Proofgold Asset

Import Signature 6aee2..
Param 3b93e..^decode_b : ι → ι → ι → ι
Param 28572..^encode_b : ι → CT2 ι
Known 8d10c..^{decode_encode_b} : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ decode_b (encode_b x0 x1) x2 x3 = x1 x2 x3
Known d0102..^encode_b_ext : ∀ x0 . ∀ x1 x2 : ι → ι → ι . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = x2 x3 x4) ⟶ encode_b x0 x1 = encode_b x0 x2
Param e9507..^decode_p : ι → ι → ο
Known d5832..^{decode_encode_p} : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ decode_p (Sep x0 x1) x2 = x1 x2
Known c1030..^encode_p_ext : ∀ x0 . ∀ x1 x2 : ι → ο . (∀ x3 . x3 ∈ x0 ⟶ iff (x1 x3) (x2 x3)) ⟶ Sep x0 x1 = Sep x0 x2
Param e6e31..^encode_r : ι → (ι → ι → ο) → ι
Param b93a6..^decode_r : ι → ι → ι → ο
Known 08a8a..^{decode_encode_r} : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ decode_r (encode_r x0 x1) x2 x3 = x1 x2 x3
Known fa3b1..^encode_r_ext : ∀ x0 . ∀ x1 x2 : ι → ι → ο . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ iff (x1 x3 x4) (x2 x3 x4)) ⟶ encode_r x0 x1 = encode_r x0 x2
Param abcb4..^encode_c : ι → ((ι → ο) → ο) → ι
Param 3edd4..^decode_c : ι → (ι → ο) → ο
Known 880bf..^{decode_encode_c} : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ο . (∀ x3 . x2 x3 ⟶ x3 ∈ x0) ⟶ decode_c (encode_c x0 x1) x2 = x1 x2
Known 0c548..^encode_c_ext : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . (∀ x3 : ι → ο . (∀ x4 . x3 x4 ⟶ x4 ∈ x0) ⟶ iff (x1 x3) (x2 x3)) ⟶ encode_c x0 x1 = encode_c x0 x2
Known 8ab1d..^{ordinal_ordsucc_In} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ∈ ordsucc x0
Known b3f1e..^{ordinal_famunion} : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . x2 ∈ x0 ⟶ ordinal (x1 x2)) ⟶ ordinal (famunion x0 x1)
Known b2498..^{ordinal_binunion} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ ordinal (binunion x0 x1)
Param 327f9..^pack_p : ι → (ι → ο) → ι
Known fac8f..^pack_p_0_eq2 : ∀ x0 . ∀ x1 : ι → ο . x0 = ap (pack_p x0 x1) 0
Param 65334..^pack_r : ι → (ι → ι → ο) → ι
Known 09249..^pack_r_0_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι → ο . x2 x0 (ap (pack_r x0 x1) 0) ⟶ x2 (ap (pack_r x0 x1) 0) x0
Param 33166..^int : ι
Known be344..^nat_p_int : ∀ x0 . nat_p x0 ⟶ x0 ∈ int
Known b5182..^beta : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ ap (lam x0 x1) x2 = x1 x2
Known 1affc..^ap0_Sigma : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ ap x2 0 ∈ x0
Known 520af..^ap1_Sigma : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ ap x2 1 ∈ x1 (ap x2 0)
Known 3733f..^lam_Pi : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1 x3) ⟶ lam x0 x2 ∈ Pi x0 x1
Known cb962..^ap_Pi : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ Pi x0 x1 ⟶ x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3
Param c16a7..^SNo_ : ι → ι → ο
Param 12b5f..^SNoEq_ : ι → ι → ι → ο
Param 116b7..^SNo : ι → ο
Param 83dce..^SNoLev : ι → ι
Param 8c8aa..^SNoLt : ι → ι → ο
Param 54f40..^SNoLe : ι → ι → ο
Known f0e9f..^{SNoLev_ordinal} : ∀ x0 . SNo x0 ⟶ ordinal (SNoLev x0)
Known 49cd5..^SNoLev_ : ∀ x0 . SNo x0 ⟶ SNo_ (SNoLev x0) x0
Known bf4ac..^SNo_eq : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x0 = SNoLev x1 ⟶ SNoEq_ (SNoLev x0) x0 x1 ⟶ x0 = x1
Known f0983..^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Known 23cc4..^SNoLeE : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ or (SNoLt x0 x1) (x0 = x1)
Known f756d..^SNoEq_sym_ : ∀ x0 x1 x2 . SNoEq_ x0 x1 x2 ⟶ SNoEq_ x0 x2 x1
Known d12fc..^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 06fac..^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Known 7588e..^{SNoLt_trichotomy_or_impred} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 : ο . (SNoLt x0 x1 ⟶ x2) ⟶ (x0 = x1 ⟶ x2) ⟶ (SNoLt x1 x0 ⟶ x2) ⟶ x2
Known ccfdf..^SNoLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known 83bd2..^SNoLe_ref : ∀ x0 . SNoLe x0 x0
Known 3abee..^{SNoLe_antisym} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x0 ⟶ x0 = x1
Known ba854..^SNoLtLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLt x0 x2
Known 4a77a..^SNoLeLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known 03198..^SNoLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLe x0 x2
Known 8965d..^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 x0)
Param 77512..^SNoCutP : ι → ι → ο
Param 295bc..^SNoCut : ι → ι → ι
Known 5bcc8..^{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
Param 4680c..^SNoS_ : ι → ι
Known 04f24..^SNoS_I : ∀ x0 . ordinal x0 ⟶ ∀ x1 x2 . x2 ∈ x0 ⟶ SNo_ x2 x1 ⟶ x1 ∈ SNoS_ x0
Known b4042..^SNoS_I2 : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x0 ∈ SNoLev x1 ⟶ x0 ∈ SNoS_ (SNoLev x1)
Known 62b49..^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
Param 48bcf..^SNoL : ι → ι
Param 0cfe8..^SNoR : ι → ι
Known 7aba5..^{SNoCutP_SNoL_SNoR} : ∀ x0 . SNo x0 ⟶ SNoCutP (SNoL x0) (SNoR x0)
Known 7ccd8..^SNoL_E : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoL x0 ⟶ ∀ x2 : ο . (SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x1 x0 ⟶ x2) ⟶ x2
Known 7dca0..^SNoR_E : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoR x0 ⟶ ∀ x2 : ο . (SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x0 x1 ⟶ x2) ⟶ x2
Known 4f3d0..^SNoL_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x1 x0 ⟶ x1 ∈ SNoL x0
Known 383e8..^SNoR_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x0 x1 ⟶ x1 ∈ SNoR x0
Known dc927..^SNo_eta : ∀ x0 . SNo x0 ⟶ x0 = SNoCut (SNoL x0) (SNoR x0)
Known f6661..^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
Known 1eea3..^ordinal_SNo : ∀ x0 . ordinal x0 ⟶ SNo x0
Known fea5f..^{ordinal_SNoLev} : ∀ x0 . ordinal x0 ⟶ SNoLev x0 = x0
Known 5fd93..^nat_p_SNo : ∀ x0 . nat_p x0 ⟶ SNo x0
Known d0f9c..^omega_SNo : ∀ x0 . x0 ∈ omega ⟶ SNo x0
Known 2ccf7..^{omega_SNoS_omega} : omega ⊆ SNoS_ omega
Known ced78..^{ordinal_In_SNoLt} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ SNoLt x1 x0
Known c9a73..^{ordinal_SNoLev_max_2} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ ordsucc x0 ⟶ SNoLe x1 x0
Known 15c58..^{ordinal_Subq_SNoLe} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ x0 ⊆ x1 ⟶ SNoLe x0 x1
Known de775..^{ordinal_SNoLt_In} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ SNoLt x0 x1 ⟶ x0 ∈ x1
Known 94a31..^SNo_0 : SNo 0
Known 9a733..^SNo_1 : SNo 1
Known c524a..^SNo_2 : SNo 2
Known 3527a..^SNoLev_0 : SNoLev 0 = 0
Param ffee2..^eps_ : ι → ι
Known d073f..^eps_0_1 : eps_ 0 = 1
Known 006b0..^SNo_eps_ : ∀ x0 . x0 ∈ omega ⟶ SNo (eps_ x0)
Known 760b4..^SNo_eps_pos : ∀ x0 . x0 ∈ omega ⟶ SNoLt 0 (eps_ x0)
Known 851a8..^SNoLev_ind : ∀ x0 : ι → ο . (∀ x1 . SNo x1 ⟶ (∀ x2 . x2 ∈ SNoS_ (SNoLev x1) ⟶ x0 x2) ⟶ x0 x1) ⟶ ∀ x1 . SNo x1 ⟶ x0 x1
Known db7b5..^SNo_omega : SNo omega
Known cb639..^SNoLt_0_1 : SNoLt 0 1
Known f4693..^SNoLt_0_2 : SNoLt 0 2
Known 19d5e..^SNoLt_1_2 : SNoLt 1 2
Param 8a6d8..^minus_SNo : ι → ι
Known 744cc..^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Known 3ebf6..^{minus_SNo_Lt_contra} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 x1 ⟶ SNoLt (minus_SNo x1) (minus_SNo x0)
Known f8c89..^{minus_SNo_Le_contra} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ SNoLe (minus_SNo x1) (minus_SNo x0)
Known cc64e..^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
Known 476ae..^{minus_SNo_Lev} : ∀ x0 . SNo x0 ⟶ SNoLev (minus_SNo x0) = SNoLev x0
Known 29872..^{minus_SNo_Lt_contra1} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt (minus_SNo x0) x1 ⟶ SNoLt (minus_SNo x1) x0
Known f4196..^{minus_SNo_Lt_contra2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 (minus_SNo x1) ⟶ SNoLt x1 (minus_SNo x0)
Known 0c19f..^{minus_SNo_SNoS_omega} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ minus_SNo x0 ∈ SNoS_ omega
Param f62f9..^add_SNo : ι → ι → ι
Known a5709..^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Known f2d78..^{SNo_add_SNo_3} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo (add_SNo x0 (add_SNo x1 x2))
Known 42037..^{SNo_add_SNo_4} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo (add_SNo x0 (add_SNo x1 (add_SNo x2 x3)))
Known cb073..^add_SNo_Lt1 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x2 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x1)
Known abcf7..^add_SNo_Le1 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x2 ⟶ SNoLe (add_SNo x0 x1) (add_SNo x2 x1)
Known 6f094..^add_SNo_Lt2 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x1 x2 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x0 x2)
Known 3f52b..^add_SNo_Le2 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x1 x2 ⟶ SNoLe (add_SNo x0 x1) (add_SNo x0 x2)
Known 55bda..^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)
Known be66c..^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)
Known f017e..^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Known 446ba..^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Known 1a7fa..^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
Known 44e25..^{add_SNo_minus_SNo_linv} : ∀ x0 . SNo x0 ⟶ add_SNo (minus_SNo x0) x0 = 0
Known b2f9c..^{add_SNo_minus_SNo_rinv} : ∀ x0 . SNo x0 ⟶ add_SNo x0 (minus_SNo x0) = 0
Known 9df65..^{add_nat_add_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ add_nat x0 x1 = add_SNo x0 x1
Known eb637..^{add_SNo_In_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ add_SNo x0 x1 ∈ omega
Known 2b829..^{add_SNo_1_1_2} : add_SNo 1 1 = 2
Known bc15a..^{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
Known 91f4e..^{add_SNo_minus_R2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo (add_SNo x0 x1) (minus_SNo x1) = x0
Known 43b4d..^{add_SNo_minus_R2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo (add_SNo x0 (minus_SNo x1)) x1 = x0
Known 598ee..^{add_SNo_minus_L2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo (minus_SNo x0) (add_SNo x0 x1) = x1
Known 78125..^{add_SNo_cancel_L} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 x1 = add_SNo x0 x2 ⟶ x1 = x2
Known e6280..^{add_SNo_cancel_R} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 x1 = add_SNo x2 x1 ⟶ x0 = x2
Known 3fcc5..^minus_SNo_0 : minus_SNo 0 = 0
Known da8d5..^{minus_add_SNo_distr} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ minus_SNo (add_SNo x0 x1) = add_SNo (minus_SNo x0) (minus_SNo x1)
Known 6d16a..^{add_SNo_SNoS_omega} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ ∀ x1 . x1 ∈ SNoS_ omega ⟶ add_SNo x0 x1 ∈ SNoS_ omega
Known 2ec8a..^{add_SNo_Lt1_cancel} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x1) ⟶ SNoLt x0 x2
Known f2ec8..^{add_SNo_assoc_4} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ add_SNo x0 (add_SNo x1 (add_SNo x2 x3)) = add_SNo (add_SNo x0 (add_SNo x1 x2)) x3
Known 3c9cc..^{add_SNo_com_3_0_1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 (add_SNo x1 x2) = add_SNo x1 (add_SNo x0 x2)
Known e25ee..^{add_SNo_com_4_inner_mid} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ add_SNo (add_SNo x0 x1) (add_SNo x2 x3) = add_SNo (add_SNo x0 x2) (add_SNo x1 x3)
Known 34064..^{add_SNo_rotate_3_1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 (add_SNo x1 x2) = add_SNo x2 (add_SNo x0 x1)
Known a8450..^{add_SNo_minus_L2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 (add_SNo (minus_SNo x0) x1) = x1
Known 4033d..^{add_SNo_minus_Lt1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt (add_SNo x0 (minus_SNo x1)) x2 ⟶ SNoLt x0 (add_SNo x2 x1)
Known 36214..^{add_SNo_minus_Lt2} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x2 (add_SNo x0 (minus_SNo x1)) ⟶ SNoLt (add_SNo x2 x1) x0
Known 0b5cc..^{add_SNo_minus_Lt1b} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 (add_SNo x2 x1) ⟶ SNoLt (add_SNo x0 (minus_SNo x1)) x2
Known 68419..^{add_SNo_minus_Lt2b} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt (add_SNo x2 x1) x0 ⟶ SNoLt x2 (add_SNo x0 (minus_SNo x1))
Known d815d..^{add_SNo_minus_Le2b} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe (add_SNo x2 x1) x0 ⟶ SNoLe x2 (add_SNo x0 (minus_SNo x1))
Known 36bf6..^{ordinal_ordsucc_SNo_eq} : ∀ x0 . ordinal x0 ⟶ ordsucc x0 = add_SNo 1 x0
Known 2c799..^{SNoLt_minus_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 x1 ⟶ SNoLt 0 (add_SNo x1 (minus_SNo x0))
Param 2990e..^abs_SNo : ι → ι
Known 88568..^pos_abs_SNo : ∀ x0 . SNoLt 0 x0 ⟶ abs_SNo x0 = x0
Known fd708..^{abs_SNo_dist_swap} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ abs_SNo (add_SNo x0 (minus_SNo x1)) = abs_SNo (add_SNo x1 (minus_SNo x0))
Param b1473..^mul_SNo : ι → ι → ι
Known 9fd08..^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Known c2320..^mul_SNo_Lt : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLt x2 x0 ⟶ SNoLt x3 x1 ⟶ SNoLt (add_SNo (mul_SNo x2 x1) (mul_SNo x0 x3)) (add_SNo (mul_SNo x0 x1) (mul_SNo x2 x3))
Known a4f4a..^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Known 5477a..^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = x0
Known 83d5f..^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
Known 3d0cd..^{mul_SNo_minus_distrL} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo (minus_SNo x0) x1 = minus_SNo (mul_SNo x0 x1)
Known 2e747..^{mul_SNo_minus_distrR} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 (minus_SNo x1) = minus_SNo (mul_SNo x0 x1)
Known 5f038..^{mul_SNo_distrR} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo (add_SNo x0 x1) x2 = add_SNo (mul_SNo x0 x2) (mul_SNo x1 x2)
Known 2a483..^{mul_SNo_distrL} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 (add_SNo x1 x2) = add_SNo (mul_SNo x0 x1) (mul_SNo x0 x2)
Known d322c..^{mul_SNo_assoc} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 (mul_SNo x1 x2) = mul_SNo (mul_SNo x0 x1) x2
Known eb633..^{mul_nat_mul_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_nat x0 x1 = mul_SNo x0 x1
Known 8435a..^{mul_SNo_In_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_SNo x0 x1 ∈ omega
Known aaf0d..^{mul_SNo_zeroL} : ∀ x0 . SNo x0 ⟶ mul_SNo 0 x0 = 0
Known 1b25c..^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 x0 = x0
Known 0e949..^{pos_mul_SNo_Lt} : ∀ x0 x1 x2 . SNo x0 ⟶ SNoLt 0 x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x1 x2 ⟶ SNoLt (mul_SNo x0 x1) (mul_SNo x0 x2)
Known ae94b..^{nonneg_mul_SNo_Le} : ∀ x0 x1 x2 . SNo x0 ⟶ SNoLe 0 x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x1 x2 ⟶ SNoLe (mul_SNo x0 x1) (mul_SNo x0 x2)
Known d984d..^{mul_SNo_pos_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt 0 x0 ⟶ SNoLt 0 x1 ⟶ SNoLt 0 (mul_SNo x0 x1)
Known fc765..^{mul_SNo_nonneg_nonneg} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe 0 x0 ⟶ SNoLe 0 x1 ⟶ SNoLe 0 (mul_SNo x0 x1)
Known cda6e..^{mul_SNo_com_3_0_1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 (mul_SNo x1 x2) = mul_SNo x1 (mul_SNo x0 x2)
Known e23f1..^{mul_SNo_com_4_inner_mid} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ mul_SNo (mul_SNo x0 x1) (mul_SNo x2 x3) = mul_SNo (mul_SNo x0 x2) (mul_SNo x1 x3)
Known 035c5..^{mul_SNo_nonzero_cancel} : ∀ x0 x1 x2 . SNo x0 ⟶ (x0 = 0 ⟶ ∀ x3 : ο . x3) ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 x1 = mul_SNo x0 x2 ⟶ x1 = x2
Known 14278..^{int_SNo_cases} : ∀ x0 : ι → ο . (∀ x1 . x1 ∈ omega ⟶ x0 x1) ⟶ (∀ x1 . x1 ∈ omega ⟶ x0 (minus_SNo x1)) ⟶ ∀ x1 . x1 ∈ int ⟶ x0 x1
Known 6089e..^int_SNo : ∀ x0 . x0 ∈ int ⟶ SNo x0
Known b1bb2..^{Subq_omega_int} : omega ⊆ int
Known 1036d..^int_add_SNo : ∀ x0 . x0 ∈ int ⟶ ∀ x1 . x1 ∈ int ⟶ add_SNo x0 x1 ∈ int
Known b2c59..^{int_minus_SNo} : ∀ x0 . x0 ∈ int ⟶ minus_SNo x0 ∈ int
Known f4056..^int_mul_SNo : ∀ x0 . x0 ∈ int ⟶ ∀ x1 . x1 ∈ int ⟶ mul_SNo x0 x1 ∈ int
Param ea810..^exp_SNo_nat : ι → ι → ι
Known f317b..^{SNo_exp_SNo_nat} : ∀ x0 . SNo x0 ⟶ ∀ x1 . nat_p x1 ⟶ SNo (exp_SNo_nat x0 x1)
Known 39327..^{nat_exp_SNo_nat} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (exp_SNo_nat x0 x1)
Known 1a029..^{eps_ordsucc_half_add} : ∀ x0 . nat_p x0 ⟶ add_SNo (eps_ (ordsucc x0)) (eps_ (ordsucc x0)) = eps_ x0
Known e69e4..^{mul_SNo_eps_power_2} : ∀ x0 . nat_p x0 ⟶ mul_SNo (eps_ x0) (exp_SNo_nat 2 x0) = 1
Param 8b12f..^div_SNo : ι → ι → ι
Known c761b..^SNo_div_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (div_SNo x0 x1)
Param 5277e..^real : ι
Known 9f45f..^real_I : ∀ x0 . x0 ∈ SNoS_ (ordsucc omega) ⟶ (x0 = omega ⟶ ∀ x1 : ο . x1) ⟶ (x0 = minus_SNo omega ⟶ ∀ x1 : ο . x1) ⟶ (∀ x1 . x1 ∈ SNoS_ omega ⟶ (∀ x2 . x2 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x1 (minus_SNo x0))) (eps_ x2)) ⟶ x1 = x0) ⟶ x0 ∈ real
Known 447bb..^real_E : ∀ x0 . x0 ∈ real ⟶ ∀ x1 : ο . (SNo x0 ⟶ SNoLev x0 ∈ ordsucc omega ⟶ x0 ∈ SNoS_ (ordsucc omega) ⟶ SNoLt (minus_SNo omega) x0 ⟶ SNoLt x0 omega ⟶ (∀ x2 . x2 ∈ SNoS_ omega ⟶ (∀ x3 . x3 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x2 (minus_SNo x0))) (eps_ x3)) ⟶ x2 = x0) ⟶ (∀ x2 . x2 ∈ omega ⟶ ∃ x3 . and (x3 ∈ SNoS_ omega) (and (SNoLt x3 x0) (SNoLt x0 (add_SNo x3 (eps_ x2))))) ⟶ x1) ⟶ x1
Known d4cbb..^real_SNo : ∀ x0 . x0 ∈ real ⟶ SNo x0
Known d9411..^{SNoS_omega_real} : SNoS_ omega ⊆ real
Known 49313..^real_0 : 0 ∈ real
Known 790b4..^real_1 : 1 ∈ real
Known 4b837..^{real_minus_SNo} : ∀ x0 . x0 ∈ real ⟶ minus_SNo x0 ∈ real
Known 3c841..^real_add_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ add_SNo x0 x1 ∈ real
Known 51dc4..^real_mul_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ mul_SNo x0 x1 ∈ real