Theorem: collapseToxicWaste_check_poly
Proof state before:
F : Type
inst✝¹ : Field F
𝓟 : AGMProofSystemInstantiation F
d : ℕ
inst✝ : DecidableEq 𝓟.Sample
sample_removed sample_target : 𝓟.Sample
agm : AGMProofSystemInstantiation.Prover F (collapseToxicWaste 𝓟 d sample_removed sample_target)
stmt : (collapseToxicWaste 𝓟 d sample_removed sample_target).Stmt
check_idx : (collapseToxicWaste 𝓟 d sample_removed sample_target).EqualityChecks
⊢ (collapseToxicWaste 𝓟 d sample_removed sample_target).check_poly agm stmt check_idx =
    (MvPolynomial.bind₁ fun x => if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
      (𝓟.check_poly agm stmt check_idx)
Tactic:
unfold collapseToxicWaste AGMProofSystemInstantiation.check_poly AGMProofSystemInstantiation.pairing_poly AGMProofSystemInstantiation.proof_element_G1_as_poly AGMProofSystemInstantiation.proof_element_G2_as_poly
Proof state after:
F : Type
inst✝¹ : Field F
𝓟 : AGMProofSystemInstantiation F
d : ℕ
inst✝ : DecidableEq 𝓟.Sample
sample_removed sample_target : 𝓟.Sample
agm : AGMProofSystemInstantiation.Prover F (collapseToxicWaste 𝓟 d sample_removed sample_target)
stmt : (collapseToxicWaste 𝓟 d sample_removed sample_target).Stmt
check_idx : (collapseToxicWaste 𝓟 d sample_removed sample_target).EqualityChecks
⊢ (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 elem =>
                                (MvPolynomial.bind₁ fun x =>
                                    if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                                  (𝓟.SRSElementValue_G1 elem),
                              SRSElementValue_G2 := fun elem =>
                                (MvPolynomial.bind₁ fun x =>
                                    if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                                  (𝓟.SRSElementValue_G2 elem),
                              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 elem =>
                                      (MvPolynomial.bind₁ fun x =>
                                          if x = sample_removed then MvPolynomial.X sample_target ^ d
                                          else MvPolynomial.X x)
                                        (𝓟.SRSElementValue_G1 elem),
                                    SRSElementValue_G2 := fun elem =>
                                      (MvPolynomial.bind₁ fun x =>
                                          if x = sample_removed then MvPolynomial.X sample_target ^ d
                                          else MvPolynomial.X x)
                                        (𝓟.SRSElementValue_G2 elem),
                                    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 elem =>
                                (MvPolynomial.bind₁ fun x =>
                                    if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                                  (𝓟.SRSElementValue_G1 elem),
                              SRSElementValue_G2 := fun elem =>
                                (MvPolynomial.bind₁ fun x =>
                                    if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                                  (𝓟.SRSElementValue_G2 elem),
                              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 elem =>
                        (MvPolynomial.bind₁ fun x =>
                            if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                          (𝓟.SRSElementValue_G1 elem),
                      SRSElementValue_G2 := fun elem =>
                        (MvPolynomial.bind₁ fun x =>
                            if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                          (𝓟.SRSElementValue_G2 elem),
                      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 elem =>
                                (MvPolynomial.bind₁ fun x =>
                                    if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                                  (𝓟.SRSElementValue_G1 elem),
                              SRSElementValue_G2 := fun elem =>
                                (MvPolynomial.bind₁ fun x =>
                                    if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                                  (𝓟.SRSElementValue_G2 elem),
                              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 elem =>
                              (MvPolynomial.bind₁ fun x =>
                                  if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                                (𝓟.SRSElementValue_G1 elem),
                            SRSElementValue_G2 := fun elem =>
                              (MvPolynomial.bind₁ fun x =>
                                  if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                                (𝓟.SRSElementValue_G2 elem),
                            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 elem =>
                        (MvPolynomial.bind₁ fun x =>
                            if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                          (𝓟.SRSElementValue_G1 elem),
                      SRSElementValue_G2 := fun elem =>
                        (MvPolynomial.bind₁ fun x =>
                            if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                          (𝓟.SRSElementValue_G2 elem),
                      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 elem =>
                                (MvPolynomial.bind₁ fun x =>
                                    if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                                  (𝓟.SRSElementValue_G1 elem),
                              SRSElementValue_G2 := fun elem =>
                                (MvPolynomial.bind₁ fun x =>
                                    if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                                  (𝓟.SRSElementValue_G2 elem),
                              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 elem =>
                                      (MvPolynomial.bind₁ fun x =>
                                          if x = sample_removed then MvPolynomial.X sample_target ^ d
                                          else MvPolynomial.X x)
                                        (𝓟.SRSElementValue_G1 elem),
                                    SRSElementValue_G2 := fun elem =>
                                      (MvPolynomial.bind₁ fun x =>
                                          if x = sample_removed then MvPolynomial.X sample_target ^ d
                                          else MvPolynomial.X x)
                                        (𝓟.SRSElementValue_G2 elem),
                                    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 elem =>
                                (MvPolynomial.bind₁ fun x =>
                                    if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                                  (𝓟.SRSElementValue_G1 elem),
                              SRSElementValue_G2 := fun elem =>
                                (MvPolynomial.bind₁ fun x =>
                                    if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                                  (𝓟.SRSElementValue_G2 elem),
                              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 elem =>
                        (MvPolynomial.bind₁ fun x =>
                            if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                          (𝓟.SRSElementValue_G1 elem),
                      SRSElementValue_G2 := fun elem =>
                        (MvPolynomial.bind₁ fun x =>
                            if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                          (𝓟.SRSElementValue_G2 elem),
                      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 elem =>
                                (MvPolynomial.bind₁ fun x =>
                                    if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                                  (𝓟.SRSElementValue_G1 elem),
                              SRSElementValue_G2 := fun elem =>
                                (MvPolynomial.bind₁ fun x =>
                                    if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                                  (𝓟.SRSElementValue_G2 elem),
                              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 elem =>
                              (MvPolynomial.bind₁ fun x =>
                                  if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                                (𝓟.SRSElementValue_G1 elem),
                            SRSElementValue_G2 := fun elem =>
                              (MvPolynomial.bind₁ fun x =>
                                  if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                                (𝓟.SRSElementValue_G2 elem),
                            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 elem =>
                        (MvPolynomial.bind₁ fun x =>
                            if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                          (𝓟.SRSElementValue_G1 elem),
                      SRSElementValue_G2 := fun elem =>
                        (MvPolynomial.bind₁ fun x =>
                            if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                          (𝓟.SRSElementValue_G2 elem),
                      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 elem =>
                (MvPolynomial.bind₁ fun x =>
                    if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                  (𝓟.SRSElementValue_G1 elem),
              SRSElementValue_G2 := fun elem =>
                (MvPolynomial.bind₁ fun x =>
                    if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
                  (𝓟.SRSElementValue_G2 elem),
              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 =
    (MvPolynomial.bind₁ fun x => if x = sample_removed then MvPolynomial.X sample_target ^ d else MvPolynomial.X x)
      (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