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
⊢ (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 =
    (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 =>
                            if SRS_elem = twin1 then
                              MvPolynomial.C (agm.1 pf_elem SRS_elem) *
                                (𝓟.SRSElementValue_G1 twin1 + 𝓟.SRSElementValue_G1 twin2)
                            else
                              if SRS_elem = twin2 then 0
                              else 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 =>
                    if SRS_elem = twin1 then
                      MvPolynomial.C (𝓟.verificationPairingSRS_G1 stmt check_idx pairing SRS_elem) *
                        (𝓟.SRSElementValue_G1 twin1 + 𝓟.SRSElementValue_G1 twin2)
                    else
                      if SRS_elem = twin2 then 0
                      else
                        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
Tactic:
congr
Proof state after:
case intro.a.left.e_a.e_f
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
⊢ (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)) =
    fun pairing =>
    ((List.map
            (fun pf_elem =>
              MvPolynomial.C (𝓟.verificationPairingProof_G1 stmt check_idx pairing pf_elem) *
                (List.map
                    (fun SRS_elem =>
                      if SRS_elem = twin1 then
                        MvPolynomial.C (agm.1 pf_elem SRS_elem) *
                          (𝓟.SRSElementValue_G1 twin1 + 𝓟.SRSElementValue_G1 twin2)
                      else
                        if SRS_elem = twin2 then 0
                        else 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 =>
              if SRS_elem = twin1 then
                MvPolynomial.C (𝓟.verificationPairingSRS_G1 stmt check_idx pairing SRS_elem) *
                  (𝓟.SRSElementValue_G1 twin1 + 𝓟.SRSElementValue_G1 twin2)
              else
                if SRS_elem = twin2 then 0
                else
                  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)