Proofgold Address

Param famunion^famunion : ι → (ι → ι) → ι
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known Empty_Subq_eq^{Empty_Subq_eq} : ∀ x0 . x0 ⊆ 0 ⟶ x0 = 0
Known famunionE_impred^{famunionE_impred} : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ famunion x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ x0 ⟶ x2 ∈ x1 x4 ⟶ x3) ⟶ x3
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Theorem famunion_Empty^{famunion_Empty} : ∀ x0 : ι → ι . famunion 0 x0 = 0 (proof)
Param SNo^SNo : ι → ο
Param and^and : ο → ο → ο
Param SNoLt^SNoLt : ι → ι → ο
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)
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Theorem SNoCutP_L_0^SNoCutP_L_0 : ∀ x0 . (∀ x1 . x1 ∈ x0 ⟶ SNo x1) ⟶ SNoCutP x0 0 (proof)
Theorem SNoCutP_0_R^SNoCutP_0_R : ∀ x0 . (∀ x1 . x1 ∈ x0 ⟶ SNo x1) ⟶ SNoCutP 0 x0 (proof)
Theorem SNoCutP_0_0^SNoCutP_0_0 : SNoCutP 0 0 (proof)
Param SNoCut^SNoCut : ι → ι → ι
Param SNoLev^SNoLev : ι → ι
Param ordsucc^ordsucc : ι → ι
Param binunion^binunion : ι → ι → ι
Param SNoEq_^SNoEq_ : ι → ι → ι → ο
Known 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
Known binunion_idl^binunion_idl : ∀ x0 . binunion 0 x0 = x0
Known SNo_eq^SNo_eq : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x0 = SNoLev x1 ⟶ SNoEq_ (SNoLev x0) x0 x1 ⟶ x0 = x1
Known SNo_0^SNo_0 : SNo 0
Known SNoLev_0^SNoLev_0 : SNoLev 0 = 0
Param iff^iff : ο → ο → ο
Known SNoEq_I^SNoEq_I : ∀ x0 x1 x2 . (∀ x3 . x3 ∈ x0 ⟶ iff (x3 ∈ x1) (x3 ∈ x2)) ⟶ SNoEq_ x0 x1 x2
Known cases_1^cases_1 : ∀ x0 . x0 ∈ 1 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 x0
Theorem SNoCut_0_0^SNoCut_0_0 : SNoCut 0 0 = 0 (proof)
Param SNoL^SNoL : ι → ι
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
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 SNo_1^SNo_1 : SNo 1
Param ordinal^ordinal : ι → ο
Known ordinal_SNoLev^{ordinal_SNoLev} : ∀ x0 . ordinal x0 ⟶ SNoLev x0 = x0
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
Known In_0_1^In_0_1 : 0 ∈ 1
Known SNoL_I^SNoL_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x1 x0 ⟶ x1 ∈ SNoL x0
Known ordinal_In_SNoLt^{ordinal_In_SNoLt} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ SNoLt x1 x0
Theorem SNoL_1^SNoL_1 : SNoL 1 = 1 (proof)
Param SNoR^SNoR : ι → ι
Known ordinal_SNoR^ordinal_SNoR : ∀ x0 . ordinal x0 ⟶ SNoR x0 = 0
Theorem SNoR_1^SNoR_1 : SNoR 1 = 0 (proof)
Param add_SNo^add_SNo : ι → ι → ι
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Theorem SNo_add_SNo_3^{SNo_add_SNo_3} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo (add_SNo x0 (add_SNo x1 x2)) (proof)
Param minus_SNo^minus_SNo : ι → ι
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Theorem SNo_add_SNo_3c^{SNo_add_SNo_3c} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo (add_SNo x0 (add_SNo x1 (minus_SNo x2))) (proof)
Known 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)
Theorem minus_add_SNo_distr_3^{minus_add_SNo_distr_3} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ minus_SNo (add_SNo x0 (add_SNo x1 x2)) = add_SNo (minus_SNo x0) (add_SNo (minus_SNo x1) (minus_SNo x2)) (proof)
Known 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
Known 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)
Theorem add_SNo_3_3_3_Lt1^{add_SNo_3_3_3_Lt1} : ∀ x0 x1 x2 x3 x4 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x3) ⟶ SNoLt (add_SNo x0 (add_SNo x1 x4)) (add_SNo x2 (add_SNo x3 x4)) (proof)
Known add_SNo_rotate_3_1^{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)
Theorem add_SNo_3_2_3_Lt1^{add_SNo_3_2_3_Lt1} : ∀ x0 x1 x2 x3 x4 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ SNoLt (add_SNo x1 x0) (add_SNo x2 x3) ⟶ SNoLt (add_SNo x0 (add_SNo x4 x1)) (add_SNo x2 (add_SNo x3 x4)) (proof)
Known add_SNo_Lt1_cancel^{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 add_SNo_minus_SNo_linv^{add_SNo_minus_SNo_linv} : ∀ x0 . SNo x0 ⟶ add_SNo (minus_SNo x0) x0 = 0
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
Theorem add_SNo_minus_Lt1b^{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 (proof)
Theorem add_SNo_minus_Lt2b^{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)) (proof)
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Theorem add_SNo_minus_Lt12b3^{add_SNo_minus_Lt12b3} : ∀ x0 x1 x2 x3 x4 x5 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ SNo x5 ⟶ SNoLt (add_SNo x0 (add_SNo x1 x5)) (add_SNo x3 (add_SNo x4 x2)) ⟶ SNoLt (add_SNo x0 (add_SNo x1 (minus_SNo x2))) (add_SNo x3 (add_SNo x4 (minus_SNo x5))) (proof)
Param mul_SNo^mul_SNo : ι → ι → ι
Known mul_SNo_prop_1^{mul_SNo_prop_1} : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ ∀ x2 : ο . (SNo (mul_SNo x0 x1) ⟶ (∀ x3 . x3 ∈ SNoL x0 ⟶ ∀ x4 . x4 ∈ SNoL x1 ⟶ SNoLt (add_SNo (mul_SNo x3 x1) (mul_SNo x0 x4)) (add_SNo (mul_SNo x0 x1) (mul_SNo x3 x4))) ⟶ (∀ x3 . x3 ∈ SNoR x0 ⟶ ∀ x4 . x4 ∈ SNoR x1 ⟶ SNoLt (add_SNo (mul_SNo x3 x1) (mul_SNo x0 x4)) (add_SNo (mul_SNo x0 x1) (mul_SNo x3 x4))) ⟶ (∀ x3 . x3 ∈ SNoL x0 ⟶ ∀ x4 . x4 ∈ SNoR x1 ⟶ SNoLt (add_SNo (mul_SNo x0 x1) (mul_SNo x3 x4)) (add_SNo (mul_SNo x3 x1) (mul_SNo x0 x4))) ⟶ (∀ x3 . x3 ∈ SNoR x0 ⟶ ∀ x4 . x4 ∈ SNoL x1 ⟶ SNoLt (add_SNo (mul_SNo x0 x1) (mul_SNo x3 x4)) (add_SNo (mul_SNo x3 x1) (mul_SNo x0 x4))) ⟶ x2) ⟶ x2
Theorem SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1) (proof)
Theorem SNo_mul_SNo_lem^{SNo_mul_SNo_lem} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo (add_SNo (mul_SNo x2 x1) (add_SNo (mul_SNo x0 x3) (minus_SNo (mul_SNo x2 x3)))) (proof)
Known mul_SNo_eq_2^mul_SNo_eq_2 : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 : ο . (∀ x3 x4 . (∀ x5 . x5 ∈ x3 ⟶ ∀ x6 : ο . (∀ x7 . x7 ∈ SNoL x0 ⟶ ∀ x8 . x8 ∈ SNoL x1 ⟶ x5 = add_SNo (mul_SNo x7 x1) (add_SNo (mul_SNo x0 x8) (minus_SNo (mul_SNo x7 x8))) ⟶ x6) ⟶ (∀ x7 . x7 ∈ SNoR x0 ⟶ ∀ x8 . x8 ∈ SNoR x1 ⟶ x5 = add_SNo (mul_SNo x7 x1) (add_SNo (mul_SNo x0 x8) (minus_SNo (mul_SNo x7 x8))) ⟶ x6) ⟶ x6) ⟶ (∀ x5 . x5 ∈ SNoL x0 ⟶ ∀ x6 . x6 ∈ SNoL x1 ⟶ add_SNo (mul_SNo x5 x1) (add_SNo (mul_SNo x0 x6) (minus_SNo (mul_SNo x5 x6))) ∈ x3) ⟶ (∀ x5 . x5 ∈ SNoR x0 ⟶ ∀ x6 . x6 ∈ SNoR x1 ⟶ add_SNo (mul_SNo x5 x1) (add_SNo (mul_SNo x0 x6) (minus_SNo (mul_SNo x5 x6))) ∈ x3) ⟶ (∀ x5 . x5 ∈ x4 ⟶ ∀ x6 : ο . (∀ x7 . x7 ∈ SNoL x0 ⟶ ∀ x8 . x8 ∈ SNoR x1 ⟶ x5 = add_SNo (mul_SNo x7 x1) (add_SNo (mul_SNo x0 x8) (minus_SNo (mul_SNo x7 x8))) ⟶ x6) ⟶ (∀ x7 . x7 ∈ SNoR x0 ⟶ ∀ x8 . x8 ∈ SNoL x1 ⟶ x5 = add_SNo (mul_SNo x7 x1) (add_SNo (mul_SNo x0 x8) (minus_SNo (mul_SNo x7 x8))) ⟶ x6) ⟶ x6) ⟶ (∀ x5 . x5 ∈ SNoL x0 ⟶ ∀ x6 . x6 ∈ SNoR x1 ⟶ add_SNo (mul_SNo x5 x1) (add_SNo (mul_SNo x0 x6) (minus_SNo (mul_SNo x5 x6))) ∈ x4) ⟶ (∀ x5 . x5 ∈ SNoR x0 ⟶ ∀ x6 . x6 ∈ SNoL x1 ⟶ add_SNo (mul_SNo x5 x1) (add_SNo (mul_SNo x0 x6) (minus_SNo (mul_SNo x5 x6))) ∈ x4) ⟶ mul_SNo x0 x1 = SNoCut x3 x4 ⟶ x2) ⟶ x2
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 mul_SNo_eq_3^mul_SNo_eq_3 : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 : ο . (∀ x3 x4 . SNoCutP x3 x4 ⟶ (∀ x5 . x5 ∈ x3 ⟶ ∀ x6 : ο . (∀ x7 . x7 ∈ SNoL x0 ⟶ ∀ x8 . x8 ∈ SNoL x1 ⟶ x5 = add_SNo (mul_SNo x7 x1) (add_SNo (mul_SNo x0 x8) (minus_SNo (mul_SNo x7 x8))) ⟶ x6) ⟶ (∀ x7 . x7 ∈ SNoR x0 ⟶ ∀ x8 . x8 ∈ SNoR x1 ⟶ x5 = add_SNo (mul_SNo x7 x1) (add_SNo (mul_SNo x0 x8) (minus_SNo (mul_SNo x7 x8))) ⟶ x6) ⟶ x6) ⟶ (∀ x5 . x5 ∈ SNoL x0 ⟶ ∀ x6 . x6 ∈ SNoL x1 ⟶ add_SNo (mul_SNo x5 x1) (add_SNo (mul_SNo x0 x6) (minus_SNo (mul_SNo x5 x6))) ∈ x3) ⟶ (∀ x5 . x5 ∈ SNoR x0 ⟶ ∀ x6 . x6 ∈ SNoR x1 ⟶ add_SNo (mul_SNo x5 x1) (add_SNo (mul_SNo x0 x6) (minus_SNo (mul_SNo x5 x6))) ∈ x3) ⟶ (∀ x5 . x5 ∈ x4 ⟶ ∀ x6 : ο . (∀ x7 . x7 ∈ SNoL x0 ⟶ ∀ x8 . x8 ∈ SNoR x1 ⟶ x5 = add_SNo (mul_SNo x7 x1) (add_SNo (mul_SNo x0 x8) (minus_SNo (mul_SNo x7 x8))) ⟶ x6) ⟶ (∀ x7 . x7 ∈ SNoR x0 ⟶ ∀ x8 . x8 ∈ SNoL x1 ⟶ x5 = add_SNo (mul_SNo x7 x1) (add_SNo (mul_SNo x0 x8) (minus_SNo (mul_SNo x7 x8))) ⟶ x6) ⟶ x6) ⟶ (∀ x5 . x5 ∈ SNoL x0 ⟶ ∀ x6 . x6 ∈ SNoR x1 ⟶ add_SNo (mul_SNo x5 x1) (add_SNo (mul_SNo x0 x6) (minus_SNo (mul_SNo x5 x6))) ∈ x4) ⟶ (∀ x5 . x5 ∈ SNoR x0 ⟶ ∀ x6 . x6 ∈ SNoL x1 ⟶ add_SNo (mul_SNo x5 x1) (add_SNo (mul_SNo x0 x6) (minus_SNo (mul_SNo x5 x6))) ∈ x4) ⟶ mul_SNo x0 x1 = SNoCut x3 x4 ⟶ x2) ⟶ x2 (proof)
Theorem mul_SNo_Subq_lem^{mul_SNo_Subq_lem} : ∀ x0 x1 x2 x3 x4 x5 x6 x7 . (∀ x8 . x8 ∈ x6 ⟶ ∀ x9 : ο . (∀ x10 . x10 ∈ x2 ⟶ ∀ x11 . x11 ∈ x3 ⟶ x8 = add_SNo (mul_SNo x10 x1) (add_SNo (mul_SNo x0 x11) (minus_SNo (mul_SNo x10 x11))) ⟶ x9) ⟶ (∀ x10 . x10 ∈ x4 ⟶ ∀ x11 . x11 ∈ x5 ⟶ x8 = add_SNo (mul_SNo x10 x1) (add_SNo (mul_SNo x0 x11) (minus_SNo (mul_SNo x10 x11))) ⟶ x9) ⟶ x9) ⟶ (∀ x8 . x8 ∈ x2 ⟶ ∀ x9 . x9 ∈ x3 ⟶ add_SNo (mul_SNo x8 x1) (add_SNo (mul_SNo x0 x9) (minus_SNo (mul_SNo x8 x9))) ∈ x7) ⟶ (∀ x8 . x8 ∈ x4 ⟶ ∀ x9 . x9 ∈ x5 ⟶ add_SNo (mul_SNo x8 x1) (add_SNo (mul_SNo x0 x9) (minus_SNo (mul_SNo x8 x9))) ∈ x7) ⟶ x6 ⊆ x7 (proof)
Known SNoR_0^SNoR_0 : SNoR 0 = 0
Known SNoL_0^SNoL_0 : SNoL 0 = 0
Theorem mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0 (proof)
Param SNoS_^SNoS_ : ι → ι
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_eta^SNo_eta : ∀ x0 . SNo x0 ⟶ x0 = SNoCut (SNoL x0) (SNoR x0)
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
Known SNoCutP_SNoL_SNoR^{SNoCutP_SNoL_SNoR} : ∀ x0 . SNo x0 ⟶ SNoCutP (SNoL x0) (SNoR x0)
Known SNoCutP_SNoCut_L^{SNoCutP_SNoCut_L} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ ∀ x2 . x2 ∈ x0 ⟶ SNoLt x2 (SNoCut x0 x1)
Known minus_SNo_0^minus_SNo_0 : minus_SNo 0 = 0
Known SNoL_SNoS^SNoL_SNoS : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoL x0 ⟶ x1 ∈ SNoS_ (SNoLev x0)
Known SNoCutP_SNoCut_R^{SNoCutP_SNoCut_R} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ ∀ x2 . x2 ∈ x1 ⟶ SNoLt (SNoCut x0 x1) x2
Known SNoR_SNoS^SNoR_SNoS : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoR x0 ⟶ x1 ∈ SNoS_ (SNoLev x0)
Theorem mul_SNo_oneR^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = x0 (proof)
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
Known add_SNo_com_3_0_1^{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)
Theorem mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0 (proof)
Known ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Known minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
Known SNoR_I^SNoR_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x0 x1 ⟶ x1 ∈ SNoR x0
Known minus_SNo_Lev^{minus_SNo_Lev} : ∀ x0 . SNo x0 ⟶ SNoLev (minus_SNo x0) = SNoLev x0
Known minus_SNo_Lt_contra2^{minus_SNo_Lt_contra2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 (minus_SNo x1) ⟶ SNoLt x1 (minus_SNo x0)
Known minus_SNo_Lt_contra1^{minus_SNo_Lt_contra1} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt (minus_SNo x0) x1 ⟶ SNoLt (minus_SNo x1) x0
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 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_SNoCut_eq^{minus_SNoCut_eq} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ minus_SNo (SNoCut x0 x1) = SNoCut (prim5 x1 minus_SNo) (prim5 x0 minus_SNo)
Theorem mul_SNo_minus_distrL^{mul_SNo_minus_distrL} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo (minus_SNo x0) x1 = minus_SNo (mul_SNo x0 x1) (proof)