Proofgold Address

Param omega^omega : ι
Param mul_SNo^mul_SNo : ι → ι → ι
Param eps_^eps_ : ι → ι
Param add_SNo^add_SNo : ι → ι → ι
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)
Param SNo^SNo : ι → ο
Known mul_SNo_com_4_inner_mid^{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)
Param nat_p^nat_p : ι → ο
Param exp_SNo_nat^exp_SNo_nat : ι → ι → ι
Known nat_exp_SNo_nat^{nat_exp_SNo_nat} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (exp_SNo_nat x0 x1)
Param ordsucc^ordsucc : ι → ι
Known mul_SNo_eps_power_2^{mul_SNo_eps_power_2} : ∀ x0 . nat_p x0 ⟶ mul_SNo (eps_ x0) (exp_SNo_nat 2 x0) = 1
Param binunion^binunion : ι → ι → ι
Param minus_SNo^minus_SNo : ι → ι
Definition int^int := binunion omega (prim5 omega minus_SNo)
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
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
Theorem int_SNo_cases^{int_SNo_cases} : ∀ x0 : ι → ο . (∀ x1 . x1 ∈ omega ⟶ x0 x1) ⟶ (∀ x1 . x1 ∈ omega ⟶ x0 (minus_SNo x1)) ⟶ ∀ x1 . x1 ∈ int ⟶ x0 x1 (proof)
Known omega_SNo^omega_SNo : ∀ x0 . x0 ∈ omega ⟶ SNo x0
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Theorem int_SNo^int_SNo : ∀ x0 . x0 ∈ int ⟶ SNo x0 (proof)
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Theorem Subq_omega_int^{Subq_omega_int} : omega ⊆ int (proof)
Known binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
Known ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Theorem int_minus_SNo_omega^{int_minus_SNo_omega} : ∀ x0 . x0 ∈ omega ⟶ minus_SNo x0 ∈ int (proof)
Known nat_ind^nat_ind : ∀ x0 : ι → ο . x0 0 ⟶ (∀ x1 . nat_p x1 ⟶ x0 x1 ⟶ x0 (ordsucc x1)) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
Param ordinal^ordinal : ι → ο
Known ordinal_SNo^ordinal_SNo : ∀ x0 . ordinal x0 ⟶ SNo x0
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Known ordinal_ordsucc_SNo_eq^{ordinal_ordsucc_SNo_eq} : ∀ x0 . ordinal x0 ⟶ ordsucc x0 = add_SNo 1 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 SNo_1^SNo_1 : SNo 1
Known omega_ordsucc^{omega_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ ordsucc x0 ∈ omega
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known nat_inv^nat_inv : ∀ x0 . nat_p x0 ⟶ or (x0 = 0) (∀ x1 : ο . (∀ x2 . and (nat_p x2) (x0 = ordsucc x2) ⟶ x1) ⟶ x1)
Known minus_SNo_0^minus_SNo_0 : minus_SNo 0 = 0
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known nat_1^nat_1 : nat_p 1
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)
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
Theorem int_add_SNo_lem^{int_add_SNo_lem} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . nat_p x1 ⟶ add_SNo (minus_SNo x0) x1 ∈ int (proof)
Known add_SNo_In_omega^{add_SNo_In_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ add_SNo x0 x1 ∈ omega
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Theorem int_add_SNo^int_add_SNo : ∀ x0 . x0 ∈ int ⟶ ∀ x1 . x1 ∈ int ⟶ add_SNo x0 x1 ∈ int (proof)
Known minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
Theorem int_minus_SNo^{int_minus_SNo} : ∀ x0 . x0 ∈ int ⟶ minus_SNo x0 ∈ int (proof)
Known mul_SNo_In_omega^{mul_SNo_In_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_SNo x0 x1 ∈ omega
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
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 SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Theorem int_mul_SNo^int_mul_SNo : ∀ x0 . x0 ∈ int ⟶ ∀ x1 . x1 ∈ int ⟶ mul_SNo x0 x1 ∈ int (proof)
Param SNoS_^SNoS_ : ι → ι
Known nonneg_diadic_rational_p_SNoS_omega^{nonneg_diadic_rational_p_SNoS_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . nat_p x1 ⟶ mul_SNo (eps_ x0) x1 ∈ SNoS_ omega
Definition diadic_rational_p^{diadic_rational_p} := λ x0 . ∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (∀ x3 : ο . (∀ x4 . and (x4 ∈ int) (x0 = mul_SNo (eps_ x2) x4) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1
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 SNo_eps_^SNo_eps_ : ∀ x0 . x0 ∈ omega ⟶ SNo (eps_ x0)
Known minus_SNo_SNoS_omega^{minus_SNo_SNoS_omega} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ minus_SNo x0 ∈ SNoS_ omega
Theorem diadic_rational_p_SNoS_omega^{diadic_rational_p_SNoS_omega} : ∀ x0 . diadic_rational_p x0 ⟶ x0 ∈ SNoS_ omega (proof)
Param SNoLev^SNoLev : 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 omega_ordinal^{omega_ordinal} : ordinal omega
Theorem 418cf.. : ∀ x0 . diadic_rational_p x0 ⟶ SNo x0 (proof)
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known nat_0^nat_0 : nat_p 0
Known eps_0_1^eps_0_1 : eps_ 0 = 1
Known mul_SNo_oneL^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 x0 = x0
Theorem int_diadic_rational_p^{int_diadic_rational_p} : ∀ x0 . x0 ∈ int ⟶ diadic_rational_p x0 (proof)
Theorem omega_diadic_rational_p^{omega_diadic_rational_p} : ∀ x0 . x0 ∈ omega ⟶ diadic_rational_p x0 (proof)
Known mul_SNo_oneR^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = x0
Theorem eps_diadic_rational_p^{eps_diadic_rational_p} : ∀ x0 . x0 ∈ omega ⟶ diadic_rational_p (eps_ x0) (proof)
Theorem minus_SNo_diadic_rational_p^{minus_SNo_diadic_rational_p} : ∀ x0 . diadic_rational_p x0 ⟶ diadic_rational_p (minus_SNo x0) (proof)
Theorem mul_SNo_diadic_rational_p^{mul_SNo_diadic_rational_p} : ∀ x0 x1 . diadic_rational_p x0 ⟶ diadic_rational_p x1 ⟶ diadic_rational_p (mul_SNo x0 x1) (proof)
Known mul_SNo_distrL^{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 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 nat_2^nat_2 : nat_p 2
Theorem add_SNo_diadic_rational_p^{add_SNo_diadic_rational_p} : ∀ x0 x1 . diadic_rational_p x0 ⟶ diadic_rational_p x1 ⟶ diadic_rational_p (add_SNo x0 x1) (proof)
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)
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 SNoL^SNoL : ι → ι
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)
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)
Param SNoLt^SNoLt : ι → ι → ο
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
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 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 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
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known minus_SNo_Lev^{minus_SNo_Lev} : ∀ x0 . SNo x0 ⟶ SNoLev (minus_SNo x0) = SNoLev x0
Known ordinal_SNoLev^{ordinal_SNoLev} : ∀ x0 . ordinal x0 ⟶ SNoLev x0 = x0
Known SNo_max_ordinal^{SNo_max_ordinal} : ∀ x0 . SNo x0 ⟶ (∀ x1 . x1 ∈ SNoS_ (SNoLev x0) ⟶ SNoLt x1 x0) ⟶ ordinal x0
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 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 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
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Definition TransSet^TransSet := λ x0 . ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Known ordinal_TransSet^{ordinal_TransSet} : ∀ x0 . ordinal x0 ⟶ TransSet x0
Known SNoS_I^SNoS_I : ∀ x0 . ordinal x0 ⟶ ∀ x1 x2 . x2 ∈ x0 ⟶ SNo_ x2 x1 ⟶ x1 ∈ SNoS_ x0
Known SNoLev_^SNoLev_ : ∀ x0 . SNo x0 ⟶ SNo_ (SNoLev x0) x0
Theorem SNoS_omega_diadic_rational_p_lem^{SNoS_omega_diadic_rational_p_lem} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 = x0 ⟶ diadic_rational_p x1 (proof)
Theorem SNoS_omega_diadic_rational_p^{SNoS_omega_diadic_rational_p} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ diadic_rational_p x0 (proof)