Proofgold Asset

Param SNo^SNo : ι → ο
Param SNoLe^SNoLe : ι → ι → ο
Param SNoLt^SNoLt : ι → ι → ο
Param mul_SNo^mul_SNo : ι → ι → ι
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 x0)
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
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 SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
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)
Theorem 3987d.. : ∀ x0 x1 x2 . SNo x0 ⟶ SNoLe 0 x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt (mul_SNo x0 x1) (mul_SNo x0 x2) ⟶ SNoLt x1 x2 (proof)
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
Theorem e2621.. : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe 0 x2 ⟶ SNoLt (mul_SNo x0 x2) (mul_SNo x1 x2) ⟶ SNoLt x0 x1 (proof)
Param real^real : ι
Param ordsucc^ordsucc : ι → ι
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Param SNoLev^SNoLev : ι → ι
Param omega^omega : ι
Param SNoS_^SNoS_ : ι → ι
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_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
Param Sep^Sep : ι → (ι → ο) → ι
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 SNoCut^SNoCut : ι → 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 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 SNo_1^SNo_1 : SNo 1
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Known real_mul_SNo^real_mul_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ mul_SNo x0 x1 ∈ real
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known omega_SNoS_omega^{omega_SNoS_omega} : omega ⊆ SNoS_ omega
Param nat_p^nat_p : ι → ο
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known nat_1^nat_1 : nat_p 1
Known omega_ordsucc^{omega_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ ordsucc x0 ∈ omega
Param ordinal^ordinal : ι → ο
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 SNo_abs_SNo^SNo_abs_SNo : ∀ x0 . SNo x0 ⟶ SNo (abs_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 SNo_triangle2^{SNo_triangle2} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe (abs_SNo (add_SNo x0 (minus_SNo x2))) (add_SNo (abs_SNo (add_SNo x0 (minus_SNo x1))) (abs_SNo (add_SNo x1 (minus_SNo x2))))
Known eps_ordsucc_half_add^{eps_ordsucc_half_add} : ∀ x0 . nat_p x0 ⟶ add_SNo (eps_ (ordsucc x0)) (eps_ (ordsucc x0)) = eps_ x0
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Known add_SNo_Lt3^add_SNo_Lt3 : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLt x0 x2 ⟶ SNoLt x1 x3 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x3)
Known abs_SNo_intvl_bd^{abs_SNo_intvl_bd} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt x1 (add_SNo x0 x2) ⟶ SNoLt (abs_SNo (add_SNo x1 (minus_SNo x0))) x2
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known add_SNo_SNoS_omega^{add_SNo_SNoS_omega} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ ∀ x1 . x1 ∈ SNoS_ omega ⟶ add_SNo x0 x1 ∈ SNoS_ omega
Known SNo_eps_SNoS_omega^{SNo_eps_SNoS_omega} : ∀ x0 . x0 ∈ omega ⟶ eps_ x0 ∈ SNoS_ omega
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 real_SNo^real_SNo : ∀ x0 . x0 ∈ real ⟶ SNo x0
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 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 add_SNo_Lt2^add_SNo_Lt2 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x1 x2 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x0 x2)
Known mul_SNo_Lt1_pos_Lt^{mul_SNo_Lt1_pos_Lt} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 1 ⟶ SNoLt 0 x1 ⟶ SNoLt (mul_SNo x0 x1) x1
Known SNo_eps_pos^SNo_eps_pos : ∀ x0 . x0 ∈ omega ⟶ SNoLt 0 (eps_ x0)
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
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 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 binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Param TransSet^TransSet : ι → ο
Known TransSet_In_ordsucc_Subq^{TransSet_In_ordsucc_Subq} : ∀ x0 x1 . TransSet x1 ⟶ x0 ∈ ordsucc x1 ⟶ x0 ⊆ x1
Known ordinal_TransSet^{ordinal_TransSet} : ∀ x0 . ordinal x0 ⟶ TransSet x0
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 ordinal_ordsucc^{ordinal_ordsucc} : ∀ x0 . ordinal x0 ⟶ ordinal (ordsucc x0)
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Known famunionE_impred^{famunionE_impred} : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ famunion x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ x0 ⟶ x2 ∈ x1 x4 ⟶ x3) ⟶ x3
Known In_no2cycle^In_no2cycle : ∀ x0 x1 . x0 ∈ x1 ⟶ x1 ∈ x0 ⟶ False
Known SNoLev_^SNoLev_ : ∀ x0 . SNo x0 ⟶ SNo_ (SNoLev x0) x0
Known nat_0^nat_0 : nat_p 0
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Known abs_SNo_minus^{abs_SNo_minus} : ∀ x0 . SNo x0 ⟶ abs_SNo (minus_SNo x0) = abs_SNo x0
Known pos_abs_SNo^pos_abs_SNo : ∀ x0 . SNoLt 0 x0 ⟶ abs_SNo x0 = x0
Param exp_SNo_nat^exp_SNo_nat : ι → ι → ι
Known SNo_omega^SNo_omega : SNo omega
Known mul_SNo_oneR^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = x0
Known SNoLtLe_tra^SNoLtLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLt x0 x2
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
Known SNo_0^SNo_0 : SNo 0
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 mul_SNo_eps_power_2^{mul_SNo_eps_power_2} : ∀ x0 . nat_p x0 ⟶ mul_SNo (eps_ x0) (exp_SNo_nat 2 x0) = 1
Known nonneg_mul_SNo_Le'^{nonneg_mul_SNo_Le} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe 0 x2 ⟶ SNoLe x0 x1 ⟶ SNoLe (mul_SNo x0 x2) (mul_SNo x1 x2)
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 ordinal_In_SNoLt^{ordinal_In_SNoLt} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ SNoLt x1 x0
Known add_SNo_1_ordsucc^{add_SNo_1_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ add_SNo x0 1 = ordsucc x0
Known nat_p_SNo^nat_p_SNo : ∀ x0 . nat_p x0 ⟶ SNo 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 mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Known SNoLt_0_1^SNoLt_0_1 : SNoLt 0 1
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_0^minus_SNo_0 : minus_SNo 0 = 0
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 SNoLt_minus_pos^{SNoLt_minus_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 x1 ⟶ SNoLt 0 (add_SNo x1 (minus_SNo x0))
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Known add_SNo_minus_Le2^{add_SNo_minus_Le2} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x2 (add_SNo x0 (minus_SNo x1)) ⟶ SNoLe (add_SNo x2 x1) x0
Known add_SNo_minus_Lt1^{add_SNo_minus_Lt1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt (add_SNo x0 (minus_SNo x1)) x2 ⟶ SNoLt x0 (add_SNo x2 x1)
Known add_SNo_minus_Le2b^{add_SNo_minus_Le2b} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe (add_SNo x2 x1) x0 ⟶ SNoLe x2 (add_SNo x0 (minus_SNo x1))
Known minus_SNo_SNoS_omega^{minus_SNo_SNoS_omega} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ minus_SNo x0 ∈ SNoS_ omega
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 minus_SNo_Lt_contra^{minus_SNo_Lt_contra} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 x1 ⟶ SNoLt (minus_SNo x1) (minus_SNo x0)
Known add_SNo_minus_Lt1b^{add_SNo_minus_Lt1b} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 (add_SNo x2 x1) ⟶ SNoLt (add_SNo x0 (minus_SNo x1)) x2
Known SNoS_omega_real^{SNoS_omega_real} : SNoS_ omega ⊆ real
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Theorem pos_small_real_recip_ex^{pos_small_real_recip_ex} : ∀ x0 . x0 ∈ real ⟶ SNoLt 0 x0 ⟶ SNoLt x0 1 ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ real) (mul_SNo x0 x2 = 1) ⟶ x1) ⟶ x1 (proof)
Known 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
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 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
Theorem pos_real_recip_ex^{pos_real_recip_ex} : ∀ x0 . x0 ∈ real ⟶ SNoLt 0 x0 ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ real) (mul_SNo x0 x2 = 1) ⟶ x1) ⟶ x1 (proof)
Known real_minus_SNo^{real_minus_SNo} : ∀ x0 . x0 ∈ real ⟶ minus_SNo x0 ∈ real
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 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)
Theorem nonzero_real_recip_ex^{nonzero_real_recip_ex} : ∀ x0 . x0 ∈ real ⟶ (x0 = 0 ⟶ ∀ x1 : ο . x1) ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ real) (mul_SNo x0 x2 = 1) ⟶ x1) ⟶ x1 (proof)
Param explicit_Field^{explicit_Field} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → ο
Known explicit_Field_I^{explicit_Field_I} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 ∈ x0) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 (x3 x6 x7) = x3 (x3 x5 x6) x7) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 = x3 x6 x5) ⟶ x1 ∈ x0 ⟶ (∀ x5 . x5 ∈ x0 ⟶ x3 x1 x5 = x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 : ο . (∀ x7 . and (x7 ∈ x0) (x3 x5 x7 = x1) ⟶ x6) ⟶ x6) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 x6 ∈ x0) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x5 (x4 x6 x7) = x4 (x4 x5 x6) x7) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 x6 = x4 x6 x5) ⟶ x2 ∈ x0 ⟶ (x2 = x1 ⟶ ∀ x5 : ο . x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ x4 x2 x5 = x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ (x5 = x1 ⟶ ∀ x6 : ο . x6) ⟶ ∀ x6 : ο . (∀ x7 . and (x7 ∈ x0) (x4 x5 x7 = x2) ⟶ x6) ⟶ x6) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x5 (x3 x6 x7) = x3 (x4 x5 x6) (x4 x5 x7)) ⟶ explicit_Field x0 x1 x2 x3 x4
Known real_add_SNo^real_add_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ add_SNo x0 x1 ∈ real
Known add_SNo_assoc^{add_SNo_assoc} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 (add_SNo x1 x2) = add_SNo (add_SNo x0 x1) x2
Known real_0^real_0 : 0 ∈ real
Known add_SNo_minus_SNo_rinv^{add_SNo_minus_SNo_rinv} : ∀ x0 . SNo x0 ⟶ add_SNo x0 (minus_SNo x0) = 0
Known real_1^real_1 : 1 ∈ real
Known neq_1_0^neq_1_0 : 1 = 0 ⟶ ∀ x0 : ο . x0
Known mul_SNo_oneL^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 x0 = x0
Theorem explicit_Field_real : explicit_Field real 0 1 add_SNo mul_SNo (proof)
Param explicit_OrderedField^{explicit_OrderedField} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ο
Param iff^iff : ο → ο → ο
Known explicit_OrderedField_I^{explicit_OrderedField_I} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_Field x0 x1 x2 x3 x4 ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x5 x6 x7 ⟶ x5 x7 x8 ⟶ x5 x6 x8) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ iff (and (x5 x6 x7) (x5 x7 x6)) (x6 = x7)) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ or (x5 x6 x7) (x5 x7 x6)) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x5 x6 x7 ⟶ x5 (x3 x6 x8) (x3 x7 x8)) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x5 x1 x6 ⟶ x5 x1 x7 ⟶ x5 x1 (x4 x6 x7)) ⟶ explicit_OrderedField x0 x1 x2 x3 x4 x5
Known SNoLe_tra^SNoLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLe x0 x2
Known iffI^iffI : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ iff x0 x1
Known SNoLe_antisym^{SNoLe_antisym} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x0 ⟶ x0 = x1
Known SNoLe_ref^SNoLe_ref : ∀ x0 . SNoLe x0 x0
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Known SNoLeE^SNoLeE : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ or (SNoLt x0 x1) (x0 = x1)
Known mul_SNo_zeroL^{mul_SNo_zeroL} : ∀ x0 . SNo x0 ⟶ mul_SNo 0 x0 = 0
Theorem explicit_OrderedField_real : explicit_OrderedField real 0 1 add_SNo mul_SNo SNoLe (proof)
Param explicit_Nats^{explicit_Nats} : ι → ι → (ι → ι) → ο
Known explicit_Nats_E^{explicit_Nats_E} : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 : ο . (explicit_Nats x0 x1 x2 ⟶ x1 ∈ x0 ⟶ (∀ x4 . x4 ∈ x0 ⟶ x2 x4 ∈ x0) ⟶ (∀ x4 . x4 ∈ x0 ⟶ x2 x4 = x1 ⟶ ∀ x5 : ο . x5) ⟶ (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 = x2 x5 ⟶ x4 = x5) ⟶ (∀ x4 : ι → ο . x4 x1 ⟶ (∀ x5 . x4 x5 ⟶ x4 (x2 x5)) ⟶ ∀ x5 . x5 ∈ x0 ⟶ x4 x5) ⟶ x3) ⟶ explicit_Nats x0 x1 x2 ⟶ x3
Theorem 62095.. : ∀ x0 x1 . ∀ x2 : ι → ι . explicit_Nats x0 x1 x2 ⟶ ∀ x3 : ο . (x1 ∈ x0 ⟶ (∀ x4 . x4 ∈ x0 ⟶ x2 x4 ∈ x0) ⟶ (∀ x4 . x4 ∈ x0 ⟶ x2 x4 = x1 ⟶ ∀ x5 : ο . x5) ⟶ (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 = x2 x5 ⟶ x4 = x5) ⟶ (∀ x4 : ι → ο . x4 x1 ⟶ (∀ x5 . x4 x5 ⟶ x4 (x2 x5)) ⟶ ∀ x5 . x5 ∈ x0 ⟶ x4 x5) ⟶ x3) ⟶ x3 (proof)
Param natOfOrderedField_p^{natOfOrderedField_p} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ι → ο
Known explicit_Nats_natOfOrderedField^{explicit_Nats_natOfOrderedField} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ explicit_Nats (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) x1 (λ x6 . x3 x6 x2)
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
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
Theorem 2c707.. : Sep real (natOfOrderedField_p real 0 1 add_SNo mul_SNo SNoLe) = omega (proof)
Param explicit_Reals^{explicit_Reals} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ο
Definition lt^lt := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . λ x5 : ι → ι → ο . λ x6 x7 . and (x5 x6 x7) (x6 = x7 ⟶ ∀ x8 : ο . x8)
Param setexp^setexp : ι → ι → ι
Param ap^ap : ι → ι → ι
Known explicit_Reals_I^{explicit_Reals_I} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ lt x0 x1 x2 x3 x4 x5 x1 x6 ⟶ x5 x1 x7 ⟶ ∀ x8 : ο . (∀ x9 . and (x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) (x5 x7 (x4 x9 x6)) ⟶ x8) ⟶ x8) ⟶ (∀ x6 . x6 ∈ setexp x0 (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ ∀ x7 . x7 ∈ setexp x0 (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ (∀ x8 . x8 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) ⟶ and (and (x5 (ap x6 x8) (ap x7 x8)) (x5 (ap x6 x8) (ap x6 (x3 x8 x2)))) (x5 (ap x7 (x3 x8 x2)) (ap x7 x8))) ⟶ ∀ x8 : ο . (∀ x9 . and (x9 ∈ x0) (∀ x10 . x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) ⟶ and (x5 (ap x6 x10) x9) (x5 x9 (ap x7 x10))) ⟶ x8) ⟶ x8) ⟶ explicit_Reals x0 x1 x2 x3 x4 x5
Known 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
Known 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
Theorem explicit_Reals_real : explicit_Reals real 0 1 add_SNo mul_SNo SNoLe (proof)
Param pack_b_b_r_e_e^{pack_b_b_r_e_e} : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ι → ι → ι
Definition real_struct := pack_b_b_r_e_e real add_SNo mul_SNo SNoLe 0 1
Param struct_b_b_r_e_e^{struct_b_b_r_e_e} : ι → ο
Param unpack_b_b_r_e_e_o^{unpack_b_b_r_e_e_o} : ι → (ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ι → ι → ο) → ο
Definition RealsStruct^RealsStruct := λ x0 . and (struct_b_b_r_e_e x0) (unpack_b_b_r_e_e_o x0 (λ x1 . λ x2 x3 : ι → ι → ι . λ x4 : ι → ι → ο . λ x5 x6 . explicit_Reals x1 x5 x6 x2 x3 x4))
Known pack_struct_b_b_r_e_e_I^{pack_struct_b_b_r_e_e_I} : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ∈ x0) ⟶ ∀ x2 : ι → ι → ι . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 x4 ∈ x0) ⟶ ∀ x3 : ι → ι → ο . ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ struct_b_b_r_e_e (pack_b_b_r_e_e x0 x1 x2 x3 x4 x5)
Known RealsStruct_unpack_eq^{RealsStruct_unpack_eq} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . unpack_b_b_r_e_e_o (pack_b_b_r_e_e x0 x1 x2 x3 x4 x5) (λ x7 . λ x8 x9 : ι → ι → ι . λ x10 : ι → ι → ο . λ x11 x12 . explicit_Reals x7 x11 x12 x8 x9 x10) = explicit_Reals x0 x4 x5 x1 x2 x3
Theorem e87bd.. : RealsStruct real_struct (proof)