Proofgold Address

Param SNo^SNo : ι → ο
Param SNoLt^SNoLt : ι → ι → ο
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param recip_SNo_pos^{recip_SNo_pos} : ι → ι
Param mul_SNo^mul_SNo : ι → ι → ι
Param ordsucc^ordsucc : ι → ι
Param SNoS_^SNoS_ : ι → ι
Param SNoLev^SNoLev : ι → ι
Known SNoLev_ind^SNoLev_ind : ∀ x0 : ι → ο . (∀ x1 . SNo x1 ⟶ (∀ x2 . x2 ∈ SNoS_ (SNoLev x1) ⟶ x0 x2) ⟶ x0 x1) ⟶ ∀ x1 . SNo x1 ⟶ x0 x1
Param SNoCut^SNoCut : ι → ι → ι
Param famunion^famunion : ι → (ι → ι) → ι
Param omega^omega : ι
Param ap^ap : ι → ι → ι
Param SNo_recipaux^SNo_recipaux : ι → (ι → ι) → ι → ι
Known recip_SNo_pos_eq^{recip_SNo_pos_eq} : ∀ x0 . SNo x0 ⟶ recip_SNo_pos x0 = SNoCut (famunion omega (λ x2 . ap (SNo_recipaux x0 recip_SNo_pos x2) 0)) (famunion omega (λ x2 . ap (SNo_recipaux x0 recip_SNo_pos x2) 1))
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)
Known SNo_recipaux_lem2^{SNo_recipaux_lem2} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ ∀ x1 : ι → ι . (∀ x2 . x2 ∈ SNoS_ (SNoLev x0) ⟶ SNoLt 0 x2 ⟶ and (SNo (x1 x2)) (mul_SNo x2 (x1 x2) = 1)) ⟶ SNoCutP (famunion omega (λ x2 . ap (SNo_recipaux x0 x1 x2) 0)) (famunion omega (λ x2 . ap (SNo_recipaux x0 x1 x2) 1))
Param binunion^binunion : ι → ι → ι
Param Subq^Subq : 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 andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and 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 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 SNo_1^SNo_1 : SNo 1
Param SNoLe^SNoLe : ι → ι → ο
Param add_SNo^add_SNo : ι → ι → ι
Param minus_SNo^minus_SNo : ι → ι
Param SNoR^SNoR : ι → ι
Param SNoL^SNoL : ι → ι
Known mul_SNo_SNoR_interpolate_impred^{mul_SNo_SNoR_interpolate_impred} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 . x2 ∈ SNoR (mul_SNo x0 x1) ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ SNoL x0 ⟶ ∀ x5 . x5 ∈ SNoR x1 ⟶ SNoLe (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x5)) (add_SNo x2 (mul_SNo x4 x5)) ⟶ x3) ⟶ (∀ x4 . x4 ∈ SNoR x0 ⟶ ∀ x5 . x5 ∈ SNoL x1 ⟶ SNoLe (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x5)) (add_SNo x2 (mul_SNo x4 x5)) ⟶ x3) ⟶ x3
Known SNoR_I^SNoR_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x0 x1 ⟶ x1 ∈ SNoR x0
Param ordinal^ordinal : ι → ο
Known ordinal_SNoLev^{ordinal_SNoLev} : ∀ x0 . ordinal x0 ⟶ SNoLev x0 = x0
Known ordinal_1^ordinal_1 : ordinal 1
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
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 FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known pos_low_eq_one^{pos_low_eq_one} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ SNoLev x0 ⊆ 1 ⟶ x0 = 1
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 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_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_SNoCutP_ex^{SNoR_SNoCutP_ex} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ ∀ x2 . x2 ∈ SNoR (SNoCut x0 x1) ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ x1) (SNoLe x4 x2) ⟶ x3) ⟶ x3
Known SNoS_I2^SNoS_I2 : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x0 ∈ SNoLev x1 ⟶ x0 ∈ SNoS_ (SNoLev x1)
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 famunionI^famunionI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ x0 ⟶ x3 ∈ x1 x2 ⟶ x3 ∈ famunion x0 x1
Known omega_ordsucc^{omega_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ ordsucc x0 ∈ omega
Param nat_p^nat_p : ι → ο
Param lam^Sigma : ι → (ι → ι) → ι
Param If_i^If_i : ο → ι → ι → ι
Param SNo_recipauxset^{SNo_recipauxset} : ι → ι → ι → (ι → ι) → ι
Param Sep^Sep : ι → (ι → ο) → ι
Definition SNoL_pos^SNoL_pos := λ x0 . Sep (SNoL x0) (SNoLt 0)
Known SNo_recipaux_S^{SNo_recipaux_S} : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . nat_p x2 ⟶ SNo_recipaux x0 x1 (ordsucc x2) = lam 2 (λ x4 . If_i (x4 = 0) (binunion (binunion (ap (SNo_recipaux x0 x1 x2) 0) (SNo_recipauxset (ap (SNo_recipaux x0 x1 x2) 0) x0 (SNoR x0) x1)) (SNo_recipauxset (ap (SNo_recipaux x0 x1 x2) 1) x0 (SNoL_pos x0) x1)) (binunion (binunion (ap (SNo_recipaux x0 x1 x2) 1) (SNo_recipauxset (ap (SNo_recipaux x0 x1 x2) 0) x0 (SNoL_pos x0) x1)) (SNo_recipauxset (ap (SNo_recipaux x0 x1 x2) 1) x0 (SNoR x0) x1)))
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Known tuple_2_0_eq^tuple_2_0_eq : ∀ x0 x1 . ap (lam 2 (λ x3 . If_i (x3 = 0) x0 x1)) 0 = x0
Known binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
Known SNo_recipauxset_I^{SNo_recipauxset_I} : ∀ x0 x1 x2 . ∀ x3 : ι → ι . ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x2 ⟶ mul_SNo (add_SNo 1 (mul_SNo (add_SNo x5 (minus_SNo x1)) x4)) (x3 x5) ∈ SNo_recipauxset x0 x1 x2 x3
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep 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)
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 add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
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 SNo_0^SNo_0 : SNo 0
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 x0)
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 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 SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Known SNo_recipaux_lem1^{SNo_recipaux_lem1} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ ∀ x1 : ι → ι . (∀ x2 . x2 ∈ SNoS_ (SNoLev x0) ⟶ SNoLt 0 x2 ⟶ and (SNo (x1 x2)) (mul_SNo x2 (x1 x2) = 1)) ⟶ ∀ x2 . nat_p x2 ⟶ and (∀ x3 . x3 ∈ ap (SNo_recipaux x0 x1 x2) 0 ⟶ and (SNo x3) (SNoLt (mul_SNo x0 x3) 1)) (∀ x3 . x3 ∈ ap (SNo_recipaux x0 x1 x2) 1 ⟶ and (SNo x3) (SNoLt 1 (mul_SNo x0 x3)))
Known SNoLe_tra^SNoLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLe x0 x2
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_nonpos_neg^{mul_SNo_nonpos_neg} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 0 ⟶ SNoLt x1 0 ⟶ SNoLe 0 (mul_SNo x0 x1)
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 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_com_3b_1_2^{add_SNo_com_3b_1_2} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo (add_SNo x0 x1) x2 = add_SNo (add_SNo x0 x2) x1
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 mul_SNo_minus_distrR^{mul_SNo_minus_distrR} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 (minus_SNo x1) = minus_SNo (mul_SNo x0 x1)
Known minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
Known SNoL_SNoCutP_ex^{SNoL_SNoCutP_ex} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ ∀ x2 . x2 ∈ SNoL (SNoCut x0 x1) ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ x0) (SNoLe x2 x4) ⟶ x3) ⟶ x3
Known SNoLt_tra^SNoLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Known nonpos_mul_SNo_Le^{nonpos_mul_SNo_Le} : ∀ x0 x1 x2 . SNo x0 ⟶ SNoLe x0 0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x2 x1 ⟶ SNoLe (mul_SNo x0 x1) (mul_SNo x0 x2)
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
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 SNoLt_minus_pos^{SNoLt_minus_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 x1 ⟶ SNoLt 0 (add_SNo x1 (minus_SNo x0))
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
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 mul_SNo_oneL^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 x0 = x0
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_rotate_3_1^{add_SNo_rotate_3_1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 (add_SNo x1 x2) = add_SNo x2 (add_SNo x0 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
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 mul_SNo_distrR^{mul_SNo_distrR} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo (add_SNo x0 x1) x2 = add_SNo (mul_SNo x0 x2) (mul_SNo x1 x2)
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 mul_SNo_SNoL_interpolate_impred^{mul_SNo_SNoL_interpolate_impred} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 . x2 ∈ SNoL (mul_SNo x0 x1) ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ SNoL x0 ⟶ ∀ x5 . x5 ∈ SNoL x1 ⟶ SNoLe (add_SNo x2 (mul_SNo x4 x5)) (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x5)) ⟶ x3) ⟶ (∀ x4 . x4 ∈ SNoR x0 ⟶ ∀ x5 . x5 ∈ SNoR x1 ⟶ SNoLe (add_SNo x2 (mul_SNo x4 x5)) (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x5)) ⟶ x3) ⟶ x3
Known SNoL_I^SNoL_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x1 x0 ⟶ x1 ∈ SNoL x0
Known tuple_2_1_eq^tuple_2_1_eq : ∀ x0 x1 . ap (lam 2 (λ x3 . If_i (x3 = 0) x0 x1)) 1 = x1
Known SNoLeLt_tra^SNoLeLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known mul_SNo_nonpos_pos^{mul_SNo_nonpos_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 0 ⟶ SNoLt 0 x1 ⟶ SNoLe (mul_SNo x0 x1) 0
Known add_SNo_minus_Lt2b^{add_SNo_minus_Lt2b} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt (add_SNo x2 x1) x0 ⟶ SNoLt x2 (add_SNo x0 (minus_SNo x1))
Known mul_SNo_pos_neg^{mul_SNo_pos_neg} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt 0 x0 ⟶ SNoLt x1 0 ⟶ SNoLt (mul_SNo x0 x1) 0
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known nat_0^nat_0 : nat_p 0
Param Sing^Sing : ι → ι
Known SNo_recipaux_0^{SNo_recipaux_0} : ∀ x0 . ∀ x1 : ι → ι . SNo_recipaux x0 x1 0 = lam 2 (λ x3 . If_i (x3 = 0) (Sing 0) 0)
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Theorem recip_SNo_pos_prop1^{recip_SNo_pos_prop1} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ and (SNo (recip_SNo_pos x0)) (mul_SNo x0 (recip_SNo_pos x0) = 1) (proof)