Proofgold Asset

Param Pi^Pi : ι → (ι → ι) → ι
Definition setexp^setexp := λ x0 x1 . Pi x1 (λ x2 . x0)
Param real^real : ι
Param omega^omega : ι
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param SNoLe^SNoLe : ι → ι → ο
Param ap^ap : ι → ι → ι
Param ordsucc^ordsucc : ι → ι
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Param SNo^SNo : ι → ο
Param SNoLt^SNoLt : ι → ι → ο
Definition SNoCutP^SNoCutP := λ x0 x1 . and (and (∀ x2 . x2 ∈ x0 ⟶ SNo x2) (∀ x2 . x2 ∈ x1 ⟶ SNo x2)) (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x1 ⟶ SNoLt x2 x3)
Param Sep^Sep : ι → (ι → ο) → ι
Param SNoS_^SNoS_ : ι → ι
Param SNoCut^SNoCut : ι → ι → ι
Param SNoLev^SNoLev : ι → ι
Param binunion^binunion : ι → ι → ι
Param famunion^famunion : ι → (ι → ι) → ι
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
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 andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Param minus_SNo^minus_SNo : ι → ι
Param abs_SNo^abs_SNo : ι → ι
Param add_SNo^add_SNo : ι → ι → ι
Param eps_^eps_ : ι → ι
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
Param SNo_^SNo_ : ι → ι → ο
Known SNoS_I^SNoS_I : ∀ x0 . ordinal x0 ⟶ ∀ x1 x2 . x2 ∈ x0 ⟶ SNo_ x2 x1 ⟶ x1 ∈ SNoS_ x0
Known ordsucc_omega_ordinal^{ordsucc_omega_ordinal} : ordinal (ordsucc omega)
Known SNoCutP_SNoCut_omega^{SNoCutP_SNoCut_omega} : ∀ x0 . x0 ⊆ SNoS_ omega ⟶ ∀ x1 . x1 ⊆ SNoS_ omega ⟶ SNoCutP x0 x1 ⟶ SNoLev (SNoCut x0 x1) ∈ ordsucc omega
Known Sep_Subq^Sep_Subq : ∀ x0 . ∀ x1 : ι → ο . Sep x0 x1 ⊆ x0
Known SNoLev_^SNoLev_ : ∀ x0 . SNo x0 ⟶ SNo_ (SNoLev x0) x0
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 ap_Pi^ap_Pi : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ Pi x0 x1 ⟶ x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3
Param nat_p^nat_p : ι → ο
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known nat_0^nat_0 : nat_p 0
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
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_tra^SNoLtLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLt x0 x2
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
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
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 x0)
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Known SNoS_omega_real^{SNoS_omega_real} : SNoS_ omega ⊆ real
Definition TransSet^TransSet := λ x0 . ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Known omega_TransSet^{omega_TransSet} : TransSet omega
Known omega_ordsucc^{omega_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ ordsucc x0 ∈ omega
Known ordinal_In_Or_Subq^{ordinal_In_Or_Subq} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ or (x0 ∈ x1) (x1 ⊆ x0)
Known SNoLev_ordinal^{SNoLev_ordinal} : ∀ x0 . SNo x0 ⟶ ordinal (SNoLev x0)
Known ordinal_ordsucc^{ordinal_ordsucc} : ∀ x0 . ordinal x0 ⟶ ordinal (ordsucc x0)
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc 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 SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Known SNo_eps_^SNo_eps_ : ∀ x0 . x0 ∈ omega ⟶ SNo (eps_ 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 minus_SNo_Le_contra^{minus_SNo_Le_contra} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ SNoLe (minus_SNo x1) (minus_SNo x0)
Known SNo_approx_real_rep^{SNo_approx_real_rep} : ∀ x0 . x0 ∈ real ⟶ ∀ x1 : ο . (∀ x2 . x2 ∈ setexp (SNoS_ omega) omega ⟶ ∀ x3 . x3 ∈ setexp (SNoS_ omega) omega ⟶ (∀ x4 . x4 ∈ omega ⟶ SNoLt (ap x2 x4) x0) ⟶ (∀ x4 . x4 ∈ omega ⟶ SNoLt x0 (add_SNo (ap x2 x4) (eps_ x4))) ⟶ (∀ x4 . x4 ∈ omega ⟶ ∀ x5 . x5 ∈ x4 ⟶ SNoLt (ap x2 x5) (ap x2 x4)) ⟶ (∀ x4 . x4 ∈ omega ⟶ SNoLt (add_SNo (ap x3 x4) (minus_SNo (eps_ x4))) x0) ⟶ (∀ x4 . x4 ∈ omega ⟶ SNoLt x0 (ap x3 x4)) ⟶ (∀ x4 . x4 ∈ omega ⟶ ∀ x5 . x5 ∈ x4 ⟶ SNoLt (ap x3 x4) (ap x3 x5)) ⟶ SNoCutP (prim5 omega (ap x2)) (prim5 omega (ap x3)) ⟶ x0 = SNoCut (prim5 omega (ap x2)) (prim5 omega (ap x3)) ⟶ x1) ⟶ x1
Known SNoCut_Le^SNoCut_Le : ∀ x0 x1 x2 x3 . SNoCutP x0 x1 ⟶ SNoCutP x2 x3 ⟶ (∀ x4 . x4 ∈ x0 ⟶ SNoLt x4 (SNoCut x2 x3)) ⟶ (∀ x4 . x4 ∈ x3 ⟶ SNoLt (SNoCut x0 x1) x4) ⟶ SNoLe (SNoCut x0 x1) (SNoCut x2 x3)
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 and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known SNoLt_tra^SNoLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
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 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 SNoLe_tra^SNoLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLe x0 x2
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 EmptyE^EmptyE : ∀ x0 . nIn x0 0
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Theorem real_complete1^{real_complete1} : ∀ x0 . x0 ∈ setexp real omega ⟶ ∀ x1 . x1 ∈ setexp real omega ⟶ (∀ x2 . x2 ∈ omega ⟶ and (and (SNoLe (ap x0 x2) (ap x1 x2)) (SNoLe (ap x0 x2) (ap x0 (ordsucc x2)))) (SNoLe (ap x1 (ordsucc x2)) (ap x1 x2))) ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ real) (∀ x4 . x4 ∈ omega ⟶ and (SNoLe (ap x0 x4) x3) (SNoLe x3 (ap x1 x4))) ⟶ x2) ⟶ x2 (proof)
Param add_nat^add_nat : ι → ι → ι
Known add_nat_add_SNo^{add_nat_add_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ add_nat x0 x1 = add_SNo x0 x1
Known nat_1^nat_1 : nat_p 1
Known add_nat_SR^add_nat_SR : ∀ x0 x1 . nat_p x1 ⟶ add_nat x0 (ordsucc x1) = ordsucc (add_nat x0 x1)
Known add_nat_0R^add_nat_0R : ∀ x0 . add_nat x0 0 = x0
Theorem real_complete2^{real_complete2} : ∀ x0 . x0 ∈ setexp real omega ⟶ ∀ x1 . x1 ∈ setexp real omega ⟶ (∀ x2 . x2 ∈ omega ⟶ and (and (SNoLe (ap x0 x2) (ap x1 x2)) (SNoLe (ap x0 x2) (ap x0 (add_SNo x2 1)))) (SNoLe (ap x1 (add_SNo x2 1)) (ap x1 x2))) ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ real) (∀ x4 . x4 ∈ omega ⟶ and (SNoLe (ap x0 x4) x3) (SNoLe x3 (ap x1 x4))) ⟶ x2) ⟶ x2 (proof)