Proofgold Address

Param real^real : ι
Param minus_SNo^minus_SNo : ι → ι
Param SNo^SNo : ι → ο
Param SNoLev^SNoLev : ι → ι
Param ordsucc^ordsucc : ι → ι
Param omega^omega : ι
Param SNoS_^SNoS_ : ι → ι
Param SNoLt^SNoLt : ι → ι → ο
Param abs_SNo^abs_SNo : ι → ι
Param add_SNo^add_SNo : ι → ι → ι
Param eps_^eps_ : ι → ι
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
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
Known real_I^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
Param ordinal^ordinal : ι → ο
Known minus_SNo_SNoS_^{minus_SNo_SNoS_} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ SNoS_ x0 ⟶ minus_SNo x1 ∈ SNoS_ x0
Known ordsucc_omega_ordinal^{ordsucc_omega_ordinal} : ordinal (ordsucc omega)
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Known minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
Known SNo_omega^SNo_omega : SNo omega
Known minus_SNo_prereal_1^{minus_SNo_prereal_1} : ∀ 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 ∈ SNoS_ omega ⟶ (∀ x2 . x2 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x1 (minus_SNo (minus_SNo x0)))) (eps_ x2)) ⟶ x1 = minus_SNo x0
Theorem real_minus_SNo^{real_minus_SNo} : ∀ x0 . x0 ∈ real ⟶ minus_SNo x0 ∈ real (proof)
Param mul_SNo^mul_SNo : ι → ι → ι
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 SNoS_ordsucc_omega_bdd_above^{SNoS_ordsucc_omega_bdd_above} : ∀ x0 . x0 ∈ SNoS_ (ordsucc omega) ⟶ SNoLt x0 omega ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (SNoLt x0 x2) ⟶ x1) ⟶ x1
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 ⟶ SNoLt 0 x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x1 x2 ⟶ SNoLt (mul_SNo x0 x1) (mul_SNo x0 x2)
Known SNo_eps_pos^SNo_eps_pos : ∀ x0 . x0 ∈ omega ⟶ SNoLt 0 (eps_ x0)
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Param SNoLe^SNoLe : ι → ι → ο
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 x0)
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Param exp_SNo_nat^exp_SNo_nat : ι → ι → ι
Known SNoLeLt_tra^SNoLeLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known mul_SNo_oneL^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 x0 = x0
Param nat_p^nat_p : ι → ο
Known mul_SNo_eps_power_2'^{mul_SNo_eps_power_2} : ∀ x0 . nat_p x0 ⟶ mul_SNo (exp_SNo_nat 2 x0) (eps_ x0) = 1
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
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 exp_SNo_2_bd^exp_SNo_2_bd : ∀ x0 . nat_p x0 ⟶ SNoLt x0 (exp_SNo_nat 2 x0)
Known mul_SNo_oneR^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = x0
Known nonneg_mul_SNo_Le^{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 SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Known exp_SNo_nat_pos^{exp_SNo_nat_pos} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ ∀ x1 . nat_p x1 ⟶ SNoLt 0 (exp_SNo_nat x0 x1)
Known SNo_2^SNo_2 : SNo 2
Known SNoLt_0_2^SNoLt_0_2 : SNoLt 0 2
Known SNo_exp_SNo_nat^{SNo_exp_SNo_nat} : ∀ x0 . SNo x0 ⟶ ∀ x1 . nat_p x1 ⟶ SNo (exp_SNo_nat x0 x1)
Known omega_SNo^omega_SNo : ∀ x0 . x0 ∈ omega ⟶ SNo x0
Theorem 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 (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
Known SNo_0^SNo_0 : SNo 0
Known pos_abs_SNo^pos_abs_SNo : ∀ x0 . SNoLt 0 x0 ⟶ abs_SNo x0 = x0
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 nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known nat_0^nat_0 : nat_p 0
Known mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Known abs_SNo_0^abs_SNo_0 : abs_SNo 0 = 0
Known SNoLt_0_1^SNoLt_0_1 : SNoLt 0 1
Known mul_SNo_minus_distrR^{mul_minus_SNo_distrR} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 (minus_SNo x1) = minus_SNo (mul_SNo x0 x1)
Known neg_abs_SNo^neg_abs_SNo : ∀ x0 . SNo x0 ⟶ SNoLt x0 0 ⟶ abs_SNo x0 = minus_SNo x0
Known mul_SNo_pos_neg^{mul_SNo_pos_neg} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt 0 x0 ⟶ SNoLt x1 0 ⟶ SNoLt (mul_SNo x0 x1) 0
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 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
Theorem a3f00.. : ∀ x0 . x0 ∈ SNoS_ (ordsucc omega) ⟶ SNoLt (minus_SNo omega) x0 ⟶ SNoLt x0 omega ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (SNoLt (abs_SNo (mul_SNo (eps_ x2) x0)) 1) ⟶ x1) ⟶ x1 (proof)
Param mul_nat^mul_nat : ι → ι → ι
Known mul_nat_mul_SNo^{mul_nat_mul_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_nat x0 x1 = mul_SNo x0 x1
Known mul_nat_p^mul_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (mul_nat x0 x1)
Known SNoLe_tra^SNoLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLe x0 x2
Known ordinal_SNoLev_max_2^{ordinal_SNoLev_max_2} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ ordsucc x0 ⟶ SNoLe x1 x0
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known SNoLev_0^SNoLev_0 : SNoLev 0 = 0
Known nat_0_in_ordsucc^{nat_0_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ 0 ∈ ordsucc x0
Known nat_exp_SNo_nat^{nat_exp_SNo_nat} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (exp_SNo_nat x0 x1)
Known nat_2^nat_2 : nat_p 2
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known omega_SNoS_omega^{omega_SNoS_omega} : omega ⊆ SNoS_ omega
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Known abs_SNo_minus^{abs_SNo_minus} : ∀ x0 . SNo x0 ⟶ abs_SNo (minus_SNo x0) = abs_SNo x0
Theorem real_Archimedean^{real_Archimedean} : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ SNoLt 0 x0 ⟶ SNoLe 0 x1 ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ omega) (SNoLe x1 (mul_SNo x3 x0)) ⟶ x2) ⟶ x2 (proof)