Proofgold Signed Transaction

Theorem eq_i_tra^eq_i_tra : ∀ x0 x1 x2 . x0 = x1 ⟶ x1 = x2 ⟶ x0 = x2 (proof)
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Definition TransSet^TransSet := λ x0 . ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Param ordsucc^ordsucc : ι → ι
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Known Subq_ref^Subq_ref : ∀ x0 . x0 ⊆ x0
Theorem TransSet_In_ordsucc_Subq^{TransSet_In_ordsucc_Subq} : ∀ x0 x1 . TransSet x1 ⟶ x0 ∈ ordsucc x1 ⟶ x0 ⊆ x1 (proof)
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Known ReplE_impred^ReplE_impred : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ prim5 x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ x0 ⟶ x2 = x1 x4 ⟶ x3) ⟶ x3
Known ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Theorem inv_Repl_eq^inv_Repl_eq : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x1 (x2 x3) = x3) ⟶ prim5 (prim5 x0 x2) x1 = x0 (proof)
Theorem invol_Repl_eq^{invol_Repl_eq} : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . x2 ∈ x0 ⟶ x1 (x1 x2) = x2) ⟶ prim5 (prim5 x0 x1) x1 = x0 (proof)
Param SNo^SNo : ι → ο
Param SNoLt^SNoLt : ι → ι → ο
Known SNoLt_trichotomy_or^{SNoLt_trichotomy_or} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (or (SNoLt x0 x1) (x0 = x1)) (SNoLt x1 x0)
Theorem SNoLt_trichotomy_or_impred^{SNoLt_trichotomy_or_impred} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 : ο . (SNoLt x0 x1 ⟶ x2) ⟶ (x0 = x1 ⟶ x2) ⟶ (SNoLt x1 x0 ⟶ x2) ⟶ x2 (proof)
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition SNoCutP^SNoCutP := λ x0 x1 . and (and (∀ x2 . x2 ∈ x0 ⟶ SNo x2) (∀ x2 . x2 ∈ x1 ⟶ SNo x2)) (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x1 ⟶ SNoLt x2 x3)
Param SNoCut^SNoCut : ι → ι → ι
Param SNoLev^SNoLev : ι → ι
Param binunion^binunion : ι → ι → ι
Param famunion^famunion : ι → (ι → ι) → ι
Param SNoEq_^SNoEq_ : ι → ι → ι → ο
Known SNoCutP_SNoCut^{SNoCutP_SNoCut} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ and (and (and (and (SNo (SNoCut x0 x1)) (SNoLev (SNoCut x0 x1) ∈ ordsucc (binunion (famunion x0 (λ x2 . ordsucc (SNoLev x2))) (famunion x1 (λ x2 . ordsucc (SNoLev x2)))))) (∀ x2 . x2 ∈ x0 ⟶ SNoLt x2 (SNoCut x0 x1))) (∀ x2 . x2 ∈ x1 ⟶ SNoLt (SNoCut x0 x1) x2)) (∀ x2 . SNo x2 ⟶ (∀ x3 . x3 ∈ x0 ⟶ SNoLt x3 x2) ⟶ (∀ x3 . x3 ∈ x1 ⟶ SNoLt x2 x3) ⟶ and (SNoLev (SNoCut x0 x1) ⊆ SNoLev x2) (SNoEq_ (SNoLev (SNoCut x0 x1)) (SNoCut x0 x1) x2))
Theorem SNoCutP_SNoCut_impred^{SNoCutP_SNoCut_impred} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ ∀ x2 : ο . (SNo (SNoCut x0 x1) ⟶ SNoLev (SNoCut x0 x1) ∈ ordsucc (binunion (famunion x0 (λ x3 . ordsucc (SNoLev x3))) (famunion x1 (λ x3 . ordsucc (SNoLev x3)))) ⟶ (∀ x3 . x3 ∈ x0 ⟶ SNoLt x3 (SNoCut x0 x1)) ⟶ (∀ x3 . x3 ∈ x1 ⟶ SNoLt (SNoCut x0 x1) x3) ⟶ (∀ x3 . SNo x3 ⟶ (∀ x4 . x4 ∈ x0 ⟶ SNoLt x4 x3) ⟶ (∀ x4 . x4 ∈ x1 ⟶ SNoLt x3 x4) ⟶ and (SNoLev (SNoCut x0 x1) ⊆ SNoLev x3) (SNoEq_ (SNoLev (SNoCut x0 x1)) (SNoCut x0 x1) x3)) ⟶ x2) ⟶ x2 (proof)
Definition ordinal^ordinal := λ x0 . and (TransSet x0) (∀ x1 . x1 ∈ x0 ⟶ TransSet x1)
Known ordinal_In_Or_Subq^{ordinal_In_Or_Subq} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ or (x0 ∈ x1) (x1 ⊆ x0)
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Param SNoLe^SNoLe : ι → ι → ο
Known SNoLtLe_tra^SNoLtLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLt x0 x2
Known ordinal_SNo^ordinal_SNo : ∀ x0 . ordinal x0 ⟶ SNo x0
Known ordinal_Subq_SNoLe^{ordinal_Subq_SNoLe} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ x0 ⊆ x1 ⟶ SNoLe x0 x1
Theorem ordinal_SNoLt_In^{ordinal_SNoLt_In} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ SNoLt x0 x1 ⟶ x0 ∈ x1 (proof)
Known SNoLeLt_tra^SNoLeLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known ordinal_In_SNoLt^{ordinal_In_SNoLt} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ SNoLt x1 x0
Theorem ordinal_SNoLe_Subq^{ordinal_SNoLe_Subq} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ SNoLe x0 x1 ⟶ x0 ⊆ x1 (proof)
Param Sep^Sep : ι → (ι → ο) → ι
Param SNoS_^SNoS_ : ι → ι
Definition SNoL^SNoL := λ x0 . {x1 ∈ SNoS_ (SNoLev x0)|SNoLt x1 x0}
Known Sep_Subq^Sep_Subq : ∀ x0 . ∀ x1 : ι → ο . Sep x0 x1 ⊆ x0
Theorem SNoL_SNoS_^SNoL_SNoS_ : ∀ x0 . SNoL x0 ⊆ SNoS_ (SNoLev x0) (proof)
Definition SNoR^SNoR := λ x0 . Sep (SNoS_ (SNoLev x0)) (SNoLt x0)
Theorem SNoR_SNoS_^SNoR_SNoS_ : ∀ x0 . SNoR x0 ⊆ SNoS_ (SNoLev x0) (proof)
Known SNoL_E^SNoL_E : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoL x0 ⟶ ∀ x2 : ο . (SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x1 x0 ⟶ x2) ⟶ x2
Known SNoS_I2^SNoS_I2 : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x0 ∈ SNoLev x1 ⟶ x0 ∈ SNoS_ (SNoLev x1)
Param SNo_^SNo_ : ι → ι → ο
Known SNoS_E2^SNoS_E2 : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ SNoS_ x0 ⟶ ∀ x2 : ο . (SNoLev x1 ∈ x0 ⟶ ordinal (SNoLev x1) ⟶ SNo x1 ⟶ SNo_ (SNoLev x1) x1 ⟶ x2) ⟶ x2
Known SNoL_I^SNoL_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x1 x0 ⟶ x1 ∈ SNoL x0
Known ordinal_SNoLev_max^{ordinal_SNoLev_max} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ x0 ⟶ SNoLt x1 x0
Known ordinal_SNoLev^{ordinal_SNoLev} : ∀ x0 . ordinal x0 ⟶ SNoLev x0 = x0
Theorem ordinal_SNoL^ordinal_SNoL : ∀ x0 . ordinal x0 ⟶ SNoL x0 = SNoS_ x0 (proof)
Known Empty_Subq_eq^{Empty_Subq_eq} : ∀ x0 . x0 ⊆ 0 ⟶ x0 = 0
Known SNoR_E^SNoR_E : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoR x0 ⟶ ∀ x2 : ο . (SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x0 x1 ⟶ x2) ⟶ x2
Known SNoLt_tra^SNoLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Theorem ordinal_SNoR^ordinal_SNoR : ∀ x0 . ordinal x0 ⟶ SNoR x0 = 0 (proof)
Known SNoCutP_SNoL_SNoR^{SNoCutP_SNoL_SNoR} : ∀ x0 . SNo x0 ⟶ SNoCutP (SNoL x0) (SNoR x0)
Theorem ordinal_SNoCutP^{ordinal_SNoCutP} : ∀ x0 . ordinal x0 ⟶ SNoCutP (SNoS_ x0) 0 (proof)
Known SNo_eta^SNo_eta : ∀ x0 . SNo x0 ⟶ x0 = SNoCut (SNoL x0) (SNoR x0)
Theorem ordinal_SNoCut_eta^{ordinal_SNoCut_eta} : ∀ x0 . ordinal x0 ⟶ x0 = SNoCut (SNoS_ x0) 0 (proof)
Known ordinal_Empty^{ordinal_Empty} : ordinal 0
Theorem SNo_0^SNo_0 : SNo 0 (proof)
Theorem SNoLev_0^SNoLev_0 : SNoLev 0 = 0 (proof)
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Theorem SNoL_0^SNoL_0 : SNoL 0 = 0 (proof)
Theorem SNoR_0^SNoR_0 : SNoR 0 = 0 (proof)
Known SNoLeE^SNoLeE : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ or (SNoLt x0 x1) (x0 = x1)
Param binintersect^binintersect : ι → ι → ι
Known SNoLtE^SNoLtE : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 x1 ⟶ ∀ x2 : ο . (∀ x3 . SNo x3 ⟶ SNoLev x3 ∈ binintersect (SNoLev x0) (SNoLev x1) ⟶ SNoEq_ (SNoLev x3) x3 x0 ⟶ SNoEq_ (SNoLev x3) x3 x1 ⟶ SNoLt x0 x3 ⟶ SNoLt x3 x1 ⟶ nIn (SNoLev x3) x0 ⟶ SNoLev x3 ∈ x1 ⟶ x2) ⟶ (SNoLev x0 ∈ SNoLev x1 ⟶ SNoEq_ (SNoLev x0) x0 x1 ⟶ SNoLev x0 ∈ x1 ⟶ x2) ⟶ (SNoLev x1 ∈ SNoLev x0 ⟶ SNoEq_ (SNoLev x1) x0 x1 ⟶ nIn (SNoLev x1) x0 ⟶ x2) ⟶ x2
Known SNoS_I^SNoS_I : ∀ x0 . ordinal x0 ⟶ ∀ x1 x2 . x2 ∈ x0 ⟶ SNo_ x2 x1 ⟶ x1 ∈ SNoS_ x0
Known SNoLev_^SNoLev_ : ∀ x0 . SNo x0 ⟶ SNo_ (SNoLev x0) x0
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Known ordinal_SNoLev_max_2^{ordinal_SNoLev_max_2} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ ordsucc x0 ⟶ SNoLe x1 x0
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Known SNoLev_ordinal^{SNoLev_ordinal} : ∀ x0 . SNo x0 ⟶ ordinal (SNoLev x0)
Theorem SNo_max_SNoLev^{SNo_max_SNoLev} : ∀ x0 . SNo x0 ⟶ (∀ x1 . x1 ∈ SNoS_ (SNoLev x0) ⟶ SNoLt x1 x0) ⟶ SNoLev x0 = x0 (proof)
Theorem SNo_max_ordinal^{SNo_max_ordinal} : ∀ x0 . SNo x0 ⟶ (∀ x1 . x1 ∈ SNoS_ (SNoLev x0) ⟶ SNoLt x1 x0) ⟶ ordinal x0 (proof)
Param SNo_extend0^SNo_extend0 : ι → ι
Known SNoLtI3^SNoLtI3 : ∀ x0 x1 . SNoLev x1 ∈ SNoLev x0 ⟶ SNoEq_ (SNoLev x1) x0 x1 ⟶ nIn (SNoLev x1) x0 ⟶ SNoLt x0 x1
Known SNo_extend0_SNoLev^{SNo_extend0_SNoLev} : ∀ x0 . SNo x0 ⟶ SNoLev (SNo_extend0 x0) = ordsucc (SNoLev x0)
Known SNo_extend0_SNoEq^{SNo_extend0_SNoEq} : ∀ x0 . SNo x0 ⟶ SNoEq_ (SNoLev x0) (SNo_extend0 x0) x0
Known SNo_extend0_nIn^{SNo_extend0_nIn} : ∀ x0 . SNo x0 ⟶ nIn (SNoLev x0) (SNo_extend0 x0)
Theorem SNo_extend0_Lt^{SNo_extend0_Lt} : ∀ x0 . SNo x0 ⟶ SNoLt (SNo_extend0 x0) x0 (proof)
Param SNo_extend1^SNo_extend1 : ι → ι
Known SNoLtI2^SNoLtI2 : ∀ x0 x1 . SNoLev x0 ∈ SNoLev x1 ⟶ SNoEq_ (SNoLev x0) x0 x1 ⟶ SNoLev x0 ∈ x1 ⟶ SNoLt x0 x1
Known SNo_extend1_SNoLev^{SNo_extend1_SNoLev} : ∀ x0 . SNo x0 ⟶ SNoLev (SNo_extend1 x0) = ordsucc (SNoLev x0)
Known SNoEq_sym_^SNoEq_sym_ : ∀ x0 x1 x2 . SNoEq_ x0 x1 x2 ⟶ SNoEq_ x0 x2 x1
Known SNo_extend1_SNoEq^{SNo_extend1_SNoEq} : ∀ x0 . SNo x0 ⟶ SNoEq_ (SNoLev x0) (SNo_extend1 x0) x0
Known SNo_extend1_In^{SNo_extend1_In} : ∀ x0 . SNo x0 ⟶ SNoLev x0 ∈ SNo_extend1 x0
Theorem SNo_extend1_Gt^{SNo_extend1_Gt} : ∀ x0 . SNo x0 ⟶ SNoLt x0 (SNo_extend1 x0) (proof)
Param In_rec_ii^In_rec_ii : (ι → (ι → ι → ι) → ι → ι) → ι → ι → ι
Param If_ii^If_ii : ο → (ι → ι) → (ι → ι) → ι → ι
Param If_i^If_i : ο → ι → ι → ι
Param True^True : ο
Definition SNo_rec_i^SNo_rec_i := λ x0 : ι → (ι → ι) → ι . λ x1 . In_rec_ii (λ x2 . λ x3 : ι → ι → ι . If_ii (ordinal x2) (λ x4 . If_i (x4 ∈ SNoS_ (ordsucc x2)) (x0 x4 (λ x5 . x3 (SNoLev x5) x5)) (prim0 (λ x5 . True))) (λ x4 . prim0 (λ x5 . True))) (SNoLev x1) x1
Known In_rec_ii_eq^In_rec_ii_eq : ∀ x0 : ι → (ι → ι → ι) → ι → ι . (∀ x1 . ∀ x2 x3 : ι → ι → ι . (∀ x4 . x4 ∈ x1 ⟶ x2 x4 = x3 x4) ⟶ x0 x1 x2 = x0 x1 x3) ⟶ ∀ x1 . In_rec_ii x0 x1 = x0 x1 (In_rec_ii x0)
Known If_ii_1^If_ii_1 : ∀ x0 : ο . ∀ x1 x2 : ι → ι . x0 ⟶ If_ii x0 x1 x2 = x1
Known If_i_1^If_i_1 : ∀ x0 : ο . ∀ x1 x2 . x0 ⟶ If_i x0 x1 x2 = x1
Known SNoS_SNoLev^SNoS_SNoLev : ∀ x0 . SNo x0 ⟶ x0 ∈ SNoS_ (ordsucc (SNoLev x0))
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known ordinal_ordsucc^{ordinal_ordsucc} : ∀ x0 . ordinal x0 ⟶ ordinal (ordsucc x0)
Known If_i_0^If_i_0 : ∀ x0 : ο . ∀ x1 x2 . not x0 ⟶ If_i x0 x1 x2 = x2
Known If_ii_0^If_ii_0 : ∀ x0 : ο . ∀ x1 x2 : ι → ι . not x0 ⟶ If_ii x0 x1 x2 = x2
Theorem SNo_rec_i_eq^SNo_rec_i_eq : ∀ x0 : ι → (ι → ι) → ι . (∀ x1 . SNo x1 ⟶ ∀ x2 x3 : ι → ι . (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ x2 x4 = x3 x4) ⟶ x0 x1 x2 = x0 x1 x3) ⟶ ∀ x1 . SNo x1 ⟶ SNo_rec_i x0 x1 = x0 x1 (SNo_rec_i x0) (proof)
Param In_rec_iii^In_rec_iii : (ι → (ι → ι → ι → ι) → ι → ι → ι) → ι → ι → ι → ι
Param If_iii^If_iii : ο → (ι → ι → ι) → (ι → ι → ι) → ι → ι → ι
Param Descr_ii^Descr_ii : ((ι → ι) → ο) → ι → ι
Definition SNo_rec_ii^SNo_rec_ii := λ x0 : ι → (ι → ι → ι) → ι → ι . λ x1 . In_rec_iii (λ x2 . λ x3 : ι → ι → ι → ι . If_iii (ordinal x2) (λ x4 . If_ii (x4 ∈ SNoS_ (ordsucc x2)) (x0 x4 (λ x5 . x3 (SNoLev x5) x5)) (Descr_ii (λ x5 : ι → ι . True))) (λ x4 . Descr_ii (λ x5 : ι → ι . True))) (SNoLev x1) x1
Known In_rec_iii_eq^{In_rec_iii_eq} : ∀ x0 : ι → (ι → ι → ι → ι) → ι → ι → ι . (∀ x1 . ∀ x2 x3 : ι → ι → ι → ι . (∀ x4 . x4 ∈ x1 ⟶ x2 x4 = x3 x4) ⟶ x0 x1 x2 = x0 x1 x3) ⟶ ∀ x1 . In_rec_iii x0 x1 = x0 x1 (In_rec_iii x0)
Known If_iii_1^If_iii_1 : ∀ x0 : ο . ∀ x1 x2 : ι → ι → ι . x0 ⟶ If_iii x0 x1 x2 = x1
Known If_iii_0^If_iii_0 : ∀ x0 : ο . ∀ x1 x2 : ι → ι → ι . not x0 ⟶ If_iii x0 x1 x2 = x2
Theorem SNo_rec_ii_eq^{SNo_rec_ii_eq} : ∀ x0 : ι → (ι → ι → ι) → ι → ι . (∀ x1 . SNo x1 ⟶ ∀ x2 x3 : ι → ι → ι . (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ x2 x4 = x3 x4) ⟶ x0 x1 x2 = x0 x1 x3) ⟶ ∀ x1 . SNo x1 ⟶ SNo_rec_ii x0 x1 = x0 x1 (SNo_rec_ii x0) (proof)
Known SNoS_In_neq^SNoS_In_neq : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoS_ (SNoLev x0) ⟶ x1 = x0 ⟶ ∀ x2 : ο . x2
Theorem SNo_rec2_G_prop^{SNo_rec2_G_prop} : ∀ x0 : ι → ι → (ι → ι → ι) → ι . (∀ x1 . SNo x1 ⟶ ∀ x2 . SNo x2 ⟶ ∀ x3 x4 : ι → ι → ι . (∀ x5 . x5 ∈ SNoS_ (SNoLev x1) ⟶ ∀ x6 . SNo x6 ⟶ x3 x5 x6 = x4 x5 x6) ⟶ (∀ x5 . x5 ∈ SNoS_ (SNoLev x2) ⟶ x3 x1 x5 = x4 x1 x5) ⟶ x0 x1 x2 x3 = x0 x1 x2 x4) ⟶ ∀ x1 . SNo x1 ⟶ ∀ x2 x3 : ι → ι → ι . (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ x2 x4 = x3 x4) ⟶ ∀ x4 . SNo x4 ⟶ ∀ x5 x6 : ι → ι . (∀ x7 . x7 ∈ SNoS_ (SNoLev x4) ⟶ x5 x7 = x6 x7) ⟶ x0 x1 x4 (λ x8 x9 . If_i (x8 = x1) (x5 x9) (x2 x8 x9)) = x0 x1 x4 (λ x8 x9 . If_i (x8 = x1) (x6 x9) (x3 x8 x9)) (proof)
Theorem SNo_rec2_eq_1^{SNo_rec2_eq_1} : ∀ x0 : ι → ι → (ι → ι → ι) → ι . (∀ x1 . SNo x1 ⟶ ∀ x2 . SNo x2 ⟶ ∀ x3 x4 : ι → ι → ι . (∀ x5 . x5 ∈ SNoS_ (SNoLev x1) ⟶ ∀ x6 . SNo x6 ⟶ x3 x5 x6 = x4 x5 x6) ⟶ (∀ x5 . x5 ∈ SNoS_ (SNoLev x2) ⟶ x3 x1 x5 = x4 x1 x5) ⟶ x0 x1 x2 x3 = x0 x1 x2 x4) ⟶ ∀ x1 . SNo x1 ⟶ ∀ x2 : ι → ι → ι . ∀ x3 . SNo x3 ⟶ SNo_rec_i (λ x5 . λ x6 : ι → ι . x0 x1 x5 (λ x7 x8 . If_i (x7 = x1) (x6 x8) (x2 x7 x8))) x3 = x0 x1 x3 (λ x5 x6 . If_i (x5 = x1) (SNo_rec_i (λ x7 . λ x8 : ι → ι . x0 x1 x7 (λ x9 x10 . If_i (x9 = x1) (x8 x10) (x2 x9 x10))) x6) (x2 x5 x6)) (proof)
Definition SNo_rec2^SNo_rec2 := λ x0 : ι → ι → (ι → ι → ι) → ι . SNo_rec_ii (λ x1 . λ x2 : ι → ι → ι . λ x3 . If_i (SNo x3) (SNo_rec_i (λ x4 . λ x5 : ι → ι . x0 x1 x4 (λ x6 x7 . If_i (x6 = x1) (x5 x7) (x2 x6 x7))) x3) 0)
Known ordinal_ind^ordinal_ind : ∀ x0 : ι → ο . (∀ x1 . ordinal x1 ⟶ (∀ x2 . x2 ∈ x1 ⟶ x0 x2) ⟶ x0 x1) ⟶ ∀ x1 . ordinal x1 ⟶ x0 x1
Theorem SNo_rec2_eq^SNo_rec2_eq : ∀ x0 : ι → ι → (ι → ι → ι) → ι . (∀ x1 . SNo x1 ⟶ ∀ x2 . SNo x2 ⟶ ∀ x3 x4 : ι → ι → ι . (∀ x5 . x5 ∈ SNoS_ (SNoLev x1) ⟶ ∀ x6 . SNo x6 ⟶ x3 x5 x6 = x4 x5 x6) ⟶ (∀ x5 . x5 ∈ SNoS_ (SNoLev x2) ⟶ x3 x1 x5 = x4 x1 x5) ⟶ x0 x1 x2 x3 = x0 x1 x2 x4) ⟶ ∀ x1 . SNo x1 ⟶ ∀ x2 . SNo x2 ⟶ SNo_rec2 x0 x1 x2 = x0 x1 x2 (SNo_rec2 x0) (proof)
Theorem SNo_ordinal_ind^{SNo_ordinal_ind} : ∀ x0 : ι → ο . (∀ x1 . ordinal x1 ⟶ ∀ x2 . x2 ∈ SNoS_ x1 ⟶ x0 x2) ⟶ ∀ x1 . SNo x1 ⟶ x0 x1 (proof)
Theorem SNo_ordinal_ind2^{SNo_ordinal_ind2} : ∀ x0 : ι → ι → ο . (∀ x1 . ordinal x1 ⟶ ∀ x2 . ordinal x2 ⟶ ∀ x3 . x3 ∈ SNoS_ x1 ⟶ ∀ x4 . x4 ∈ SNoS_ x2 ⟶ x0 x3 x4) ⟶ ∀ x1 x2 . SNo x1 ⟶ SNo x2 ⟶ x0 x1 x2 (proof)
Theorem SNo_ordinal_ind3^{SNo_ordinal_ind3} : ∀ x0 : ι → ι → ι → ο . (∀ x1 . ordinal x1 ⟶ ∀ x2 . ordinal x2 ⟶ ∀ x3 . ordinal x3 ⟶ ∀ x4 . x4 ∈ SNoS_ x1 ⟶ ∀ x5 . x5 ∈ SNoS_ x2 ⟶ ∀ x6 . x6 ∈ SNoS_ x3 ⟶ x0 x4 x5 x6) ⟶ ∀ x1 x2 x3 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ x0 x1 x2 x3 (proof)
Theorem SNoLev_ind^SNoLev_ind : ∀ x0 : ι → ο . (∀ x1 . SNo x1 ⟶ (∀ x2 . x2 ∈ SNoS_ (SNoLev x1) ⟶ x0 x2) ⟶ x0 x1) ⟶ ∀ x1 . SNo x1 ⟶ x0 x1 (proof)
Theorem SNoLev_ind2^SNoLev_ind2 : ∀ x0 : ι → ι → ο . (∀ x1 x2 . SNo x1 ⟶ SNo x2 ⟶ (∀ x3 . x3 ∈ SNoS_ (SNoLev x1) ⟶ x0 x3 x2) ⟶ (∀ x3 . x3 ∈ SNoS_ (SNoLev x2) ⟶ x0 x1 x3) ⟶ (∀ x3 . x3 ∈ SNoS_ (SNoLev x1) ⟶ ∀ x4 . x4 ∈ SNoS_ (SNoLev x2) ⟶ x0 x3 x4) ⟶ x0 x1 x2) ⟶ ∀ x1 x2 . SNo x1 ⟶ SNo x2 ⟶ x0 x1 x2 (proof)
Theorem SNoLev_ind3^SNoLev_ind3 : ∀ x0 : ι → ι → ι → ο . (∀ x1 x2 x3 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ x0 x4 x2 x3) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x2) ⟶ x0 x1 x4 x3) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x3) ⟶ x0 x1 x2 x4) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ ∀ x5 . x5 ∈ SNoS_ (SNoLev x2) ⟶ x0 x4 x5 x3) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ ∀ x5 . x5 ∈ SNoS_ (SNoLev x3) ⟶ x0 x4 x2 x5) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x2) ⟶ ∀ x5 . x5 ∈ SNoS_ (SNoLev x3) ⟶ x0 x1 x4 x5) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ ∀ x5 . x5 ∈ SNoS_ (SNoLev x2) ⟶ ∀ x6 . x6 ∈ SNoS_ (SNoLev x3) ⟶ x0 x4 x5 x6) ⟶ x0 x1 x2 x3) ⟶ ∀ x1 x2 x3 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ x0 x1 x2 x3 (proof)
Param nat_p^nat_p : ι → ο
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known nat_1^nat_1 : nat_p 1
Theorem SNo_1^SNo_1 : SNo 1 (proof)
Known nat_2^nat_2 : nat_p 2
Theorem SNo_2^SNo_2 : SNo 2 (proof)
Param omega^omega : ι
Known omega_ordinal^{omega_ordinal} : ordinal omega
Theorem SNo_omega^SNo_omega : SNo omega (proof)
Known ordinal_1^ordinal_1 : ordinal 1
Known In_0_1^In_0_1 : 0 ∈ 1
Theorem SNoLt_0_1^SNoLt_0_1 : SNoLt 0 1 (proof)
Known ordinal_2^ordinal_2 : ordinal 2
Known In_0_2^In_0_2 : 0 ∈ 2
Theorem SNoLt_0_2^SNoLt_0_2 : SNoLt 0 2 (proof)
Known In_1_2^In_1_2 : 1 ∈ 2
Theorem SNoLt_1_2^SNoLt_1_2 : SNoLt 1 2 (proof)
Definition minus_SNo^minus_SNo := SNo_rec_i (λ x0 . λ x1 : ι → ι . SNoCut (prim5 (SNoR x0) x1) (prim5 (SNoL x0) x1))
Known ReplEq_ext^ReplEq_ext : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x1 x3 = x2 x3) ⟶ prim5 x0 x1 = prim5 x0 x2
Known SNoL_SNoS^SNoL_SNoS : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoL x0 ⟶ x1 ∈ SNoS_ (SNoLev x0)
Known SNoR_SNoS^SNoR_SNoS : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoR x0 ⟶ x1 ∈ SNoS_ (SNoLev x0)
Theorem minus_SNo_eq^minus_SNo_eq : ∀ x0 . SNo x0 ⟶ minus_SNo x0 = SNoCut (prim5 (SNoR x0) minus_SNo) (prim5 (SNoL x0) minus_SNo) (proof)
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known SNoCutP_SNoCut_R^{SNoCutP_SNoCut_R} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ ∀ x2 . x2 ∈ x1 ⟶ SNoLt (SNoCut x0 x1) x2
Known SNoCutP_SNoCut_L^{SNoCutP_SNoCut_L} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ ∀ x2 . x2 ∈ x0 ⟶ SNoLt x2 (SNoCut x0 x1)
Known SNoCutP_SNo_SNoCut^{SNoCutP_SNo_SNoCut} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ SNo (SNoCut x0 x1)
Known SNoLev_prop^SNoLev_prop : ∀ x0 . SNo x0 ⟶ and (ordinal (SNoLev x0)) (SNo_ (SNoLev x0) x0)
Known binintersectE^{binintersectE} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ and (x2 ∈ x0) (x2 ∈ x1)
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Theorem minus_SNo_prop1^{minus_SNo_prop1} : ∀ x0 . SNo x0 ⟶ and (and (and (SNo (minus_SNo x0)) (∀ x1 . x1 ∈ SNoL x0 ⟶ SNoLt (minus_SNo x0) (minus_SNo x1))) (∀ x1 . x1 ∈ SNoR x0 ⟶ SNoLt (minus_SNo x1) (minus_SNo x0))) (SNoCutP (prim5 (SNoR x0) minus_SNo) (prim5 (SNoL x0) minus_SNo)) (proof)
Theorem SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0) (proof)
Theorem minus_SNo_Lt_contra^{minus_SNo_Lt_contra} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 x1 ⟶ SNoLt (minus_SNo x1) (minus_SNo x0) (proof)
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Known SNoLe_ref^SNoLe_ref : ∀ x0 . SNoLe x0 x0
Theorem minus_SNo_Le_contra^{minus_SNo_Le_contra} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ SNoLe (minus_SNo x1) (minus_SNo x0) (proof)
Theorem minus_SNo_SNoCutP^{minus_SNo_SNoCutP} : ∀ x0 . SNo x0 ⟶ SNoCutP (prim5 (SNoR x0) minus_SNo) (prim5 (SNoL x0) minus_SNo) (proof)
Theorem minus_SNo_SNoCutP_gen^{minus_SNo_SNoCutP_gen} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ SNoCutP (prim5 x1 minus_SNo) (prim5 x0 minus_SNo) (proof)
Known binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Known famunionE^famunionE : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ famunion x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ x0) (x2 ∈ x1 x4) ⟶ x3) ⟶ x3
Known In_no2cycle^In_no2cycle : ∀ x0 x1 . x0 ∈ x1 ⟶ x1 ∈ x0 ⟶ False
Known ordinal_Hered^{ordinal_Hered} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordinal x1
Known ordinal_ordsucc_In_eq^{ordinal_ordsucc_In_eq} : ∀ x0 x1 . ordinal x0 ⟶ x1 ∈ x0 ⟶ or (ordsucc x1 ∈ x0) (x0 = ordsucc x1)
Known ordinal_binunion^{ordinal_binunion} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ ordinal (binunion x0 x1)
Known ordinal_famunion^{ordinal_famunion} : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . x2 ∈ x0 ⟶ ordinal (x1 x2)) ⟶ ordinal (famunion x0 x1)
Theorem minus_SNo_Lev_lem1^{minus_SNo_Lev_lem1} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ SNoS_ x0 ⟶ SNoLev (minus_SNo x1) ⊆ SNoLev x1 (proof)
Theorem minus_SNo_Lev_lem2^{minus_SNo_Lev_lem2} : ∀ x0 . SNo x0 ⟶ SNoLev (minus_SNo x0) ⊆ SNoLev x0 (proof)
Known SNo_ind^SNo_ind : ∀ x0 : ι → ο . (∀ x1 x2 . SNoCutP x1 x2 ⟶ (∀ x3 . x3 ∈ x1 ⟶ x0 x3) ⟶ (∀ x3 . x3 ∈ x2 ⟶ x0 x3) ⟶ x0 (SNoCut x1 x2)) ⟶ ∀ x1 . SNo x1 ⟶ x0 x1
Known SNo_eq^SNo_eq : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x0 = SNoLev x1 ⟶ SNoEq_ (SNoLev x0) x0 x1 ⟶ x0 = x1
Known Subq_tra^Subq_tra : ∀ x0 x1 x2 . x0 ⊆ x1 ⟶ x1 ⊆ x2 ⟶ x0 ⊆ x2
Known SNoCutP_SNoCut_fst^{SNoCutP_SNoCut_fst} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ ∀ x2 . SNo x2 ⟶ (∀ x3 . x3 ∈ x0 ⟶ SNoLt x3 x2) ⟶ (∀ x3 . x3 ∈ x1 ⟶ SNoLt x2 x3) ⟶ and (SNoLev (SNoCut x0 x1) ⊆ SNoLev x2) (SNoEq_ (SNoLev (SNoCut x0 x1)) (SNoCut x0 x1) x2)
Theorem minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0 (proof)
Theorem minus_SNo_Lev^{minus_SNo_Lev} : ∀ x0 . SNo x0 ⟶ SNoLev (minus_SNo x0) = SNoLev x0 (proof)
Known SNoLev_uniq2^SNoLev_uniq2 : ∀ x0 . ordinal x0 ⟶ ∀ x1 . SNo_ x0 x1 ⟶ SNoLev x1 = x0
Known SNo_SNo^SNo_SNo : ∀ x0 . ordinal x0 ⟶ ∀ x1 . SNo_ x0 x1 ⟶ SNo x1
Theorem minus_SNo_SNo_^{minus_SNo_SNo_} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . SNo_ x0 x1 ⟶ SNo_ x0 (minus_SNo x1) (proof)
Theorem minus_SNo_SNoS_^{minus_SNo_SNoS_} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ SNoS_ x0 ⟶ minus_SNo x1 ∈ SNoS_ x0 (proof)
Theorem minus_SNoCut_eq_lem^{minus_SNoCut_eq_lem} : ∀ x0 . SNo x0 ⟶ ∀ x1 x2 . SNoCutP x1 x2 ⟶ x0 = SNoCut x1 x2 ⟶ minus_SNo x0 = SNoCut (prim5 x2 minus_SNo) (prim5 x1 minus_SNo) (proof)
Theorem minus_SNoCut_eq^{minus_SNoCut_eq} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ minus_SNo (SNoCut x0 x1) = SNoCut (prim5 x1 minus_SNo) (prim5 x0 minus_SNo) (proof)
Theorem minus_SNo_Lt_contra1^{minus_SNo_Lt_contra1} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt (minus_SNo x0) x1 ⟶ SNoLt (minus_SNo x1) x0 (proof)
Theorem minus_SNo_Lt_contra2^{minus_SNo_Lt_contra2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 (minus_SNo x1) ⟶ SNoLt x1 (minus_SNo x0) (proof)
Theorem minus_SNo_Lt_contra3^{minus_SNo_Lt_contra3} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt (minus_SNo x0) (minus_SNo x1) ⟶ SNoLt x1 x0 (proof)
Theorem SNo_momega^SNo_momega : SNo (minus_SNo omega) (proof)
Theorem mordinal_SNo^mordinal_SNo : ∀ x0 . ordinal x0 ⟶ SNo (minus_SNo x0) (proof)
Theorem mordinal_SNoLev^{mordinal_SNoLev} : ∀ x0 . ordinal x0 ⟶ SNoLev (minus_SNo x0) = x0 (proof)
Theorem mordinal_SNoLev_min^{mordinal_SNoLev_min} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ x0 ⟶ SNoLt (minus_SNo x0) x1 (proof)
Theorem mordinal_SNoLev_min_2^{mordinal_SNoLev_min_2} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ ordsucc x0 ⟶ SNoLe (minus_SNo x0) x1 (proof)
Definition add_SNo^add_SNo := SNo_rec2 (λ x0 x1 . λ x2 : ι → ι → ι . SNoCut (binunion {x2 x3 x1|x3 ∈ SNoL x0} (prim5 (SNoL x1) (x2 x0))) (binunion {x2 x3 x1|x3 ∈ SNoR x0} (prim5 (SNoR x1) (x2 x0))))
Theorem add_SNo_eq^add_SNo_eq : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ add_SNo x0 x1 = SNoCut (binunion {add_SNo x3 x1|x3 ∈ SNoL x0} (prim5 (SNoL x1) (add_SNo x0))) (binunion {add_SNo x3 x1|x3 ∈ SNoR x0} (prim5 (SNoR x1) (add_SNo x0))) (proof)
Param setprod^setprod : ι → ι → ι
Param ap^ap : ι → ι → ι
Definition mul_SNo^mul_SNo := SNo_rec2 (λ x0 x1 . λ x2 : ι → ι → ι . SNoCut (binunion {add_SNo (x2 (ap x3 0) x1) (add_SNo (x2 x0 (ap x3 1)) (minus_SNo (x2 (ap x3 0) (ap x3 1))))|x3 ∈ setprod (SNoL x0) (SNoL x1)} {add_SNo (x2 (ap x3 0) x1) (add_SNo (x2 x0 (ap x3 1)) (minus_SNo (x2 (ap x3 0) (ap x3 1))))|x3 ∈ setprod (SNoR x0) (SNoR x1)}) (binunion {add_SNo (x2 (ap x3 0) x1) (add_SNo (x2 x0 (ap x3 1)) (minus_SNo (x2 (ap x3 0) (ap x3 1))))|x3 ∈ setprod (SNoL x0) (SNoR x1)} {add_SNo (x2 (ap x3 0) x1) (add_SNo (x2 x0 (ap x3 1)) (minus_SNo (x2 (ap x3 0) (ap x3 1))))|x3 ∈ setprod (SNoR x0) (SNoL x1)}))
Known ReplEq_setprod_ext^{ReplEq_setprod_ext} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x1 ⟶ x2 x4 x5 = x3 x4 x5) ⟶ {x2 (ap x5 0) (ap x5 1)|x5 ∈ setprod x0 x1} = {x3 (ap x5 0) (ap x5 1)|x5 ∈ setprod x0 x1}
Theorem mul_SNo_eq^mul_SNo_eq : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ mul_SNo x0 x1 = SNoCut (binunion {add_SNo (mul_SNo (ap x3 0) x1) (add_SNo (mul_SNo x0 (ap x3 1)) (minus_SNo (mul_SNo (ap x3 0) (ap x3 1))))|x3 ∈ setprod (SNoL x0) (SNoL x1)} {add_SNo (mul_SNo (ap x3 0) x1) (add_SNo (mul_SNo x0 (ap x3 1)) (minus_SNo (mul_SNo (ap x3 0) (ap x3 1))))|x3 ∈ setprod (SNoR x0) (SNoR x1)}) (binunion {add_SNo (mul_SNo (ap x3 0) x1) (add_SNo (mul_SNo x0 (ap x3 1)) (minus_SNo (mul_SNo (ap x3 0) (ap x3 1))))|x3 ∈ setprod (SNoL x0) (SNoR x1)} {add_SNo (mul_SNo (ap x3 0) x1) (add_SNo (mul_SNo x0 (ap x3 1)) (minus_SNo (mul_SNo (ap x3 0) (ap x3 1))))|x3 ∈ setprod (SNoR x0) (SNoL x1)}) (proof)
Param explicit_Reals^{explicit_Reals} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ο
Definition Eps_i_realset := prim0 (λ x0 . and (∀ x1 . x1 ∈ x0 ⟶ SNo x1) (explicit_Reals x0 0 1 add_SNo mul_SNo SNoLe))