Proofgold Address

Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param Sep^Sep : ι → (ι → ο) → ι
Param SNoS_^SNoS_ : ι → ι
Param SNoLev^SNoLev : ι → ι
Param SNoLt^SNoLt : ι → ι → ο
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
Known SNoL_SNoS_^SNoL_SNoS_ : ∀ x0 . SNoL x0 ⊆ SNoS_ (SNoLev x0)
Definition SNoR^SNoR := λ x0 . Sep (SNoS_ (SNoLev x0)) (SNoLt x0)
Known SNoR_SNoS_^SNoR_SNoS_ : ∀ x0 . SNoR x0 ⊆ SNoS_ (SNoLev x0)
Param SNo^SNo : ι → ο
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param add_SNo^add_SNo : ι → ι → ι
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 binunion^binunion : ι → ι → ι
Known 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
Definition TransSet^TransSet := λ x0 . ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Definition ordinal^ordinal := λ x0 . and (TransSet x0) (∀ x1 . x1 ∈ x0 ⟶ TransSet x1)
Param SNoCut^SNoCut : ι → ι → ι
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known 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)))
Known and5I^and5I : ∀ x0 x1 x2 x3 x4 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ and (and (and (and x0 x1) x2) x3) x4
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 SNoCutP_SNoCut_L^{SNoCutP_SNoCut_L} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ ∀ x2 . x2 ∈ x0 ⟶ SNoLt x2 (SNoCut x0 x1)
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Known ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
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 SNoCutP_SNoCut_R^{SNoCutP_SNoCut_R} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ ∀ x2 . x2 ∈ x1 ⟶ SNoLt (SNoCut x0 x1) x2
Known binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
Known SNoCutP_SNo_SNoCut^{SNoCutP_SNo_SNoCut} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ SNo (SNoCut x0 x1)
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ 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
Param binintersect^binintersect : ι → ι → ι
Param SNoEq_^SNoEq_ : ι → ι → ι → ο
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
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 binintersectE^{binintersectE} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ and (x2 ∈ x0) (x2 ∈ x1)
Known SNoLt_tra^SNoLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known SNoR_I^SNoR_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x0 x1 ⟶ x1 ∈ SNoR x0
Known SNoL_I^SNoL_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x1 x0 ⟶ x1 ∈ SNoL x0
Known SNoS_I2^SNoS_I2 : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x0 ∈ SNoLev x1 ⟶ x0 ∈ SNoS_ (SNoLev x1)
Known ordinal_TransSet^{ordinal_TransSet} : ∀ x0 . ordinal x0 ⟶ TransSet x0
Known SNoLev_ordinal^{SNoLev_ordinal} : ∀ x0 . SNo x0 ⟶ ordinal (SNoLev x0)
Theorem add_SNo_prop1^{add_SNo_prop1} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ and (and (and (and (and (SNo (add_SNo x0 x1)) (∀ x2 . x2 ∈ SNoL x0 ⟶ SNoLt (add_SNo x2 x1) (add_SNo x0 x1))) (∀ x2 . x2 ∈ SNoR x0 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x1))) (∀ x2 . x2 ∈ SNoL x1 ⟶ SNoLt (add_SNo x0 x2) (add_SNo x0 x1))) (∀ x2 . x2 ∈ SNoR x1 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x0 x2))) (SNoCutP (binunion {add_SNo x2 x1|x2 ∈ SNoL x0} (prim5 (SNoL x1) (add_SNo x0))) (binunion {add_SNo x2 x1|x2 ∈ SNoR x0} (prim5 (SNoR x1) (add_SNo x0)))) (proof)
Theorem SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1) (proof)
Theorem add_SNo_Lt1^add_SNo_Lt1 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x2 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x1) (proof)
Param SNoLe^SNoLe : ι → ι → ο
Known SNoLeE^SNoLeE : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ or (SNoLt x0 x1) (x0 = x1)
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Known SNoLe_ref^SNoLe_ref : ∀ x0 . SNoLe x0 x0
Theorem add_SNo_Le1^add_SNo_Le1 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x2 ⟶ SNoLe (add_SNo x0 x1) (add_SNo x2 x1) (proof)
Theorem add_SNo_Lt2^add_SNo_Lt2 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x1 x2 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x0 x2) (proof)
Theorem add_SNo_Le2^add_SNo_Le2 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x1 x2 ⟶ SNoLe (add_SNo x0 x1) (add_SNo x0 x2) (proof)
Known SNoLtLe_tra^SNoLtLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLt x0 x2
Theorem add_SNo_Lt3a^add_SNo_Lt3a : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLt x0 x2 ⟶ SNoLe x1 x3 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x3) (proof)
Known SNoLeLt_tra^SNoLeLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Theorem add_SNo_Lt3b^add_SNo_Lt3b : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLe x0 x2 ⟶ SNoLt x1 x3 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x3) (proof)
Theorem add_SNo_Lt3^add_SNo_Lt3 : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLt x0 x2 ⟶ SNoLt x1 x3 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x3) (proof)
Known SNoLe_tra^SNoLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLe x0 x2
Theorem add_SNo_Le3^add_SNo_Le3 : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLe x0 x2 ⟶ SNoLe x1 x3 ⟶ SNoLe (add_SNo x0 x1) (add_SNo x2 x3) (proof)
Theorem add_SNo_SNoCutP^{add_SNo_SNoCutP} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoCutP (binunion {add_SNo x2 x1|x2 ∈ SNoL x0} (prim5 (SNoL x1) (add_SNo x0))) (binunion {add_SNo x2 x1|x2 ∈ SNoR x0} (prim5 (SNoR x1) (add_SNo x0))) (proof)
Theorem add_SNo_SNoCutP_gen^{add_SNo_SNoCutP_gen} : ∀ x0 x1 x2 x3 . SNoCutP x0 x1 ⟶ SNoCutP x2 x3 ⟶ SNoCutP (binunion {add_SNo x4 (SNoCut x2 x3)|x4 ∈ x0} (prim5 x2 (add_SNo (SNoCut x0 x1)))) (binunion {add_SNo x4 (SNoCut x2 x3)|x4 ∈ x1} (prim5 x3 (add_SNo (SNoCut x0 x1)))) (proof)
Known binunion_com^binunion_com : ∀ x0 x1 . binunion x0 x1 = binunion x1 x0
Known ReplEq_ext^ReplEq_ext : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x1 x3 = x2 x3) ⟶ prim5 x0 x1 = prim5 x0 x2
Theorem add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0 (proof)
Known SNoLev_ind^SNoLev_ind : ∀ x0 : ι → ο . (∀ x1 . SNo x1 ⟶ (∀ x2 . x2 ∈ SNoS_ (SNoLev x1) ⟶ x0 x2) ⟶ x0 x1) ⟶ ∀ x1 . SNo x1 ⟶ x0 x1
Known SNo_0^SNo_0 : SNo 0
Known SNo_eta^SNo_eta : ∀ x0 . SNo x0 ⟶ x0 = SNoCut (SNoL x0) (SNoR x0)
Known SNoR_0^SNoR_0 : SNoR 0 = 0
Known Repl_Empty^Repl_Empty : ∀ x0 : ι → ι . prim5 0 x0 = 0
Known binunion_idl^binunion_idl : ∀ x0 . binunion 0 x0 = x0
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Known SNoL_0^SNoL_0 : SNoL 0 = 0
Theorem add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0 (proof)
Theorem add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0 (proof)
Param minus_SNo^minus_SNo : ι → ι
Known SNo_eq^SNo_eq : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x0 = SNoLev x1 ⟶ SNoEq_ (SNoLev x0) x0 x1 ⟶ x0 = x1
Known ordinal_SNoLev^{ordinal_SNoLev} : ∀ x0 . ordinal x0 ⟶ SNoLev x0 = x0
Known ordinal_Empty^{ordinal_Empty} : ordinal 0
Known Subq_Empty^Subq_Empty : ∀ x0 . 0 ⊆ x0
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)
Known minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
Known minus_SNo_Lt_contra^{minus_SNo_Lt_contra} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 x1 ⟶ SNoLt (minus_SNo x1) (minus_SNo x0)
Known minus_SNo_Lev^{minus_SNo_Lev} : ∀ x0 . SNo x0 ⟶ SNoLev (minus_SNo x0) = SNoLev x0
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Theorem add_SNo_minus_SNo_linv^{add_SNo_minus_SNo_linv} : ∀ x0 . SNo x0 ⟶ add_SNo (minus_SNo x0) x0 = 0 (proof)
Theorem add_SNo_minus_SNo_rinv^{add_SNo_minus_SNo_rinv} : ∀ x0 . SNo x0 ⟶ add_SNo x0 (minus_SNo x0) = 0 (proof)
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 ordinal_SNo^ordinal_SNo : ∀ x0 . ordinal x0 ⟶ SNo x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Theorem add_SNo_ordinal_SNoCutP^{add_SNo_ordinal_SNoCutP} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ SNoCutP (binunion {add_SNo x2 x1|x2 ∈ SNoS_ x0} (prim5 (SNoS_ x1) (add_SNo x0))) 0 (proof)
Known ordinal_SNoL^ordinal_SNoL : ∀ x0 . ordinal x0 ⟶ SNoL x0 = SNoS_ x0
Known ordinal_SNoR^ordinal_SNoR : ∀ x0 . ordinal x0 ⟶ SNoR x0 = 0
Theorem add_SNo_ordinal_eq^{add_SNo_ordinal_eq} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ add_SNo x0 x1 = SNoCut (binunion {add_SNo x3 x1|x3 ∈ SNoS_ x0} (prim5 (SNoS_ x1) (add_SNo x0))) 0 (proof)
Known SNo_max_ordinal^{SNo_max_ordinal} : ∀ x0 . SNo x0 ⟶ (∀ x1 . x1 ∈ SNoS_ (SNoLev x0) ⟶ SNoLt x1 x0) ⟶ ordinal x0
Known SNoLt_trichotomy_or^{SNoLt_trichotomy_or} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (or (SNoLt x0 x1) (x0 = x1)) (SNoLt x1 x0)
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Param ordsucc^ordsucc : ι → ι
Param famunion^famunion : ι → (ι → ι) → ι
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 add_SNo_ordinal_ordinal^{add_SNo_ordinal_ordinal} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ ordinal (add_SNo x0 x1) (proof)
Known ordinal_ind^ordinal_ind : ∀ x0 : ι → ο . (∀ x1 . ordinal x1 ⟶ (∀ x2 . x2 ∈ x1 ⟶ x0 x2) ⟶ x0 x1) ⟶ ∀ x1 . ordinal x1 ⟶ x0 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_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 ordinal_Subq_SNoLe^{ordinal_Subq_SNoLe} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ x0 ⊆ x1 ⟶ SNoLe x0 x1
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Known Subq_ref^Subq_ref : ∀ x0 . x0 ⊆ x0
Known ordinal_In_SNoLt^{ordinal_In_SNoLt} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ SNoLt x1 x0
Known ordinal_ordsucc_In^{ordinal_ordsucc_In} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ∈ ordsucc x0
Known ordinal_SNoLt_In^{ordinal_SNoLt_In} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ SNoLt x0 x1 ⟶ x0 ∈ x1
Known ordinal_ordsucc^{ordinal_ordsucc} : ∀ x0 . ordinal x0 ⟶ ordinal (ordsucc x0)
Known SNoS_I^SNoS_I : ∀ x0 . ordinal x0 ⟶ ∀ x1 x2 . x2 ∈ x0 ⟶ SNo_ x2 x1 ⟶ x1 ∈ SNoS_ x0
Known ordinal_SNo_^ordinal_SNo_ : ∀ x0 . ordinal x0 ⟶ SNo_ x0 x0
Theorem add_SNo_ordinal_SL^{add_SNo_ordinal_SL} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ add_SNo (ordsucc x0) x1 = ordsucc (add_SNo x0 x1) (proof)
Theorem add_SNo_ordinal_SR^{add_SNo_ordinal_SR} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ add_SNo x0 (ordsucc x1) = ordsucc (add_SNo x0 x1) (proof)
Known ordinal_Hered^{ordinal_Hered} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordinal x1
Theorem add_SNo_ordinal_InL^{add_SNo_ordinal_InL} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ ∀ x2 . x2 ∈ x0 ⟶ add_SNo x2 x1 ∈ add_SNo x0 x1 (proof)
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Known SNoCut_Le^SNoCut_Le : ∀ x0 x1 x2 x3 . SNoCutP x0 x1 ⟶ SNoCutP x2 x3 ⟶ (∀ x4 . x4 ∈ x0 ⟶ SNoLt x4 (SNoCut x2 x3)) ⟶ (∀ x4 . x4 ∈ x3 ⟶ SNoLt (SNoCut x0 x1) x4) ⟶ SNoLe (SNoCut x0 x1) (SNoCut x2 x3)
Known SNoCutP_SNoL_SNoR^{SNoCutP_SNoL_SNoR} : ∀ x0 . SNo x0 ⟶ SNoCutP (SNoL x0) (SNoR x0)
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 x0)
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Known In_no2cycle^In_no2cycle : ∀ x0 x1 . x0 ∈ x1 ⟶ x1 ∈ x0 ⟶ False
Theorem add_SNo_SNoL_interpolate^{add_SNo_SNoL_interpolate} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 . x2 ∈ SNoL (add_SNo x0 x1) ⟶ or (∀ x3 : ο . (∀ x4 . and (x4 ∈ SNoL x0) (SNoLe x2 (add_SNo x4 x1)) ⟶ x3) ⟶ x3) (∀ x3 : ο . (∀ x4 . and (x4 ∈ SNoL x1) (SNoLe x2 (add_SNo x0 x4)) ⟶ x3) ⟶ x3) (proof)
Theorem add_SNo_SNoR_interpolate^{add_SNo_SNoR_interpolate} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 . x2 ∈ SNoR (add_SNo x0 x1) ⟶ or (∀ x3 : ο . (∀ x4 . and (x4 ∈ SNoR x0) (SNoLe (add_SNo x4 x1) x2) ⟶ x3) ⟶ x3) (∀ x3 : ο . (∀ x4 . and (x4 ∈ SNoR x1) (SNoLe (add_SNo x0 x4) x2) ⟶ x3) ⟶ x3) (proof)
Known 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
Known SNoCut_ext^SNoCut_ext : ∀ x0 x1 x2 x3 . SNoCutP x0 x1 ⟶ SNoCutP x2 x3 ⟶ (∀ x4 . x4 ∈ x0 ⟶ SNoLt x4 (SNoCut x2 x3)) ⟶ (∀ x4 . x4 ∈ x1 ⟶ SNoLt (SNoCut x2 x3) x4) ⟶ (∀ x4 . x4 ∈ x2 ⟶ SNoLt x4 (SNoCut x0 x1)) ⟶ (∀ x4 . x4 ∈ x3 ⟶ SNoLt (SNoCut x0 x1) x4) ⟶ SNoCut x0 x1 = SNoCut x2 x3
Theorem add_SNo_assoc^{add_SNo_assoc} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 (add_SNo x1 x2) = add_SNo (add_SNo x0 x1) x2 (proof)
Theorem add_SNo_cancel_L^{add_SNo_cancel_L} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 x1 = add_SNo x0 x2 ⟶ x1 = x2 (proof)
Theorem minus_SNo_0^minus_SNo_0 : minus_SNo 0 = 0 (proof)
Theorem minus_add_SNo_distr^{minus_add_SNo_distr} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ minus_SNo (add_SNo x0 x1) = add_SNo (minus_SNo x0) (minus_SNo x1) (proof)
Param If_i^If_i : ο → ι → ι → ι
Param mul_SNo^mul_SNo : ι → ι → ι
Definition div_SNo^div_SNo := λ x0 x1 . If_i (x1 = 0) 0 (prim0 (λ x2 . and (SNo x2) (mul_SNo x2 x1 = x0)))
Param SNo_pair^SNo_pair : ι → ι → ι
Param CSNo_Re^CSNo_Re : ι → ι
Param CSNo_Im^CSNo_Im : ι → ι
Definition minus_CSNo^minus_CSNo := λ x0 . SNo_pair (minus_SNo (CSNo_Re x0)) (minus_SNo (CSNo_Im x0))
Param CSNo^CSNo : ι → ο
Known CSNo_I^CSNo_I : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ CSNo (SNo_pair x0 x1)
Known CSNo_ReR^CSNo_ReR : ∀ x0 . CSNo x0 ⟶ SNo (CSNo_Re x0)
Known CSNo_ImR^CSNo_ImR : ∀ x0 . CSNo x0 ⟶ SNo (CSNo_Im x0)
Theorem CSNo_minus_CSNo^{CSNo_minus_CSNo} : ∀ x0 . CSNo x0 ⟶ CSNo (minus_CSNo x0) (proof)
Known SNo_pair_0^SNo_pair_0 : ∀ x0 . SNo_pair x0 0 = x0
Known CSNo_Re2^CSNo_Re2 : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ CSNo_Re (SNo_pair x0 x1) = x0
Theorem SNo_Re^SNo_Re : ∀ x0 . SNo x0 ⟶ CSNo_Re x0 = x0 (proof)
Known CSNo_Im2^CSNo_Im2 : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ CSNo_Im (SNo_pair x0 x1) = x1
Theorem SNo_Im^SNo_Im : ∀ x0 . SNo x0 ⟶ CSNo_Im x0 = 0 (proof)
Theorem Re_0^Re_0 : CSNo_Re 0 = 0 (proof)
Theorem Im_0^Im_0 : CSNo_Im 0 = 0 (proof)
Known SNo_1^SNo_1 : SNo 1
Theorem Re_1^Re_1 : CSNo_Re 1 = 1 (proof)
Theorem Im_1^Im_1 : CSNo_Im 1 = 0 (proof)
Definition Complex_i^Complex_i := SNo_pair 0 1
Theorem Re_i^Re_i : CSNo_Re Complex_i = 0 (proof)
Theorem Im_i^Im_i : CSNo_Im Complex_i = 1 (proof)
Definition add_CSNo^add_CSNo := λ x0 x1 . SNo_pair (add_SNo (CSNo_Re x0) (CSNo_Re x1)) (add_SNo (CSNo_Im x0) (CSNo_Im x1))
Theorem add_SNo_add_CSNo^{add_SNo_add_CSNo} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_CSNo x0 x1 (proof)
Theorem CSNo_add_CSNo^{CSNo_add_CSNo} : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo (add_CSNo x0 x1) (proof)
Known CSNo_ReIm^CSNo_ReIm : ∀ x0 . CSNo x0 ⟶ x0 = SNo_pair (CSNo_Re x0) (CSNo_Im x0)
Theorem add_CSNo_0L^add_CSNo_0L : ∀ x0 . CSNo x0 ⟶ add_CSNo 0 x0 = x0 (proof)
Theorem add_CSNo_0R^add_CSNo_0R : ∀ x0 . CSNo x0 ⟶ add_CSNo x0 0 = x0 (proof)
Theorem add_CSNo_minus_CSNo_linv^{add_CSNo_minus_CSNo_linv} : ∀ x0 . CSNo x0 ⟶ add_CSNo (minus_CSNo x0) x0 = 0 (proof)
Theorem add_CSNo_minus_CSNo_rinv^{add_CSNo_minus_CSNo_rinv} : ∀ x0 . CSNo x0 ⟶ add_CSNo x0 (minus_CSNo x0) = 0 (proof)
Theorem minus_SNo_minus_CSNo^{minus_SNo_minus_CSNo} : ∀ x0 . SNo x0 ⟶ minus_SNo x0 = minus_CSNo x0 (proof)
Param mul_CSNo^mul_CSNo : ι → ι → ι
Definition div_CSNo^div_CSNo := λ x0 x1 . If_i (x1 = 0) 0 (prim0 (λ x2 . and (CSNo x2) (mul_CSNo x2 x1 = x0)))