Proofgold Signed Transaction

Param nat_p^nat_p : ι → ο
Param SNo^SNo : ι → ο
Param SNoLev^SNoLev : ι → ι
Param diadic_rational_alt1_p : ι → ο
Known nat_complete_ind^{nat_complete_ind} : ∀ x0 : ι → ο . (∀ x1 . nat_p x1 ⟶ (∀ x2 . x2 ∈ x1 ⟶ x0 x2) ⟶ x0 x1) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Param SNoS_^SNoS_ : ι → ι
Param omega^omega : ι
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param SNoLe^SNoLe : ι → ι → ο
Definition SNo_max_of^SNo_max_of := λ x0 x1 . and (and (x1 ∈ x0) (SNo x1)) (∀ x2 . x2 ∈ x0 ⟶ SNo x2 ⟶ SNoLe x2 x1)
Param SNoL^SNoL : ι → ι
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known SNoS_omega_SNoL_max_exists^{SNoS_omega_SNoL_max_exists} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ or (SNoL x0 = 0) (∀ x1 : ο . (∀ x2 . SNo_max_of (SNoL x0) x2 ⟶ x1) ⟶ x1)
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Param minus_SNo^minus_SNo : ι → ι
Known minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
Known 3801d.. : ∀ x0 . diadic_rational_alt1_p x0 ⟶ diadic_rational_alt1_p (minus_SNo x0)
Known minus_SNo_Lev^{minus_SNo_Lev} : ∀ x0 . SNo x0 ⟶ SNoLev (minus_SNo x0) = SNoLev x0
Param ordinal^ordinal : ι → ο
Known ordinal_SNoLev^{ordinal_SNoLev} : ∀ x0 . ordinal x0 ⟶ SNoLev x0 = x0
Param SNoLt^SNoLt : ι → ι → ο
Known SNo_max_ordinal^{SNo_max_ordinal} : ∀ x0 . SNo x0 ⟶ (∀ x1 . x1 ∈ SNoS_ (SNoLev x0) ⟶ SNoLt x1 x0) ⟶ ordinal x0
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
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 SNoLev_ordinal^{SNoLev_ordinal} : ∀ x0 . SNo x0 ⟶ ordinal (SNoLev x0)
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
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Known SNoL_I^SNoL_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x1 x0 ⟶ x1 ∈ SNoL 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 a89df.. : ∀ x0 . x0 ∈ omega ⟶ diadic_rational_alt1_p x0
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
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
Definition SNo_min_of^SNo_min_of := λ x0 x1 . and (and (x1 ∈ x0) (SNo x1)) (∀ x2 . x2 ∈ x0 ⟶ SNo x2 ⟶ SNoLe x1 x2)
Param SNoR^SNoR : ι → ι
Known SNoS_omega_SNoR_min_exists^{SNoS_omega_SNoR_min_exists} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ or (SNoR x0 = 0) (∀ x1 : ο . (∀ x2 . SNo_min_of (SNoR x0) x2 ⟶ x1) ⟶ x1)
Known SNoR_I^SNoR_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x0 x1 ⟶ x1 ∈ SNoR x0
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
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)
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Known double_SNo_min_1^{double_SNo_min_1} : ∀ x0 x1 . SNo x0 ⟶ SNo_min_of (SNoR x0) x1 ⟶ ∀ x2 . SNo x2 ⟶ SNoLt x2 x0 ⟶ SNoLt (add_SNo x0 x0) (add_SNo x1 x2) ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ SNoL x2) (add_SNo x1 x4 = add_SNo x0 x0) ⟶ x3) ⟶ x3
Param mul_SNo^mul_SNo : ι → ι → ι
Param eps_^eps_ : ι → ι
Param ordsucc^ordsucc : ι → ι
Known double_eps_1^double_eps_1 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 x0 = add_SNo x1 x2 ⟶ x0 = mul_SNo (eps_ 1) (add_SNo x1 x2)
Known 858c0.. : ∀ x0 x1 . diadic_rational_alt1_p x0 ⟶ diadic_rational_alt1_p x1 ⟶ diadic_rational_alt1_p (mul_SNo x0 x1)
Known 5042a.. : ∀ x0 . x0 ∈ omega ⟶ diadic_rational_alt1_p (eps_ x0)
Known nat_1^nat_1 : nat_p 1
Known 9e46f.. : ∀ x0 x1 . diadic_rational_alt1_p x0 ⟶ diadic_rational_alt1_p x1 ⟶ diadic_rational_alt1_p (add_SNo x0 x1)
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 nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known double_SNo_max_1^{double_SNo_max_1} : ∀ x0 x1 . SNo x0 ⟶ SNo_max_of (SNoL x0) x1 ⟶ ∀ x2 . SNo x2 ⟶ SNoLt x0 x2 ⟶ SNoLt (add_SNo x1 x2) (add_SNo x0 x0) ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ SNoR x2) (add_SNo x1 x4 = add_SNo x0 x0) ⟶ x3) ⟶ x3
Known SNoS_I^SNoS_I : ∀ x0 . ordinal x0 ⟶ ∀ x1 x2 . x2 ∈ x0 ⟶ SNo_ x2 x1 ⟶ x1 ∈ SNoS_ x0
Known omega_ordinal^{omega_ordinal} : ordinal omega
Known SNoLev_^SNoLev_ : ∀ x0 . SNo x0 ⟶ SNo_ (SNoLev x0) x0
Theorem b9dc3.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 = x0 ⟶ diadic_rational_alt1_p x1 (proof)
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Theorem 5c296.. : ∀ x0 . x0 ∈ SNoS_ omega ⟶ diadic_rational_alt1_p x0 (proof)
Known f05cb.. : ∀ x0 . diadic_rational_alt1_p x0 ⟶ x0 ∈ SNoS_ omega
Theorem mul_SNo_SNoS_omega^{mul_SNo_SNoS_omega} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ ∀ x1 . x1 ∈ SNoS_ omega ⟶ mul_SNo x0 x1 ∈ SNoS_ omega (proof)
Known SNo_foil^SNo_foil : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ mul_SNo (add_SNo x0 x1) (add_SNo x2 x3) = add_SNo (mul_SNo x0 x2) (add_SNo (mul_SNo x0 x3) (add_SNo (mul_SNo x1 x2) (mul_SNo x1 x3)))
Known mul_SNo_minus_minus^{mul_SNo_minus_minus} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo (minus_SNo x0) (minus_SNo x1) = mul_SNo x0 x1
Known 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)
Known mul_SNo_minus_distrR^{mul_SNo_minus_distrR} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 (minus_SNo x1) = minus_SNo (mul_SNo x0 x1)
Theorem SNo_foil_mm^SNo_foil_mm : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ mul_SNo (add_SNo x0 (minus_SNo x1)) (add_SNo x2 (minus_SNo x3)) = add_SNo (mul_SNo x0 x2) (add_SNo (minus_SNo (mul_SNo x0 x3)) (add_SNo (minus_SNo (mul_SNo x1 x2)) (mul_SNo x1 x3))) (proof)
Known 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))
Known 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)
Theorem add_SNo_3a_2b^{add_SNo_3a_2b} : ∀ x0 x1 x2 x3 x4 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ add_SNo (add_SNo x0 (add_SNo x1 x2)) (add_SNo x3 x4) = add_SNo (add_SNo x4 (add_SNo x1 x2)) (add_SNo x3 x0) (proof)
Param real^real : ι
Param abs_SNo^abs_SNo : ι → ι
Known real_E^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 : ο . (∀ x4 . and (x4 ∈ SNoS_ omega) (and (SNoLt x4 x0) (SNoLt x0 (add_SNo x4 (eps_ x2)))) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1
Param SNoCutP^SNoCutP : ι → ι → ο
Param SNoCut^SNoCut : ι → ι → ι
Known 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
Known SNoS_ordsucc_omega_bdd_eps_pos^{SNoS_ordsucc_omega_bdd_eps_pos} : ∀ x0 . x0 ∈ SNoS_ (ordsucc omega) ⟶ SNoLt 0 x0 ⟶ SNoLt x0 omega ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (SNoLt (mul_SNo (eps_ x2) x0) 1) ⟶ x1) ⟶ x1
Known 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
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 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 SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Known SNo_eps_^SNo_eps_ : ∀ x0 . x0 ∈ omega ⟶ SNo (eps_ x0)
Known SNo_1^SNo_1 : SNo 1
Known pos_mul_SNo_Lt'^{pos_mul_SNo_Lt} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt 0 x2 ⟶ SNoLt x0 x1 ⟶ SNoLt (mul_SNo x0 x2) (mul_SNo x1 x2)
Known SNo_eps_decr^SNo_eps_decr : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ x0 ⟶ SNoLt (eps_ x0) (eps_ x1)
Param Pi^Pi : ι → (ι → ι) → ι
Definition setexp^setexp := λ x0 x1 . Pi x1 (λ x2 . x0)
Param ap^ap : ι → ι → ι
Known SNo_prereal_incr_lower_approx^{SNo_prereal_incr_lower_approx} : ∀ x0 . SNo x0 ⟶ (∀ x1 . x1 ∈ SNoS_ omega ⟶ (∀ x2 . x2 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x1 (minus_SNo x0))) (eps_ x2)) ⟶ x1 = x0) ⟶ (∀ x1 . x1 ∈ omega ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ SNoS_ omega) (and (SNoLt x3 x0) (SNoLt x0 (add_SNo x3 (eps_ x1)))) ⟶ x2) ⟶ x2) ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ setexp (SNoS_ omega) omega) (∀ x3 . x3 ∈ omega ⟶ and (and (SNoLt (ap x2 x3) x0) (SNoLt x0 (add_SNo (ap x2 x3) (eps_ x3)))) (∀ x4 . x4 ∈ x3 ⟶ SNoLt (ap x2 x4) (ap x2 x3))) ⟶ x1) ⟶ x1
Known SNo_prereal_decr_upper_approx^{SNo_prereal_decr_upper_approx} : ∀ x0 . SNo x0 ⟶ (∀ x1 . x1 ∈ SNoS_ omega ⟶ (∀ x2 . x2 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x1 (minus_SNo x0))) (eps_ x2)) ⟶ x1 = x0) ⟶ (∀ x1 . x1 ∈ omega ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ SNoS_ omega) (and (SNoLt x3 x0) (SNoLt x0 (add_SNo x3 (eps_ x1)))) ⟶ x2) ⟶ x2) ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ setexp (SNoS_ omega) omega) (∀ x3 . x3 ∈ omega ⟶ and (and (SNoLt (add_SNo (ap x2 x3) (minus_SNo (eps_ x3))) x0) (SNoLt x0 (ap x2 x3))) (∀ x4 . x4 ∈ x3 ⟶ SNoLt (ap x2 x3) (ap x2 x4))) ⟶ x1) ⟶ x1
Known SNo_approx_real_lem^{SNo_approx_real_lem} : ∀ x0 . x0 ∈ setexp (SNoS_ omega) omega ⟶ ∀ x1 . x1 ∈ setexp (SNoS_ omega) omega ⟶ (∀ x2 . x2 ∈ omega ⟶ ∀ x3 . x3 ∈ omega ⟶ SNoLt (ap x0 x2) (ap x1 x3)) ⟶ ∀ x2 : ο . (SNoCutP (prim5 omega (ap x0)) (prim5 omega (ap x1)) ⟶ SNo (SNoCut (prim5 omega (ap x0)) (prim5 omega (ap x1))) ⟶ SNoLev (SNoCut (prim5 omega (ap x0)) (prim5 omega (ap x1))) ∈ ordsucc omega ⟶ SNoCut (prim5 omega (ap x0)) (prim5 omega (ap x1)) ∈ SNoS_ (ordsucc omega) ⟶ (∀ x3 . x3 ∈ omega ⟶ SNoLt (ap x0 x3) (SNoCut (prim5 omega (ap x0)) (prim5 omega (ap x1)))) ⟶ (∀ x3 . x3 ∈ omega ⟶ SNoLt (SNoCut (prim5 omega (ap x0)) (prim5 omega (ap x1))) (ap x1 x3)) ⟶ x2) ⟶ x2
Known SNo_approx_real^{SNo_approx_real} : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ setexp (SNoS_ omega) omega ⟶ ∀ x2 . x2 ∈ setexp (SNoS_ omega) omega ⟶ (∀ x3 . x3 ∈ omega ⟶ SNoLt (ap x1 x3) x0) ⟶ (∀ x3 . x3 ∈ omega ⟶ SNoLt x0 (add_SNo (ap x1 x3) (eps_ x3))) ⟶ (∀ x3 . x3 ∈ omega ⟶ ∀ x4 . x4 ∈ x3 ⟶ SNoLt (ap x1 x4) (ap x1 x3)) ⟶ (∀ x3 . x3 ∈ omega ⟶ SNoLt x0 (ap x2 x3)) ⟶ (∀ x3 . x3 ∈ omega ⟶ ∀ x4 . x4 ∈ x3 ⟶ SNoLt (ap x2 x3) (ap x2 x4)) ⟶ x0 = SNoCut (prim5 omega (ap x1)) (prim5 omega (ap x2)) ⟶ x0 ∈ real
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 SNoLeLt_tra^SNoLeLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 x0)
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Known SNoLtLe_tra^SNoLtLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLt x0 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)
Known add_SNo_minus_Le2^{add_SNo_minus_Le2} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x2 (add_SNo x0 (minus_SNo x1)) ⟶ SNoLe (add_SNo x2 x1) x0
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)
Known 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))
Known mul_SNo_eps_eps_add_SNo^{mul_SNo_eps_eps_add_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_SNo (eps_ x0) (eps_ x1) = eps_ (add_SNo x0 x1)
Known nonneg_mul_SNo_Le2^{nonneg_mul_SNo_Le2} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLe 0 x0 ⟶ SNoLe 0 x1 ⟶ SNoLe x0 x2 ⟶ SNoLe x1 x3 ⟶ SNoLe (mul_SNo x0 x1) (mul_SNo x2 x3)
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Known SNo_eps_pos^SNo_eps_pos : ∀ x0 . x0 ∈ omega ⟶ SNoLt 0 (eps_ x0)
Known add_SNo_In_omega^{add_SNo_In_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ add_SNo x0 x1 ∈ omega
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)
Known 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
Known add_SNo_minus_Le2b^{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 add_SNo_rotate_4_1^{add_SNo_rotate_4_1} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ add_SNo x0 (add_SNo x1 (add_SNo x2 x3)) = add_SNo x3 (add_SNo x0 (add_SNo x1 x2))
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 ap_Pi^ap_Pi : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ Pi x0 x1 ⟶ x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3
Known SNo_prereal_incr_lower_pos^{SNo_prereal_incr_lower_pos} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ (∀ x1 . x1 ∈ SNoS_ omega ⟶ (∀ x2 . x2 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x1 (minus_SNo x0))) (eps_ x2)) ⟶ x1 = x0) ⟶ (∀ x1 . x1 ∈ omega ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ SNoS_ omega) (and (SNoLt x3 x0) (SNoLt x0 (add_SNo x3 (eps_ x1)))) ⟶ x2) ⟶ x2) ⟶ ∀ x1 . x1 ∈ omega ⟶ ∀ x2 : ο . (∀ x3 . x3 ∈ SNoS_ omega ⟶ SNoLt 0 x3 ⟶ SNoLt x3 x0 ⟶ SNoLt x0 (add_SNo x3 (eps_ x1)) ⟶ x2) ⟶ x2
Known pos_mul_SNo_Lt2^{pos_mul_SNo_Lt2} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLt 0 x0 ⟶ SNoLt 0 x1 ⟶ SNoLt x0 x2 ⟶ SNoLt x1 x3 ⟶ SNoLt (mul_SNo x0 x1) (mul_SNo x2 x3)
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)
Known eps_ordsucc_half_add^{eps_ordsucc_half_add} : ∀ x0 . nat_p x0 ⟶ add_SNo (eps_ (ordsucc x0)) (eps_ (ordsucc x0)) = eps_ x0
Known nat_ordsucc^nat_ordsucc : ∀ x0 . nat_p x0 ⟶ nat_p (ordsucc x0)
Known add_SNo_1_ordsucc^{add_SNo_1_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ add_SNo x0 1 = ordsucc x0
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 omega_SNo^omega_SNo : ∀ x0 . x0 ∈ omega ⟶ SNo x0
Known add_SNo_1_1_2^{add_SNo_1_1_2} : add_SNo 1 1 = 2
Known add_SNo_Lt4^add_SNo_Lt4 : ∀ x0 x1 x2 x3 x4 x5 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ SNo x5 ⟶ SNoLt x0 x3 ⟶ SNoLt x1 x4 ⟶ SNoLt x2 x5 ⟶ SNoLt (add_SNo x0 (add_SNo x1 x2)) (add_SNo x3 (add_SNo x4 x5))
Known mul_SNo_assoc^{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 mul_SNo_Lt1_pos_Lt^{mul_SNo_Lt1_pos_Lt} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 1 ⟶ SNoLt 0 x1 ⟶ SNoLt (mul_SNo x0 x1) x1
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
Known SNoLe_tra^SNoLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLe x0 x2
Known mul_SNo_Le1_nonneg_Le^{mul_SNo_Le1_nonneg_Le} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 1 ⟶ SNoLe 0 x1 ⟶ SNoLe (mul_SNo x0 x1) x1
Known mul_SNo_oneL^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 x0 = x0
Known eps_bd_1^eps_bd_1 : ∀ x0 . x0 ∈ omega ⟶ SNoLe (eps_ x0) 1
Known mul_SNo_pos_pos^{mul_SNo_pos_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt 0 x0 ⟶ SNoLt 0 x1 ⟶ SNoLt 0 (mul_SNo x0 x1)
Known ordinal_SNoLt_In^{ordinal_SNoLt_In} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ SNoLt x0 x1 ⟶ x0 ∈ x1
Known SNo_2^SNo_2 : SNo 2
Known SNoLt_1_2^SNoLt_1_2 : SNoLt 1 2
Known nat_2^nat_2 : nat_p 2
Known mul_SNo_SNoL_interpolate_impred^{mul_SNo_SNoL_interpolate_impred} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 . x2 ∈ SNoL (mul_SNo x0 x1) ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ SNoL x0 ⟶ ∀ x5 . x5 ∈ SNoL x1 ⟶ SNoLe (add_SNo x2 (mul_SNo x4 x5)) (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x5)) ⟶ x3) ⟶ (∀ x4 . x4 ∈ SNoR x0 ⟶ ∀ x5 . x5 ∈ SNoR x1 ⟶ SNoLe (add_SNo x2 (mul_SNo x4 x5)) (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x5)) ⟶ x3) ⟶ x3
Known SNo_abs_SNo^SNo_abs_SNo : ∀ x0 . SNo x0 ⟶ SNo (abs_SNo x0)
Known abs_SNo_dist_swap^{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))
Known pos_abs_SNo^pos_abs_SNo : ∀ x0 . SNoLt 0 x0 ⟶ abs_SNo x0 = x0
Known SNoLt_minus_pos^{SNoLt_minus_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 x1 ⟶ SNoLt 0 (add_SNo x1 (minus_SNo x0))
Known 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)
Known 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)
Known 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)
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
Known SNoLe_ref^SNoLe_ref : ∀ x0 . SNoLe x0 x0
Known mul_SNo_SNoR_interpolate_impred^{mul_SNo_SNoR_interpolate_impred} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 . x2 ∈ SNoR (mul_SNo x0 x1) ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ SNoL x0 ⟶ ∀ x5 . x5 ∈ SNoR x1 ⟶ SNoLe (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x5)) (add_SNo x2 (mul_SNo x4 x5)) ⟶ x3) ⟶ (∀ x4 . x4 ∈ SNoR x0 ⟶ ∀ x5 . x5 ∈ SNoL x1 ⟶ SNoLe (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x5)) (add_SNo x2 (mul_SNo x4 x5)) ⟶ x3) ⟶ x3
Known SNo_add_SNo_4^{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 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
Known add_SNo_minus_L2^{add_SNo_minus_L2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo (minus_SNo x0) (add_SNo x0 x1) = x1
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Known omega_TransSet^{omega_TransSet} : TransSet omega
Known nat_p_trans^nat_p_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ nat_p x1
Known SNoS_omega_real^{SNoS_omega_real} : SNoS_ omega ⊆ real
Theorem real_mul_SNo_pos^{real_mul_SNo_pos} : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ SNoLt 0 x0 ⟶ SNoLt 0 x1 ⟶ mul_SNo x0 x1 ∈ real (proof)
Known SNo_0^SNo_0 : SNo 0
Known real_minus_SNo^{real_minus_SNo} : ∀ x0 . x0 ∈ real ⟶ minus_SNo x0 ∈ real
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_0^minus_SNo_0 : minus_SNo 0 = 0
Known mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Known real_0^real_0 : 0 ∈ real
Known mul_SNo_zeroL^{mul_SNo_zeroL} : ∀ x0 . SNo x0 ⟶ mul_SNo 0 x0 = 0
Known real_SNo^real_SNo : ∀ x0 . x0 ∈ real ⟶ SNo x0
Theorem real_mul_SNo^real_mul_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ mul_SNo x0 x1 ∈ real (proof)
Known nonneg_abs_SNo^{nonneg_abs_SNo} : ∀ x0 . SNoLe 0 x0 ⟶ abs_SNo x0 = 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_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
Theorem abs_SNo_intvl_bd^{abs_SNo_intvl_bd} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt x1 (add_SNo x0 x2) ⟶ SNoLt (abs_SNo (add_SNo x1 (minus_SNo x0))) x2 (proof)
Known real_SNoS_omega_prop^{real_SNoS_omega_prop} : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ SNoS_ omega ⟶ (∀ x2 . x2 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x1 (minus_SNo x0))) (eps_ x2)) ⟶ x1 = x0
Theorem 73c92.. : ∀ x0 . x0 ∈ SNoS_ omega ⟶ (∀ x1 . x1 ∈ omega ⟶ SNoLt (abs_SNo x0) (eps_ x1)) ⟶ x0 = 0 (proof)
Known real_1^real_1 : 1 ∈ real
Theorem 569c3.. : ∀ x0 . x0 ∈ SNoS_ omega ⟶ (∀ x1 . x1 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x0 (minus_SNo 1))) (eps_ x1)) ⟶ x0 = 1 (proof)