, ListPairings := 𝓟.ListPairings,
                                    verificationPairingSRS_G1 := 𝓟.verificationPairingSRS_G1,
                                    verificationPairingSRS_G2 := 𝓟.verificationPairingSRS_G2,
                                    verificationPairingProof_G1 := 𝓟.verificationPairingProof_G1,
                                    verificationPairingProof_G2 := 𝓟.verificationPairingProof_G2,
                                    Identified_Proof_Elems := 𝓟.Identified_Proof_Elems }.SRSElementValue_G1
                                SRS_elem)
                          { Stmt := 𝓟.Stmt, Sample := 𝓟.Sample, SRSElements_G1 := 𝓟.SRSElements_G1,
                              ListSRSElements_G1 := 𝓟.ListSRSElements_G1, SRSElements_G2 := 𝓟.SRSElements_G2,
                              ListSRSElements_G2 := 𝓟.ListSRSElements_G2,
                              SRSElementValue_G1 := fun srs =>
                                if srs = twin1 then 𝓟.SRSElementValue_G1 twin1 + 𝓟.SRSElementValue_G1 twin2
                                else if srs = twin2 then 0 else 𝓟.SRSElementValue_G1 srs,
                              SRSElementValue_G2 := 𝓟.SRSElementValue_G2, Proof_G1 := 𝓟.Proof_G1,
                              ListProof_G1 := 𝓟.ListProof_G1, Proof_G2 := 𝓟.Proof_G2, ListProof_G2 := 𝓟.ListProof_G2,
                              EqualityChecks := 𝓟.EqualityChecks, Pairings := 𝓟.Pairings,
                              ListPairings := 𝓟.ListPairings, verificationPairingSRS_G1 := 𝓟.verificationPairingSRS_G1,
                              verificationPairingSRS_G2 := 𝓟.verificationPairingSRS_G2,
                              verificationPairingProof_G1 := 𝓟.verificationPairingProof_G1,
                              verificationPairingProof_G2 := 𝓟.verificationPairingProof_G2,
                              Identified_Proof_Elems := 𝓟.Identified_Proof_Elems }.ListSRSElements_G1).sum)
                  { Stmt := 𝓟.Stmt, Sample := 𝓟.Sample, SRSElements_G1 := 𝓟.SRSElements_G1,
                      ListSRSElements_G1 := 𝓟.ListSRSElements_G1, SRSElements_G2 := 𝓟.SRSElements_G2,
                      ListSRSElements_G2 := 𝓟.ListSRSElements_G2,
                      SRSElementValue_G1 := fun srs =>
                        if srs = twin1 then 𝓟.SRSElementValue_G1 twin1 + 𝓟.SRSElementValue_G1 twin2
                        else if srs = twin2 then 0 else 𝓟.SRSElementValue_G1 srs,
                      SRSElementValue_G2 := 𝓟.SRSElementValue_G2, Proof_G1 := 𝓟.Proof_G1,
                      ListProof_G1 := 𝓟.ListProof_G1, Proof_G2 := 𝓟.Proof_G2, ListProof_G2 := 𝓟.ListProof_G2,
                      EqualityChecks := 𝓟.EqualityChecks, Pairings := 𝓟.Pairings, ListPairings := 𝓟.ListPairings,
                      verificationPairingSRS_G1 := 𝓟.verificationPairingSRS_G1,
                      verificationPairingSRS_G2 := 𝓟.verificationPairingSRS_G2,
                      verificationPairingProof_G1 := 𝓟.verificationPairingProof_G1,
                      verificationPairingProof_G2 := 𝓟.verificationPairingProof_G2,
                      Identified_Proof_Elems := 𝓟.Identified_Proof_Elems }.ListProof_G1).sum +
              (List.map
                  (fun SRS_elem =>
                    MvPolynomial.C
                        ({ Stmt := 𝓟.Stmt, Sample := 𝓟.Sample, SRSElements_G1 := 𝓟.SRSElements_G1,
                              ListSRSElements_G1 := 𝓟.ListSRSElements_G1, SRSElements_G2 := 𝓟.SRSElements_G2,
                              ListSRSElements_G2 := 𝓟.ListSRSElements_G2,
                              SRSElementValue_G1 := fun srs =>
                                if srs = twin1 then 𝓟.SRSElementValue_G1 twin1 + 𝓟.SRSElementValue_G1 twin2
                                else if srs = twin2 then 0 else 𝓟.SRSElementValue_G1 srs,
                              SRSElementValue_G2 := 𝓟.SRSElementValue_G2, Proof_G1 := 𝓟.Proof_G1,
                              ListProof_G1 := 𝓟.ListProof_G1, Proof_G2 := 𝓟.Proof_G2, ListProof_G2 := 𝓟.ListProof_G2,
                              EqualityChecks := 𝓟.EqualityChecks, Pairings := 𝓟.Pairings,
                              ListPairings := 𝓟.ListPairings, verificationPairingSRS_G1 := 𝓟.verificationPairingSRS_G1,
                              verificationPairingSRS_G2 := 𝓟.verificationPairingSRS_G2,
                              verificationPairingProof_G1 := 𝓟.verificationPairingProof_G1,
                              verificationPairingProof_G2 := 𝓟.verificationPairingProof_G2,
                              Identified_Proof_Elems := 𝓟.Identified_Proof_Elems }.verificationPairingSRS_G1
                          stmt check_idx pairing SRS_elem) *
                      { Stmt := 𝓟.Stmt, Sample := 𝓟.Sample, SRSElements_G1 := 𝓟.SRSElements_G1,
                            ListSRSElements_G1 := 𝓟.ListSRSElements_G1, SRSElements_G2 := 𝓟.SRSElements_G2,
                            ListSRSElements_G2 := 𝓟.ListSRSElements_G2,
                            SRSElementValue_G1 := fun srs =>
                              if srs = twin1 then 𝓟.SRSElementValue_G1 twin1 + 𝓟.SRSElementValue_G1 twin2
                              else if srs = twin2 then 0 else 𝓟.SRSElementValue_G1 srs,
                            SRSElementValue_G2 := 𝓟.SRSElementValue_G2, Proof_G1 := 𝓟.Proof_G1,
                            ListProof_G1 := 𝓟.ListProof_G1, Proof_G2 := 𝓟.Proof_G2, ListProof_G2 := 𝓟.ListProof_G2,
                            EqualityChecks := 𝓟.EqualityChecks, Pairings := 𝓟.Pairings, ListPairings := 𝓟.ListPairings,
                            verificationPairingSRS_G1 := 𝓟.verificationPairingSRS_G1,
                            verificationPairingSRS_G2 := 𝓟.verificationPairingSRS_G2,
                            verificationPairingProof_G1 := 𝓟.verificationPairingProof_G1,
                            verificationPairingProof_G2 := 𝓟.verificationPairingProof_G2,
                            Identified_Proof_Elems := 𝓟.Identified_Proof_Elems }.SRSElementValue_G1
                        SRS_elem)
                  { Stmt := 𝓟.Stmt, Sample := 𝓟.Sample, SRSElements_G1 := 𝓟.SRSElements_G1,
                      ListSRSElements_G1 := 𝓟.ListSRSElements_G1, SRSElements_G2 := 𝓟.SRSElements_G2,
                      ListSRSElements_G2 := 𝓟.ListSRSElements_G2,
                      SRSElementValue_G1 := fun srs =>
                        if srs = twin1 then 𝓟.SRSElementValue_G1 twin1 + 𝓟.SRSElementValue_G1 twin2
                        else if srs = twin2 then 0 else 𝓟.SRSElementValue_G1 srs,
                      SRSElementValue_G2 := 𝓟.SRSElementValue_G2, Proof_G1 := 𝓟.Proof_G1,
                      ListProof_G1 := 𝓟.ListProof_G1, Proof_G2 := 𝓟.Proof_G2, ListProof_G2 := 𝓟.ListProof_G2,
                      EqualityChecks := 𝓟.EqualityChecks, Pairings := 𝓟.Pairings, ListPairings := 𝓟.ListPairings,
                      verificationPairingSRS_G1 := 𝓟.verificationPairingSRS_G1,
                      verificationPairingSRS_G2 := 𝓟.verificationPairingSRS_G2,
                      verificationPairingProof_G1 := 𝓟.verificationPairingProof_G1,
                      verificationPairingProof_G2 := 𝓟.verificationPairingProof_G2,
                      Identified_Proof_Elems := 𝓟.Identified_Proof_Elems }.ListSRSElements_G1).sum) *
            ((List.map
                  (fun pf_elem =>
                    MvPolynomial.C
                        ({ Stmt := 𝓟.Stmt, Sample := 𝓟.Sample, SRSElements_G1 := 𝓟.SRSElements_G1,
                              ListSRSElements_G1 := 𝓟.ListSRSElements_G1, SRSElements_G2 := 𝓟.SRSElements_G2,
                              ListSRSElements_G2 := 𝓟.ListSRSElements_G2,
                              SRSElementValue_G1 := fun srs =>
                                if srs = twin1 then 𝓟.SRSElementValue_G1 twin1 + 𝓟.SRSElementValue_G1 twin2
                                else if srs = twin2 then 0 else 𝓟.SRSElementValue_G1 srs,
                              SRSElementValue_G2 := 𝓟.SRSElementValue_G2, Proof_G1 := 𝓟.Proof_G1,
                              ListProof_G1 := 𝓟.ListProof_G1, Proof_G2 := 𝓟.Proof_G2, ListProof_G2 := 𝓟.ListProof_G2,
                              EqualityChecks := 𝓟.EqualityChecks, Pairings := 𝓟.Pairings,
                              ListPairings := 𝓟.ListPairings, verificationPairingSRS_G1 := 𝓟.verificationPairingSRS_G1,
                              verificationPairingSRS_G2 := 𝓟.verificationPairingSRS_G2,
                              verificationPairingProof_G1 := 𝓟.verificationPairingProof_G1,
                              verificationPairingProof_G2 := 𝓟.verificationPairingProof_G2,
                              Identified_Proof_Elems := 𝓟.Identified_Proof_Elems }.verificationPairingProof_G2
                          stmt check_idx pairing pf_elem) *
                      (List.map
                          (fun SRS_elem =>
                            MvPolynomial.C (agm.2 pf_elem SRS_elem) *
                              { Stmt := 𝓟.Stmt, Sample := 𝓟.Sample, SRSElements_G1 := 𝓟.SRSElements_G1,
                                    ListSRSElements_G1 := 𝓟.ListSRSElements_G1, SRSElements_G2 := 𝓟.SRSElements_G2,
                                    ListSRSElements_G2 := 𝓟.ListSRSElements_G2,
                                    SRSElementValue_G1 := fun srs =>
                                      if srs = twin1 then 𝓟.SRSElementValue_G1 twin1 + 𝓟.SRSElementValue_G1 twin2
                                      else if srs = twin2 then 0 else 𝓟.SRSElementValue_G1 srs,
                                    SRSElementValue_G2 := 𝓟.SRSElementValue_G2, Proof_G1 := 𝓟.Proof_G1,
                                    ListProof_G1 := 𝓟.ListProof_G1, Proof_G2 := 𝓟.Proof_G2,
                                    ListProof_G2 := 𝓟.ListProof_G2, EqualityChecks := 𝓟.EqualityChecks,
                                    Pairings := 𝓟.Pairings, ListPairings := 𝓟.ListPairings,
                                    verificationPairingSRS_G1 := 𝓟.verificationPairingSRS_G1,
                                    verificationPairingSRS_G2 := 𝓟.verificationPairingSRS_G2,
                                    verificationPairingProof_G1 := 𝓟.verificationPairingProof_G1,
                                    verificationPairingProof_G2 := 𝓟.verificationPairingProof_G2,
                                    Identified_Proof_Elems := 𝓟.Identified_Proof_Elems }.SRSElementValue_G2
                                SRS_elem)
                          { Stmt := 𝓟.Stmt, Sample := 𝓟.Sample, SRSElements_G1 := 𝓟.SRSElements_G1,
                              ListSRSElements_G1 := 𝓟.ListSRSElements_G1, SRSElements_G2 := 𝓟.SRSElements_G2,
                              ListSRSElements_G2 := 𝓟.ListSRSElements_G2,
                              SRSElementValue_G1 := fun srs =>
                                if srs = twin1 then 𝓟.SRSElementValue_G1 twin1 + 𝓟.SRSElementValue_G1 twin2
                                else if srs = twin2 then 0 else 𝓟.SRSElementValue_G1 srs,
                              SRSElementValue_G2 := 𝓟.SRSElementValue_G2, Proof_G1 := 𝓟.Proof_G1,
                              ListProof_G1 := 𝓟.ListProof_G1, Proof_G2 := 𝓟.Proof_G2, ListProof_G2 := 𝓟.ListProof_G2,
                              EqualityChecks := 𝓟.EqualityChecks, Pairings := 𝓟.Pairings,
                              ListPairings := 𝓟.ListPairings, verificationPairingSRS_G1 := 𝓟.verificationPairingSRS_G1,
                              verificationPairingSRS_G2 := 𝓟.verificationPairingSRS_G2,
                              verificationPairingProof_G1 := 𝓟.verificationPairingProof_G1,
                              verificationPairingProof_G2 := 𝓟.verificationPairingProof_G2,
                              Identified_Proof_Elems := 𝓟.Identified_Proof_Elems }.ListSRSElements_G2).sum)
                  { Stmt := 𝓟.Stmt, Sample := 𝓟.Sample, SRSElements_G1 := 𝓟.SRSElements_G1,
                      ListSRSElements_G1 := 𝓟.ListSRSElements_G1, SRSElements_G2 := 𝓟.SRSElements_G2,
                      ListSRSElements_G2 := 𝓟.ListSRSElements_G2,
                      SRSElementValue_G1 := fun srs =>
                        if srs = twin1 then 𝓟.SRSElementValue_G1 twin1 + 𝓟.SRSElementValue_G1 twin2
                        else if srs = twin2 then 0 else 𝓟.SRSElementValue_G1 srs,
                      SRSElementValue_G2 := 𝓟.SRSElementValue_G2, Proof_G1 := 𝓟.Proof_G1,
                      ListProof_G1 := 𝓟.ListProof_G1, Proof_G2 := 𝓟.Proof_G2, ListProof_G2 := 𝓟.ListProof_G2,
                      EqualityChecks := 𝓟.EqualityChecks, Pairings := 𝓟.Pairings, ListPairings := 𝓟.ListPairings,
                      verificationPairingSRS_G1 := 𝓟.verificationPairingSRS_G1,
                      verificationPairingSRS_G2 := 𝓟.verificationPairingSRS_G2,
                      verificationPairingProof_G1 := 𝓟.verificationPairingProof_G1,
                      verificationPairingProof_G2 := 𝓟.verificationPairingProof_G2,
                      Identified_Proof_Elems := 𝓟.Identified_Proof_Elems }.ListProof_G2).sum +
              (List.map
                  (fun SRS_elem =>
                    MvPolynomial.C
                        ({ Stmt := 𝓟.Stmt, Sample := 𝓟.Sample, SRSElements_G1 := 𝓟.SRSElements_G1,
                              ListSRSElements_G1 := 𝓟.ListSRSElements_G1, SRSElements_G2 := 𝓟.SRSElements_G2,
                              ListSRSElements_G2 := 𝓟.ListSRSElements_G2,
                              SRSElementValue_G1 := fun srs =>
                                if srs = twin1 then 𝓟.SRSElementValue_G1 twin1 + 𝓟.SRSElementValue_G1 twin2
                                else if srs = twin2 then 0 else 𝓟.SRSElementValue_G1 srs,
                              SRSElementValue_G2 := 𝓟.SRSElementValue_G2, Proof_G1 := 𝓟.Proof_G1,
                              ListProof_G1 := 𝓟.ListProof_G1, Proof_G2 := 𝓟.Proof_G2, ListProof_G2 := 𝓟.ListProof_G2,
                              EqualityChecks := 𝓟.EqualityChecks, Pairings := 𝓟.Pairings,
                              ListPairings := 𝓟.ListPairings, verificationPairingSRS_G1 := 𝓟.verificationPairingSRS_G1,
                              verificationPairingSRS_G2 := 𝓟.verificationPairingSRS_G2,
                              verificationPairingProof_G1 := 𝓟.verificationPairingProof_G1,
                              verificationPairingProof_G2 := 𝓟.verificationPairingProof_G2,
                              Identified_Proof_Elems := 𝓟.Identified_Proof_Elems }.verificationPairingSRS_G2
                          stmt check_idx pairing SRS_elem) *
                      { Stmt := 𝓟.Stmt, Sample := 𝓟.Sample, SRSElements_G1 := 𝓟.SRSElements_G1,
                            ListSRSElements_G1 := 𝓟.ListSRSElements_G1, SRSElements_G2 := 𝓟.SRSElements_G2,
                            ListSRSElements_G2 := 𝓟.ListSRSElements_G2,
                            SRSElementValue_G1 := fun srs =>
                              if srs = twin1 then 𝓟.SRSElementValue_G1 twin1 + 𝓟.SRSElementValue_G1 twin2
                              else if srs = twin2 then 0 else 𝓟.SRSElementValue_G1 srs,
                            SRSElementValue_G2 := 𝓟.SRSElementValue_G2, Proof_G1 := 𝓟.Proof_G1,
                            ListProof_G1 := 𝓟.ListProof_G1, Proof_G2 := 𝓟.Proof_G2, ListProof_G2 := 𝓟.ListProof_G2,
                            EqualityChecks := 𝓟.EqualityChecks, Pairings := 𝓟.Pairings, ListPairings := 𝓟.ListPairings,
                            verificationPairingSRS_G1 := 𝓟.verificationPairingSRS_G1,
                            verificationPairingSRS_G2 := 𝓟.verificationPairingSRS_G2,
                            verificationPairingProof_G1 := 𝓟.verificationPairingProof_G1,
                            verificationPairingProof_G2 := 𝓟.verificationPairingProof_G2,
                            Identified_Proof_Elems := 𝓟.Identified_Proof_Elems }.SRSElementValue_G2
                        SRS_elem)
                  { Stmt := 𝓟.Stmt, Sample := 𝓟.Sample, SRSElements_G1 := 𝓟.SRSElements_G1,
                      ListSRSElements_G1 := 𝓟.ListSRSElements_G1, SRSElements_G2 := 𝓟.SRSElements_G2,
                      ListSRSElements_G2 := 𝓟.ListSRSElements_G2,
                      SRSElementValue_G1 := fun srs =>
                        if srs = twin1 then 𝓟.SRSElementValue_G1 twin1 + 𝓟.SRSElementValue_G1 twin2
                        else if srs = twin2 then 0 else 𝓟.SRSElementValue_G1 srs,
                      SRSElementValue_G2 := 𝓟.SRSElementValue_G2, Proof_G1 := 𝓟.Proof_G1,
                      ListProof_G1 := 𝓟.ListProof_G1, Proof_G2 := 𝓟.Proof_G2, ListProof_G2 := 𝓟.ListProof_G2,
                      EqualityChecks := 𝓟.EqualityChecks, Pairings := 𝓟.Pairings, ListPairings := 𝓟.ListPairings,
                      verificationPairingSRS_G1 := 𝓟.verificationPairingSRS_G1,
                      verificationPairingSRS_G2 := 𝓟.verificationPairingSRS_G2,
                      verificationPairingProof_G1 := 𝓟.verificationPairingProof_G1,
                      verificationPairingProof_G2 := 𝓟.verificationPairingProof_G2,
                      Identified_Proof_Elems := 𝓟.Identified_Proof_Elems }.ListSRSElements_G2).sum))
        ({ Stmt := 𝓟.Stmt, Sample := 𝓟.Sample, SRSElements_G1 := 𝓟.SRSElements_G1,
              ListSRSElements_G1 := 𝓟.ListSRSElements_G1, SRSElements_G2 := 𝓟.SRSElements_G2,
              ListSRSElements_G2 := 𝓟.ListSRSElements_G2,
              SRSElementValue_G1 := fun srs =>
                if srs = twin1 then 𝓟.SRSElementValue_G1 twin1 + 𝓟.SRSElementValue_G1 twin2
                else if srs = twin2 then 0 else 𝓟.SRSElementValue_G1 srs,
              SRSElementValue_G2 := 𝓟.SRSElementValue_G2, Proof_G1 := 𝓟.Proof_G1, ListProof_G1 := 𝓟.ListProof_G1,
              Proof_G2 := 𝓟.Proof_G2, ListProof_G2 := 𝓟.ListProof_G2, EqualityChecks := 𝓟.EqualityChecks,
              Pairings := 𝓟.Pairings, ListPairings := 𝓟.ListPairings,
              verificationPairingSRS_G1 := 𝓟.verificationPairingSRS_G1,
              verificationPairingSRS_G2 := 𝓟.verificationPairingSRS_G2,
              verificationPairingProof_G1 := 𝓟.verificationPairingProof_G1,
              verificationPairingProof_G2 := 𝓟.verificationPairingProof_G2,
              Identified_Proof_Elems := 𝓟.Identified_Proof_Elems }.ListPairings
          check_idx)).sum =
    0
⊢ (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