Proofgold Signed Transaction

Param int^int : ι
Param omega^omega : ι
Param minus_SNo^minus_SNo : ι → ι
Param ordsucc^ordsucc : ι → ι
Known int_SNo_cases^{int_SNo_cases} : ∀ x0 : ι → ο . (∀ x1 . x1 ∈ omega ⟶ x0 x1) ⟶ (∀ x1 . x1 ∈ omega ⟶ x0 (minus_SNo x1)) ⟶ ∀ x1 . x1 ∈ int ⟶ x0 x1
Param nat_p^nat_p : ι → ο
Known nat_inv_impred^{nat_inv_impred} : ∀ x0 : ι → ο . x0 0 ⟶ (∀ x1 . nat_p x1 ⟶ x0 (ordsucc x1)) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Known minus_SNo_0^minus_SNo_0 : minus_SNo 0 = 0
Theorem int_3_cases^int_3_cases : ∀ x0 . x0 ∈ int ⟶ ∀ x1 : ο . (∀ x2 . x2 ∈ omega ⟶ x0 = minus_SNo (ordsucc x2) ⟶ x1) ⟶ (x0 = 0 ⟶ x1) ⟶ (∀ x2 . x2 ∈ omega ⟶ x0 = ordsucc x2 ⟶ x1) ⟶ x1 (proof)
Param SNo^SNo : ι → ο
Param real^real : ι
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
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param ap^ap : ι → ι → ι
Param SNo_recipaux^SNo_recipaux : ι → (ι → ι) → 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 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 mul_SNo^mul_SNo : ι → ι → ι
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Param famunion^famunion : ι → (ι → ι) → ι
Param SNoCutP^SNoCutP : ι → ι → ο
Param SNoCut^SNoCut : ι → ι → ι
Param binunion^binunion : ι → ι → ι
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 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))
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Param exp_SNo_nat^exp_SNo_nat : ι → ι → ι
Known recip_SNo_pos_eps_^{recip_SNo_pos_eps_} : ∀ x0 . nat_p x0 ⟶ recip_SNo_pos (eps_ x0) = exp_SNo_nat 2 x0
Known SNoS_omega_real^{SNoS_omega_real} : SNoS_ omega ⊆ real
Known omega_SNoS_omega^{omega_SNoS_omega} : omega ⊆ SNoS_ omega
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 recip_SNo_pos_2^{recip_SNo_pos_2} : recip_SNo_pos 2 = eps_ 1
Known SNo_eps_SNoS_omega^{SNo_eps_SNoS_omega} : ∀ x0 . x0 ∈ omega ⟶ eps_ x0 ∈ SNoS_ omega
Known nat_1^nat_1 : nat_p 1
Param Sep^Sep : ι → (ι → ο) → ι
Param SNoL^SNoL : ι → ι
Definition SNoL_pos^SNoL_pos := λ x0 . Sep (SNoL x0) (SNoLt 0)
Known pos_real_left_approx_double^{pos_real_left_approx_double} : ∀ x0 . x0 ∈ real ⟶ SNoLt 0 x0 ⟶ (x0 = 2 ⟶ ∀ x1 : ο . x1) ⟶ (∀ x1 . x1 ∈ omega ⟶ x0 = eps_ x1 ⟶ ∀ x2 : ο . x2) ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ SNoL_pos x0) (SNoLt x0 (mul_SNo 2 x2)) ⟶ x1) ⟶ x1
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
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
Param recip_SNo^recip_SNo : ι → ι
Definition div_SNo^div_SNo := λ x0 x1 . mul_SNo x0 (recip_SNo x1)
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))
Known real_SNoCut^real_SNoCut : ∀ x0 . x0 ⊆ real ⟶ ∀ x1 . x1 ⊆ real ⟶ SNoCutP x0 x1 ⟶ (x0 = 0 ⟶ ∀ x2 : ο . x2) ⟶ (x1 = 0 ⟶ ∀ x2 : ο . x2) ⟶ (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ x0) (SNoLt x2 x4) ⟶ x3) ⟶ x3) ⟶ (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ x1) (SNoLt x4 x2) ⟶ x3) ⟶ x3) ⟶ SNoCut x0 x1 ∈ real
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Known nat_0^nat_0 : nat_p 0
Param lam^Sigma : ι → (ι → ι) → ι
Param If_i^If_i : ο → ι → ι → ι
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 tuple_2_0_eq^tuple_2_0_eq : ∀ x0 x1 . ap (lam 2 (λ x3 . If_i (x3 = 0) x0 x1)) 0 = x0
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Known div_SNo_pos_LtR^{div_SNo_pos_LtR} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt 0 x1 ⟶ SNoLt (mul_SNo x2 x1) x0 ⟶ SNoLt x2 (div_SNo x0 x1)
Known SNo_add_SNo_4^{SNo_add_SNo_4} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo (add_SNo x0 (add_SNo x1 (add_SNo x2 x3)))
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
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 SNo_0^SNo_0 : SNo 0
Known SNo_add_SNo_3^{SNo_add_SNo_3} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo (add_SNo x0 (add_SNo x1 x2))
Known add_SNo_minus_Lt2b3^{add_SNo_minus_Lt2b3} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLt (add_SNo x3 x2) (add_SNo x0 x1) ⟶ SNoLt x3 (add_SNo x0 (add_SNo x1 (minus_SNo x2)))
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Known mul_SNo_oneR^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = x0
Known mul_SNo_Lt^mul_SNo_Lt : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLt x2 x0 ⟶ SNoLt x3 x1 ⟶ SNoLt (add_SNo (mul_SNo x2 x1) (mul_SNo x0 x3)) (add_SNo (mul_SNo x0 x1) (mul_SNo x2 x3))
Known SNo_1^SNo_1 : SNo 1
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
Known div_SNo_pos_LtR'^{div_SNo_pos_LtR} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt 0 x1 ⟶ SNoLt x2 (div_SNo x0 x1) ⟶ SNoLt (mul_SNo x2 x1) x0
Known mul_SNo_oneL^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 x0 = x0
Known SNo_recip_SNo^{SNo_recip_SNo} : ∀ x0 . SNo x0 ⟶ SNo (recip_SNo x0)
Known recip_SNo_poscase^{recip_SNo_poscase} : ∀ x0 . SNoLt 0 x0 ⟶ recip_SNo x0 = recip_SNo_pos x0
Known real_SNo^real_SNo : ∀ x0 . x0 ∈ real ⟶ SNo x0
Known div_SNo_pos_LtL^{div_SNo_pos_LtL} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt 0 x1 ⟶ SNoLt x0 (mul_SNo x2 x1) ⟶ SNoLt (div_SNo x0 x1) x2
Known add_SNo_minus_Lt1b3^{add_SNo_minus_Lt1b3} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x3 x2) ⟶ SNoLt (add_SNo x0 (add_SNo x1 (minus_SNo x2))) x3
Known div_SNo_pos_LtL'^{div_SNo_pos_LtL} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt 0 x1 ⟶ SNoLt (div_SNo x0 x1) x2 ⟶ SNoLt x0 (mul_SNo x2 x1)
Known mul_div_SNo_nonzero_eq^{mul_div_SNo_nonzero_eq} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ (x1 = 0 ⟶ ∀ x3 : ο . x3) ⟶ x0 = mul_SNo x1 x2 ⟶ div_SNo x0 x1 = x2
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Known add_SNo_com_3_0_1^{add_SNo_com_3_0_1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 (add_SNo x1 x2) = add_SNo x1 (add_SNo x0 x2)
Known add_SNo_1_1_2^{add_SNo_1_1_2} : add_SNo 1 1 = 2
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 minus_add_SNo_distr_3^{minus_add_SNo_distr_3} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ minus_SNo (add_SNo x0 (add_SNo x1 x2)) = add_SNo (minus_SNo x0) (add_SNo (minus_SNo x1) (minus_SNo x2))
Known mul_SNo_com_3_0_1^{mul_SNo_com_3_0_1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 (mul_SNo x1 x2) = mul_SNo x1 (mul_SNo x0 x2)
Known add_SNo_assoc_4^{add_SNo_assoc_4} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ add_SNo x0 (add_SNo x1 (add_SNo x2 x3)) = add_SNo (add_SNo x0 (add_SNo x1 x2)) x3
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 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_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 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_minus_minus^{mul_SNo_minus_minus} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo (minus_SNo x0) (minus_SNo x1) = mul_SNo x0 x1
Known recip_SNo_pos_invR^{recip_SNo_pos_invR} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ mul_SNo x0 (recip_SNo_pos x0) = 1
Known SNo_2^SNo_2 : SNo 2
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 famunionE_impred^{famunionE_impred} : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ famunion x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ x0 ⟶ x2 ∈ x1 x4 ⟶ x3) ⟶ x3
Known omega_ordsucc^{omega_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ ordsucc x0 ∈ omega
Param SNo_recipauxset^{SNo_recipauxset} : ι → ι → ι → (ι → ι) → ι
Param SNoR^SNoR : ι → ι
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 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 famunionI^famunionI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ x0 ⟶ x3 ∈ x1 x2 ⟶ x3 ∈ famunion x0 x1
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 binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Known SNo_recip_SNo_pos^{SNo_recip_SNo_pos} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ SNo (recip_SNo_pos x0)
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 SNoLev_ordinal^{SNoLev_ordinal} : ∀ x0 . SNo x0 ⟶ ordinal (SNoLev x0)
Known 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)
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 SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Known real_0^real_0 : 0 ∈ real
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known binunionE'^binunionE : ∀ x0 x1 x2 . ∀ x3 : ο . (x2 ∈ x0 ⟶ x3) ⟶ (x2 ∈ x1 ⟶ x3) ⟶ x2 ∈ binunion x0 x1 ⟶ x3
Known SNo_recipauxset_E^{SNo_recipauxset_E} : ∀ x0 x1 x2 . ∀ x3 : ι → ι . ∀ x4 . x4 ∈ SNo_recipauxset x0 x1 x2 x3 ⟶ ∀ x5 : ο . (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x2 ⟶ x4 = mul_SNo (add_SNo 1 (mul_SNo (add_SNo x7 (minus_SNo x1)) x6)) (x3 x7) ⟶ x5) ⟶ x5
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 SNoS_I2^SNoS_I2 : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x0 ∈ SNoLev x1 ⟶ x0 ∈ SNoS_ (SNoLev x1)
Known SNoLt_tra^SNoLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known real_mul_SNo^real_mul_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ mul_SNo x0 x1 ∈ real
Known real_add_SNo^real_add_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ add_SNo x0 x1 ∈ real
Known real_1^real_1 : 1 ∈ real
Known real_minus_SNo^{real_minus_SNo} : ∀ x0 . x0 ∈ real ⟶ minus_SNo x0 ∈ real
Known SNoS_I^SNoS_I : ∀ x0 . ordinal x0 ⟶ ∀ x1 x2 . x2 ∈ x0 ⟶ SNo_ x2 x1 ⟶ x1 ∈ SNoS_ x0
Known omega_ordinal^{omega_ordinal} : ordinal omega
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Known nat_p_trans^nat_p_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ nat_p x1
Theorem real_recip_SNo_lem1^{real_recip_SNo_lem1} : ∀ x0 . SNo x0 ⟶ x0 ∈ real ⟶ SNoLt 0 x0 ⟶ and (recip_SNo_pos x0 ∈ real) (∀ x1 . nat_p x1 ⟶ and (ap (SNo_recipaux x0 recip_SNo_pos x1) 0 ⊆ real) (ap (SNo_recipaux x0 recip_SNo_pos x1) 1 ⊆ real)) (proof)
Theorem real_recip_SNo_pos^{real_recip_SNo_pos} : ∀ x0 . x0 ∈ real ⟶ SNoLt 0 x0 ⟶ recip_SNo_pos x0 ∈ real (proof)
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 recip_SNo_negcase^{recip_SNo_negcase} : ∀ x0 . SNo x0 ⟶ SNoLt x0 0 ⟶ recip_SNo x0 = minus_SNo (recip_SNo_pos (minus_SNo x0))
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 recip_SNo_0^recip_SNo_0 : recip_SNo 0 = 0
Theorem real_recip_SNo^{real_recip_SNo} : ∀ x0 . x0 ∈ real ⟶ recip_SNo x0 ∈ real (proof)
Theorem real_div_SNo^real_div_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ div_SNo x0 x1 ∈ real (proof)