Proofgold Asset

Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Definition TransSet^TransSet := λ x0 . ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Definition ordinal^ordinal := λ x0 . and (TransSet x0) (∀ x1 . x1 ∈ x0 ⟶ TransSet x1)
Param add_SNo^add_SNo : ι → ι → ι
Param SNo^SNo : ι → ο
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Known add_SNo_ordinal_InL^{add_SNo_ordinal_InL} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ ∀ x2 . x2 ∈ x0 ⟶ add_SNo x2 x1 ∈ add_SNo x0 x1
Known ordinal_SNo^ordinal_SNo : ∀ x0 . ordinal x0 ⟶ SNo x0
Known ordinal_Hered^{ordinal_Hered} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordinal x1
Theorem add_SNo_ordinal_InR^{add_SNo_ordinal_InR} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ ∀ x2 . x2 ∈ x1 ⟶ add_SNo x0 x2 ∈ add_SNo x0 x1 (proof)
Param omega^omega : ι
Param add_nat^add_nat : ι → ι → ι
Param nat_p^nat_p : ι → ο
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Param ordsucc^ordsucc : ι → ι
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
Known add_nat_0R^add_nat_0R : ∀ x0 . add_nat x0 0 = x0
Known add_SNo_ordinal_SR^{add_SNo_ordinal_SR} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ add_SNo x0 (ordsucc x1) = ordsucc (add_SNo x0 x1)
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known add_nat_SR^add_nat_SR : ∀ x0 x1 . nat_p x1 ⟶ add_nat x0 (ordsucc x1) = ordsucc (add_nat x0 x1)
Theorem add_nat_add_SNo^{add_nat_add_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ add_nat x0 x1 = add_SNo x0 x1 (proof)
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known add_nat_p^add_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (add_nat x0 x1)
Theorem add_SNo_In_omega^{add_SNo_In_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ add_SNo x0 x1 ∈ omega (proof)
Known add_SNo_cancel_L^{add_SNo_cancel_L} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 x1 = add_SNo x0 x2 ⟶ x1 = x2
Theorem add_SNo_cancel_R^{add_SNo_cancel_R} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 x1 = add_SNo x2 x1 ⟶ x0 = x2 (proof)
Param SNoLev^SNoLev : ι → ι
Param SNoS_^SNoS_ : ι → ι
Known SNoLev_ind2^SNoLev_ind2 : ∀ x0 : ι → ι → ο . (∀ x1 x2 . SNo x1 ⟶ SNo x2 ⟶ (∀ x3 . x3 ∈ SNoS_ (SNoLev x1) ⟶ x0 x3 x2) ⟶ (∀ x3 . x3 ∈ SNoS_ (SNoLev x2) ⟶ x0 x1 x3) ⟶ (∀ x3 . x3 ∈ SNoS_ (SNoLev x1) ⟶ ∀ x4 . x4 ∈ SNoS_ (SNoLev x2) ⟶ x0 x3 x4) ⟶ x0 x1 x2) ⟶ ∀ x1 x2 . SNo x1 ⟶ SNo x2 ⟶ x0 x1 x2
Param SNoCut^SNoCut : ι → ι → ι
Param binunion^binunion : ι → ι → ι
Param SNoL^SNoL : ι → ι
Param SNoR^SNoR : ι → ι
Known add_SNo_eq^add_SNo_eq : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ add_SNo x0 x1 = SNoCut (binunion {add_SNo x3 x1|x3 ∈ SNoL x0} (prim5 (SNoL x1) (add_SNo x0))) (binunion {add_SNo x3 x1|x3 ∈ SNoR x0} (prim5 (SNoR x1) (add_SNo x0)))
Param SNoCutP^SNoCutP : ι → ι → ο
Param famunion^famunion : ι → (ι → ι) → ι
Param SNoLt^SNoLt : ι → ι → ο
Param SNoEq_^SNoEq_ : ι → ι → ι → ο
Known SNoCutP_SNoCut_impred^{SNoCutP_SNoCut_impred} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ ∀ x2 : ο . (SNo (SNoCut x0 x1) ⟶ SNoLev (SNoCut x0 x1) ∈ ordsucc (binunion (famunion x0 (λ x3 . ordsucc (SNoLev x3))) (famunion x1 (λ x3 . ordsucc (SNoLev x3)))) ⟶ (∀ x3 . x3 ∈ x0 ⟶ SNoLt x3 (SNoCut x0 x1)) ⟶ (∀ x3 . x3 ∈ x1 ⟶ SNoLt (SNoCut x0 x1) x3) ⟶ (∀ x3 . SNo x3 ⟶ (∀ x4 . x4 ∈ x0 ⟶ SNoLt x4 x3) ⟶ (∀ x4 . x4 ∈ x1 ⟶ SNoLt x3 x4) ⟶ and (SNoLev (SNoCut x0 x1) ⊆ SNoLev x3) (SNoEq_ (SNoLev (SNoCut x0 x1)) (SNoCut x0 x1) x3)) ⟶ x2) ⟶ x2
Known Subq_tra^Subq_tra : ∀ x0 x1 x2 . x0 ⊆ x1 ⟶ x1 ⊆ x2 ⟶ x0 ⊆ x2
Known binunion_Subq_min^{binunion_Subq_min} : ∀ x0 x1 x2 . x0 ⊆ x2 ⟶ x1 ⊆ x2 ⟶ binunion x0 x1 ⊆ x2
Known famunionE_impred^{famunionE_impred} : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ famunion x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ x0 ⟶ x2 ∈ x1 x4 ⟶ x3) ⟶ x3
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 ordinal_In_Or_Subq^{ordinal_In_Or_Subq} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ or (x0 ∈ x1) (x1 ⊆ x0)
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Known SNoLev_ordinal^{SNoLev_ordinal} : ∀ x0 . SNo x0 ⟶ ordinal (SNoLev x0)
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Known In_no2cycle^In_no2cycle : ∀ x0 x1 . x0 ∈ x1 ⟶ x1 ∈ x0 ⟶ False
Known ordinal_ordsucc^{ordinal_ordsucc} : ∀ x0 . ordinal x0 ⟶ ordinal (ordsucc x0)
Known TransSet_In_ordsucc_Subq^{TransSet_In_ordsucc_Subq} : ∀ x0 x1 . TransSet x1 ⟶ x0 ∈ ordsucc x1 ⟶ x0 ⊆ x1
Known ordinal_binunion^{ordinal_binunion} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ ordinal (binunion x0 x1)
Known ordinal_famunion^{ordinal_famunion} : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . x2 ∈ x0 ⟶ ordinal (x1 x2)) ⟶ ordinal (famunion x0 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 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 SNoR_SNoS_^SNoR_SNoS_ : ∀ x0 . SNoR x0 ⊆ SNoS_ (SNoLev 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 SNoL_SNoS_^SNoL_SNoS_ : ∀ x0 . SNoL x0 ⊆ SNoS_ (SNoLev x0)
Known add_SNo_SNoCutP^{add_SNo_SNoCutP} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoCutP (binunion {add_SNo x2 x1|x2 ∈ SNoL x0} (prim5 (SNoL x1) (add_SNo x0))) (binunion {add_SNo x2 x1|x2 ∈ SNoR x0} (prim5 (SNoR x1) (add_SNo x0)))
Known add_SNo_ordinal_ordinal^{add_SNo_ordinal_ordinal} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . ordinal x1 ⟶ ordinal (add_SNo x0 x1)
Theorem add_SNo_Lev_bd^{add_SNo_Lev_bd} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev (add_SNo x0 x1) ⊆ add_SNo (SNoLev x0) (SNoLev x1) (proof)
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
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 add_SNo_SNoS_omega^{add_SNo_SNoS_omega} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ ∀ x1 . x1 ∈ SNoS_ omega ⟶ add_SNo x0 x1 ∈ SNoS_ omega (proof)
Param minus_CSNo^minus_CSNo : ι → ι
Definition int_alt1^int := binunion omega (prim5 omega minus_CSNo)
Param ReplSep2^ReplSep2 : ι → (ι → ι) → (ι → ι → ο) → CT2 ι
Param div_CSNo^div_CSNo : ι → ι → ι
Definition cab0d..^rational := ReplSep2 int_alt1 (λ x0 . omega) (λ x0 x1 . x1 = 0 ⟶ ∀ x2 : ο . x2) div_CSNo
Param nat_primrec^nat_primrec : ι → (ι → ι → ι) → ι → ι
Param If_i^If_i : ο → ι → ι → ι
Param add_CSNo^add_CSNo : ι → ι → ι
Definition Sum^Sum := λ x0 x1 . λ x2 : ι → ι . nat_primrec 0 (λ x3 x4 . If_i (x3 ∈ x0) 0 (add_CSNo (x2 x3) x4)) (ordsucc x1)
Param mul_CSNo^mul_CSNo : ι → ι → ι
Definition Prod^Prod := λ x0 x1 . λ x2 : ι → ι . nat_primrec 1 (λ x3 x4 . If_i (x3 ∈ x0) 1 (mul_CSNo (x2 x3) x4)) (ordsucc x1)