Proofgold Address

Param ordinal^ordinal : ι → ο
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known ordinal_trichotomy_or^{ordinal_trichotomy_or} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ or (or (x0 ∈ x1) (x0 = x1)) (x1 ∈ x0)
Theorem ordinal_trichotomy_or_impred^{ordinal_trichotomy_or_impred} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ ∀ x2 : ο . (x0 ∈ x1 ⟶ x2) ⟶ (x0 = x1 ⟶ x2) ⟶ (x1 ∈ x0 ⟶ x2) ⟶ x2 (proof)
Param SNo^SNo : ι → ο
Param SNoLt^SNoLt : ι → ι → ο
Param SNoL^SNoL : ι → ι
Param SNoR^SNoR : ι → ι
Param SNoLev^SNoLev : ι → ι
Param binintersect^binintersect : ι → ι → ι
Param SNoEq_^SNoEq_ : ι → ι → ι → ο
Param nIn^nIn : ι → ι → ο
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
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known binintersectE^{binintersectE} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ and (x2 ∈ x0) (x2 ∈ x1)
Known SNoL_I^SNoL_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x1 x0 ⟶ x1 ∈ SNoL x0
Known SNoR_I^SNoR_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x0 x1 ⟶ x1 ∈ SNoR x0
Theorem SNoLt_SNoL_or_SNoR_impred^{SNoLt_SNoL_or_SNoR_impred} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 x1 ⟶ ∀ x2 : ο . (∀ x3 . x3 ∈ SNoL x1 ⟶ x3 ∈ SNoR x0 ⟶ x2) ⟶ (x0 ∈ SNoL x1 ⟶ x2) ⟶ (x1 ∈ SNoR x0 ⟶ x2) ⟶ x2 (proof)
Known 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
Theorem SNoL_or_SNoR_impred^{SNoL_or_SNoR_impred} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 : ο . (x0 = x1 ⟶ x2) ⟶ (∀ x3 . x3 ∈ SNoL x1 ⟶ x3 ∈ SNoR x0 ⟶ x2) ⟶ (x0 ∈ SNoL x1 ⟶ x2) ⟶ (x1 ∈ SNoR x0 ⟶ x2) ⟶ (∀ x3 . x3 ∈ SNoR x1 ⟶ x3 ∈ SNoL x0 ⟶ x2) ⟶ (x0 ∈ SNoR x1 ⟶ x2) ⟶ (x1 ∈ SNoL x0 ⟶ x2) ⟶ x2 (proof)
Param add_SNo^add_SNo : ι → ι → ι
Param minus_SNo^minus_SNo : ι → ι
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 SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Known add_SNo_minus_SNo_rinv^{add_SNo_minus_SNo_rinv} : ∀ x0 . SNo x0 ⟶ add_SNo x0 (minus_SNo x0) = 0
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
Theorem add_SNo_minus_R2^{add_SNo_minus_R2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo (add_SNo x0 x1) (minus_SNo x1) = x0 (proof)
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)
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Theorem 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 (proof)
Theorem add_SNo_assoc_4^{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 (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_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) (proof)
Theorem add_SNo_com_4_inner_flat^{add_SNo_com_4_inner_flat} : ∀ x0 x1 x2 x3 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ add_SNo x0 (add_SNo x1 (add_SNo x2 x3)) = add_SNo x0 (add_SNo x2 (add_SNo x1 x3)) (proof)
Theorem add_SNo_com_3b_1_2^{add_SNo_com_3b_1_2} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo (add_SNo x0 x1) x2 = add_SNo (add_SNo x0 x2) x1 (proof)
Theorem add_SNo_com_4_inner_mid^{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) (proof)
Theorem 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) (proof)
Known add_SNo_minus_SNo_linv^{add_SNo_minus_SNo_linv} : ∀ x0 . SNo x0 ⟶ add_SNo (minus_SNo x0) x0 = 0
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Theorem add_SNo_minus_L2^{add_SNo_minus_L2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo (minus_SNo x0) (add_SNo x0 x1) = x1 (proof)
Theorem add_SNo_minus_SNo_prop2^{add_SNo_minus_SNo_prop2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 (add_SNo (minus_SNo x0) x1) = x1 (proof)
Theorem add_SNo_minus_SNo_prop3^{add_SNo_minus_SNo_prop3} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ add_SNo (add_SNo x0 (add_SNo x1 x2)) (add_SNo (minus_SNo x2) x3) = add_SNo x0 (add_SNo x1 x3) (proof)
Theorem add_SNo_minus_SNo_prop4^{add_SNo_minus_SNo_prop4} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ add_SNo (add_SNo x0 (add_SNo x1 x2)) (add_SNo x3 (minus_SNo x2)) = add_SNo x0 (add_SNo x1 x3) (proof)
Known minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
Theorem add_SNo_minus_SNo_prop5^{add_SNo_minus_SNo_prop5} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ add_SNo (add_SNo x0 (add_SNo x1 (minus_SNo x2))) (add_SNo x2 x3) = add_SNo x0 (add_SNo x1 x3) (proof)
Theorem add_SNo_minus_Lt1^{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) (proof)
Theorem add_SNo_minus_Lt2^{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 (proof)
Known 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)
Theorem add_SNo_Lt_subprop2^{add_SNo_Lt_subprop2} : ∀ x0 x1 x2 x3 x4 x5 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ SNo x5 ⟶ SNoLt (add_SNo x0 x4) (add_SNo x2 x5) ⟶ SNoLt (add_SNo x1 x5) (add_SNo x3 x4) ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x3) (proof)
Known SNoLt_tra^SNoLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known 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)
Theorem add_SNo_Lt_subprop3a^{add_SNo_Lt_subprop3a} : ∀ x0 x1 x2 x3 x4 x5 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ SNo x5 ⟶ SNoLt (add_SNo x0 x2) (add_SNo x3 x5) ⟶ SNoLt (add_SNo x1 x5) x4 ⟶ SNoLt (add_SNo x0 (add_SNo x1 x2)) (add_SNo x3 x4) (proof)
Known SNo_0^SNo_0 : SNo 0
Theorem add_SNo_Lt_subprop3b^{add_SNo_Lt_subprop3b} : ∀ x0 x1 x2 x3 x4 x5 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ SNo x5 ⟶ SNoLt (add_SNo x0 x5) (add_SNo x2 x4) ⟶ SNoLt x1 (add_SNo x5 x3) ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 (add_SNo x3 x4)) (proof)
Theorem add_SNo_Lt_subprop3c^{add_SNo_Lt_subprop3c} : ∀ x0 x1 x2 x3 x4 x5 x6 x7 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ SNo x5 ⟶ SNo x6 ⟶ SNo x7 ⟶ SNoLt (add_SNo x0 x5) (add_SNo x6 x7) ⟶ SNoLt (add_SNo x1 x7) x4 ⟶ SNoLt (add_SNo x6 x2) (add_SNo x3 x5) ⟶ SNoLt (add_SNo x0 (add_SNo x1 x2)) (add_SNo x3 x4) (proof)
Theorem add_SNo_Lt_subprop3d^{add_SNo_Lt_subprop3d} : ∀ x0 x1 x2 x3 x4 x5 x6 x7 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ SNo x5 ⟶ SNo x6 ⟶ SNo x7 ⟶ SNoLt (add_SNo x0 x5) (add_SNo x6 x4) ⟶ SNoLt x1 (add_SNo x7 x3) ⟶ SNoLt (add_SNo x6 x7) (add_SNo x2 x5) ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 (add_SNo x3 x4)) (proof)
Param mul_SNo^mul_SNo : ι → ι → ι
Param SNoCut^SNoCut : ι → ι → ι
Param binunion^binunion : ι → ι → ι
Param lam^Sigma : ι → (ι → ι) → ι
Definition setprod^setprod := λ x0 x1 . lam x0 (λ x2 . x1)
Param ap^ap : ι → ι → ι
Param ordsucc^ordsucc : ι → ι
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
Known ap0_Sigma^ap0_Sigma : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ ap x2 0 ∈ x0
Known ap1_Sigma^ap1_Sigma : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ ap x2 1 ∈ x1 (ap x2 0)
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Param If_i^If_i : ο → ι → ι → ι
Known tuple_2_0_eq^tuple_2_0_eq : ∀ x0 x1 . ap (lam 2 (λ x3 . If_i (x3 = 0) x0 x1)) 0 = x0
Known tuple_2_1_eq^tuple_2_1_eq : ∀ x0 x1 . ap (lam 2 (λ x3 . If_i (x3 = 0) x0 x1)) 1 = x1
Known ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Known tuple_2_setprod^{tuple_2_setprod} : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x1 ⟶ lam 2 (λ x4 . If_i (x4 = 0) x2 x3) ∈ setprod x0 x1
Known binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
Known 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)})
Theorem 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 (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
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 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 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 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
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)
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Definition TransSet^TransSet := λ x0 . ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Known ordinal_TransSet^{ordinal_TransSet} : ∀ x0 . ordinal x0 ⟶ TransSet x0
Known SNoLev_ordinal^{SNoLev_ordinal} : ∀ x0 . SNo x0 ⟶ ordinal (SNoLev x0)
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
Theorem 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 (proof)