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Proofgold Asset

asset id
711f9ee7d2e646a7f81a006490137ff98368ca59de2dd3ebfd2a73651726167e
asset hash
ba6130f0e41208a3f9f5a3ba6faec9793d573edc5f5ff700643588b1e1e0bf5a
bday / block
12443
tx
a9b67..
preasset
doc published by PrGxv..
Param realreal : ι
Definition SubqSubq := λ x0 x1 . ∀ x2 . x2x0x2x1
Param SNoS_SNoS_ : ιι
Param omegaomega : ι
Known SNoS_omega_realSNoS_omega_real : SNoS_ omegareal
Known omega_SNoS_omegaomega_SNoS_omega : omegaSNoS_ omega
Param nat_pnat_p : ιο
Known nat_p_omeganat_p_omega : ∀ x0 . nat_p x0x0omega
Known nat_0nat_0 : nat_p 0
Theorem real_0real_0 : 0real (proof)
Param ordsuccordsucc : ιι
Known nat_1nat_1 : nat_p 1
Theorem real_1real_1 : 1real (proof)
Definition andand := λ x0 x1 : ο . ∀ x2 : ο . (x0x1x2)x2
Param int_alt1int : ι
Param mul_SNomul_SNo : ιιι
Param eps_eps_ : ιι
Definition diadic_rational_alt1_p := λ x0 . ∀ x1 : ο . (∀ x2 . and (x2omega) (∀ x3 : ο . (∀ x4 . and (x4int_alt1) (x0 = mul_SNo (eps_ x2) x4)x3)x3)x1)x1
Param minus_SNominus_SNo : ιι
Known a3283..int_SNo_cases : ∀ x0 : ι → ο . (∀ x1 . x1omegax0 x1)(∀ x1 . x1omegax0 (minus_SNo x1))∀ x1 . x1int_alt1x0 x1
Known nonneg_diadic_rational_p_SNoS_omeganonneg_diadic_rational_p_SNoS_omega : ∀ x0 . x0omega∀ x1 . nat_p x1mul_SNo (eps_ x0) x1SNoS_ omega
Known omega_nat_pomega_nat_p : ∀ x0 . x0omeganat_p x0
Param SNoSNo : ιο
Known mul_SNo_minus_distrRmul_minus_SNo_distrR : ∀ x0 x1 . SNo x0SNo x1mul_SNo x0 (minus_SNo x1) = minus_SNo (mul_SNo x0 x1)
Known SNo_eps_SNo_eps_ : ∀ x0 . x0omegaSNo (eps_ x0)
Known omega_SNoomega_SNo : ∀ x0 . x0omegaSNo x0
Known minus_SNo_SNoS_omegaminus_SNo_SNoS_omega : ∀ x0 . x0SNoS_ omegaminus_SNo x0SNoS_ omega
Theorem f05cb.. : ∀ x0 . diadic_rational_alt1_p x0x0SNoS_ omega (proof)
Param ordinalordinal : ιο
Param SNoLevSNoLev : ιι
Param SNo_SNo_ : ιιο
Known SNoS_E2SNoS_E2 : ∀ x0 . ordinal x0∀ x1 . x1SNoS_ x0∀ x2 : ο . (SNoLev x1x0ordinal (SNoLev x1)SNo x1SNo_ (SNoLev x1) x1x2)x2
Known omega_ordinalomega_ordinal : ordinal omega
Theorem 455b5.. : ∀ x0 . diadic_rational_alt1_p x0SNo x0 (proof)
Known andIandI : ∀ x0 x1 : ο . x0x1and x0 x1
Known eps_0_1eps_0_1 : eps_ 0 = 1
Known mul_SNo_oneLmul_SNo_oneL : ∀ x0 . SNo x0mul_SNo 1 x0 = x0
Known 0c7a0.. : ∀ x0 . x0int_alt1SNo x0
Theorem a08ea.. : ∀ x0 . x0int_alt1diadic_rational_alt1_p x0 (proof)
Known c3d99..Subq_omega_int : omegaint_alt1
Theorem a89df.. : ∀ x0 . x0omegadiadic_rational_alt1_p x0 (proof)
Known mul_SNo_oneRmul_SNo_oneR : ∀ x0 . SNo x0mul_SNo x0 1 = x0
Theorem 5042a.. : ∀ x0 . x0omegadiadic_rational_alt1_p (eps_ x0) (proof)
Known daaad..int_minus_SNo : ∀ x0 . x0int_alt1minus_SNo x0int_alt1
Theorem 3801d.. : ∀ x0 . diadic_rational_alt1_p x0diadic_rational_alt1_p (minus_SNo x0) (proof)
Param add_SNoadd_SNo : ιιι
Known add_SNo_In_omegaadd_SNo_In_omega : ∀ x0 . x0omega∀ x1 . x1omegaadd_SNo x0 x1omega
Known eefdf..int_mul_SNo : ∀ x0 . x0int_alt1∀ x1 . x1int_alt1mul_SNo x0 x1int_alt1
Known mul_SNo_eps_eps_add_SNomul_SNo_eps_eps_add_SNo : ∀ x0 . x0omega∀ x1 . x1omegamul_SNo (eps_ x0) (eps_ x1) = eps_ (add_SNo x0 x1)
Known mul_SNo_com_4_inner_midmul_SNo_com_4_inner_mid : ∀ x0 x1 x2 x3 . SNo x0SNo x1SNo x2SNo x3mul_SNo (mul_SNo x0 x1) (mul_SNo x2 x3) = mul_SNo (mul_SNo x0 x2) (mul_SNo x1 x3)
Theorem 858c0.. : ∀ x0 x1 . diadic_rational_alt1_p x0diadic_rational_alt1_p x1diadic_rational_alt1_p (mul_SNo x0 x1) (proof)
Param exp_SNo_natexp_SNo_nat : ιιι
Known a0aa5..int_add_SNo : ∀ x0 . x0int_alt1∀ x1 . x1int_alt1add_SNo x0 x1int_alt1
Known mul_SNo_distrLmul_SNo_distrL : ∀ x0 x1 x2 . SNo x0SNo x1SNo x2mul_SNo x0 (add_SNo x1 x2) = add_SNo (mul_SNo x0 x1) (mul_SNo x0 x2)
Known SNo_mul_SNoSNo_mul_SNo : ∀ x0 x1 . SNo x0SNo x1SNo (mul_SNo x0 x1)
Known mul_SNo_assocmul_SNo_assoc : ∀ x0 x1 x2 . SNo x0SNo x1SNo x2mul_SNo x0 (mul_SNo x1 x2) = mul_SNo (mul_SNo x0 x1) x2
Known mul_SNo_eps_power_2mul_SNo_eps_power_2 : ∀ x0 . nat_p x0mul_SNo (eps_ x0) (exp_SNo_nat 2 x0) = 1
Known nat_exp_SNo_natnat_exp_SNo_nat : ∀ x0 . nat_p x0∀ x1 . nat_p x1nat_p (exp_SNo_nat x0 x1)
Known nat_2nat_2 : nat_p 2
Theorem 9e46f.. : ∀ x0 x1 . diadic_rational_alt1_p x0diadic_rational_alt1_p x1diadic_rational_alt1_p (add_SNo x0 x1) (proof)