Proofgold Signed Transaction

Param nat_p^nat_p : ι → ο
Param mul_nat^mul_nat : ι → ι → ι
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 mul_nat_1R^mul_nat_1R : ∀ x0 . mul_nat x0 1 = x0
Known In_0_2^In_0_2 : 0 ∈ 2
Param add_nat^add_nat : ι → ι → ι
Known mul_nat_SR^mul_nat_SR : ∀ x0 x1 . nat_p x1 ⟶ mul_nat x0 (ordsucc x1) = add_nat x0 (mul_nat x0 x1)
Known nat_ordsucc^nat_ordsucc : ∀ x0 . nat_p x0 ⟶ nat_p (ordsucc x0)
Known add_nat_SL^add_nat_SL : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ add_nat (ordsucc x0) x1 = ordsucc (add_nat x0 x1)
Known nat_1^nat_1 : nat_p 1
Known nat_0^nat_0 : nat_p 0
Known add_nat_0L^add_nat_0L : ∀ x0 . nat_p x0 ⟶ add_nat 0 x0 = x0
Known nat_ordsucc_in_ordsucc^{nat_ordsucc_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ∈ ordsucc x0
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known ordsuccI1^ordsuccI1 : ∀ x0 . x0 ⊆ ordsucc x0
Known mul_nat_p^mul_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (mul_nat x0 x1)
Known nat_2^nat_2 : nat_p 2
Theorem nat_In_double_S : ∀ x0 . nat_p x0 ⟶ x0 ∈ mul_nat 2 (ordsucc x0) (proof)
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param omega^omega : ι
Definition odd_nat^odd_nat := λ x0 . and (x0 ∈ omega) (∀ x1 . x1 ∈ omega ⟶ x0 = mul_nat 2 x1 ⟶ ∀ x2 : ο . x2)
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Definition False^False := ∀ x0 : ο . x0
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known nat_inv^nat_inv : ∀ x0 . nat_p x0 ⟶ or (x0 = 0) (∀ x1 : ο . (∀ x2 . and (nat_p x2) (x0 = ordsucc x2) ⟶ x1) ⟶ x1)
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Known neq_1_0^neq_1_0 : 1 = 0 ⟶ ∀ x0 : ο . x0
Known mul_nat_0R^mul_nat_0R : ∀ x0 . mul_nat x0 0 = 0
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 nat_0_in_ordsucc^{nat_0_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ 0 ∈ ordsucc x0
Theorem odd_nat_1^odd_nat_1 : odd_nat 1 (proof)
Definition even_nat^even_nat := λ x0 . and (x0 ∈ omega) (∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (x0 = mul_nat 2 x2) ⟶ x1) ⟶ x1)
Theorem even_nat_not_odd_nat^{even_nat_not_odd_nat} : ∀ x0 . even_nat x0 ⟶ not (odd_nat x0) (proof)
Known omega_ordsucc^{omega_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ ordsucc x0 ∈ omega
Theorem even_nat_S_S^even_nat_S_S : ∀ x0 . even_nat x0 ⟶ even_nat (ordsucc (ordsucc x0)) (proof)
Known neq_ordsucc_0^{neq_ordsucc_0} : ∀ x0 . ordsucc x0 = 0 ⟶ ∀ x1 : ο . x1
Known ordsucc_inj^ordsucc_inj : ∀ x0 x1 . ordsucc x0 = ordsucc x1 ⟶ x0 = x1
Theorem even_nat_S_S_inv^{even_nat_S_S_inv} : ∀ x0 . nat_p x0 ⟶ even_nat (ordsucc (ordsucc x0)) ⟶ even_nat x0 (proof)
Theorem even_nat_double^{even_nat_double} : ∀ x0 . nat_p x0 ⟶ even_nat (mul_nat 2 x0) (proof)
Param exactly1of2^exactly1of2 : ο → ο → ο
Known exactly1of2_I1^{exactly1of2_I1} : ∀ x0 x1 : ο . x0 ⟶ not x1 ⟶ exactly1of2 x0 x1
Known exactly1of2_E^{exactly1of2_E} : ∀ x0 x1 : ο . exactly1of2 x0 x1 ⟶ ∀ x2 : ο . (x0 ⟶ not x1 ⟶ x2) ⟶ (not x0 ⟶ x1 ⟶ x2) ⟶ x2
Known exactly1of2_I2^{exactly1of2_I2} : ∀ x0 x1 : ο . not x0 ⟶ x1 ⟶ exactly1of2 x0 x1
Theorem even_nat_xor_S^{even_nat_xor_S} : ∀ x0 . nat_p x0 ⟶ exactly1of2 (even_nat x0) (even_nat (ordsucc x0)) (proof)
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Theorem not_even_nat_S_double : ∀ x0 . nat_p x0 ⟶ not (even_nat (ordsucc (mul_nat 2 x0))) (proof)
Definition inj^inj := λ x0 x1 . λ x2 : ι → ι . and (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1) (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4)
Definition atleastp^atleastp := λ x0 x1 . ∀ x2 : ο . (∀ x3 : ι → ι . inj x0 x1 x3 ⟶ x2) ⟶ x2
Known form100_63_Cantor^{form100_63_injCantor} : ∀ x0 . ∀ x1 : ι → ι . not (inj (prim4 x0) x0 x1)
Theorem Cantor_atleastp : ∀ x0 . not (atleastp (prim4 x0) x0) (proof)
Param SNo^SNo : ι → ο
Param mul_SNo^mul_SNo : ι → ι → ι
Known mul_SNo_oneL^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 x0 = x0
Param eps_^eps_ : ι → ι
Known eps_1_half_eq3^{eps_1_half_eq3} : mul_SNo (eps_ 1) 2 = 1
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 SNo_eps_1^SNo_eps_1 : SNo (eps_ 1)
Known SNo_2^SNo_2 : SNo 2
Known mul_nat_mul_SNo^{mul_nat_mul_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_nat x0 x1 = mul_SNo x0 x1
Known omega_SNo^omega_SNo : ∀ x0 . x0 ∈ omega ⟶ SNo x0
Theorem double_omega_cancel : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_nat 2 x0 = mul_nat 2 x1 ⟶ x0 = x1 (proof)
Param SNoLe^SNoLe : ι → ι → ο
Known mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
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_0^SNo_0 : SNo 0
Theorem mul_SNo_nonneg_nonneg^{mul_SNo_nonneg_nonneg} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe 0 x0 ⟶ SNoLe 0 x1 ⟶ SNoLe 0 (mul_SNo x0 x1) (proof)
Param binunion^binunion : ι → ι → ι
Param minus_SNo^minus_SNo : ι → ι
Definition int^int := binunion omega (prim5 omega minus_SNo)
Definition diadic_rational_p^{diadic_rational_p} := λ x0 . ∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (∀ x3 : ο . (∀ x4 . and (x4 ∈ int) (x0 = mul_SNo (eps_ x2) x4) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1
Param SNoLt^SNoLt : ι → ι → ο
Param ordinal^ordinal : ι → ο
Param SNoS_^SNoS_ : ι → ι
Param SNoLev^SNoLev : ι → ι
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 diadic_rational_p_SNoS_omega^{diadic_rational_p_SNoS_omega} : ∀ x0 . diadic_rational_p x0 ⟶ x0 ∈ SNoS_ omega
Known binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Known SNo_eps_^SNo_eps_ : ∀ x0 . x0 ∈ omega ⟶ SNo (eps_ x0)
Known SNoLe_ref^SNoLe_ref : ∀ x0 . SNoLe x0 x0
Param add_SNo^add_SNo : ι → ι → ι
Known ordinal_ordsucc_SNo_eq^{ordinal_ordsucc_SNo_eq} : ∀ x0 . ordinal x0 ⟶ ordsucc x0 = add_SNo 1 x0
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
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 SNo_1^SNo_1 : SNo 1
Known nat_p_SNo^nat_p_SNo : ∀ x0 . nat_p x0 ⟶ SNo x0
Known mul_SNo_oneR^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = x0
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
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 SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Known SNo_eps_pos^SNo_eps_pos : ∀ x0 . x0 ∈ omega ⟶ SNoLt 0 (eps_ x0)
Known ordinal_Subq_SNoLe^{ordinal_Subq_SNoLe} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ x0 ⊆ x1 ⟶ SNoLe x0 x1
Known ordinal_Empty^{ordinal_Empty} : ordinal 0
Known Subq_Empty^Subq_Empty : ∀ x0 . 0 ⊆ x0
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 minus_SNo_0^minus_SNo_0 : minus_SNo 0 = 0
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_Le_contra^{minus_SNo_Le_contra} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ SNoLe (minus_SNo x1) (minus_SNo x0)
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
Theorem diadic_rational_p_pos_eps_between : ∀ x0 . diadic_rational_p x0 ⟶ SNoLt 0 x0 ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (SNoLe (eps_ x2) x0) ⟶ x1) ⟶ x1 (proof)
Param abs_SNo^abs_SNo : ι → ι
Param binintersect^binintersect : ι → ι → ι
Param SetAdjoin^SetAdjoin : ι → ι → ι
Param Sing^Sing : ι → ι
Definition SNoElts_^SNoElts_ := λ x0 . binunion x0 {SetAdjoin x1 (Sing 1)|x1 ∈ x0}
Param iff^iff : ο → ο → ο
Definition PNoEq_^PNoEq_ := λ x0 . λ x1 x2 : ι → ο . ∀ x3 . x3 ∈ x0 ⟶ iff (x1 x3) (x2 x3)
Definition SNoEq_^SNoEq_ := λ x0 x1 x2 . PNoEq_ x0 (λ x3 . x3 ∈ x1) (λ x3 . x3 ∈ x2)
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 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 add_SNo_Lt1^add_SNo_Lt1 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x2 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x1)
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Known SNoLtI2^SNoLtI2 : ∀ x0 x1 . SNoLev x0 ∈ SNoLev x1 ⟶ SNoEq_ (SNoLev x0) x0 x1 ⟶ SNoLev x0 ∈ x1 ⟶ SNoLt x0 x1
Known restr_SNoLev^restr_SNoLev : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoLev x0 ⟶ SNoLev (binintersect x0 (SNoElts_ x1)) = x1
Known SNoEq_tra_^SNoEq_tra_ : ∀ x0 x1 x2 x3 . SNoEq_ x0 x1 x2 ⟶ SNoEq_ x0 x2 x3 ⟶ SNoEq_ x0 x1 x3
Known iffI^iffI : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ iff x0 x1
Known binintersectI^{binintersectI} : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ x1 ⟶ x2 ∈ binintersect x0 x1
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Known nat_trans^nat_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Known binintersectE1^{binintersectE1} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ x2 ∈ x0
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ 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 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 SNoLtI3^SNoLtI3 : ∀ x0 x1 . SNoLev x1 ∈ SNoLev x0 ⟶ SNoEq_ (SNoLev x1) x0 x1 ⟶ nIn (SNoLev x1) x0 ⟶ SNoLt x0 x1
Known SNoEq_sym_^SNoEq_sym_ : ∀ x0 x1 x2 . SNoEq_ x0 x1 x2 ⟶ SNoEq_ x0 x2 x1
Known Subq_ref^Subq_ref : ∀ x0 . x0 ⊆ x0
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Known Subq_tra^Subq_tra : ∀ x0 x1 x2 . x0 ⊆ x1 ⟶ x1 ⊆ x2 ⟶ x0 ⊆ x2
Known restr_SNoEq^restr_SNoEq : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoLev x0 ⟶ SNoEq_ x1 (binintersect x0 (SNoElts_ x1)) x0
Known SNoS_omega_diadic_rational_p^{SNoS_omega_diadic_rational_p} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ diadic_rational_p x0
Known add_SNo_Lt1_cancel^{add_SNo_Lt1_cancel} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x1) ⟶ SNoLt x0 x2
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Known add_SNo_minus_R2'^{add_SNo_minus_R2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo (add_SNo x0 (minus_SNo x1)) x1 = x0
Known add_SNo_SNoS_omega^{add_SNo_SNoS_omega} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ ∀ x1 . x1 ∈ SNoS_ omega ⟶ add_SNo x0 x1 ∈ SNoS_ omega
Known minus_SNo_SNoS_omega^{minus_SNo_SNoS_omega} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ minus_SNo x0 ∈ SNoS_ omega
Known SNoS_I^SNoS_I : ∀ x0 . ordinal x0 ⟶ ∀ x1 x2 . x2 ∈ x0 ⟶ SNo_ x2 x1 ⟶ x1 ∈ SNoS_ x0
Known restr_SNo_^restr_SNo_ : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoLev x0 ⟶ SNo_ x1 (binintersect x0 (SNoElts_ x1))
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 nonneg_abs_SNo^{nonneg_abs_SNo} : ∀ x0 . SNoLe 0 x0 ⟶ abs_SNo x0 = x0
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Known SNoLev_uniq^SNoLev_uniq : ∀ x0 x1 x2 . ordinal x1 ⟶ ordinal x2 ⟶ SNo_ x1 x0 ⟶ SNo_ x2 x0 ⟶ x1 = x2
Known SNoLev_ordinal^{SNoLev_ordinal} : ∀ x0 . SNo x0 ⟶ ordinal (SNoLev x0)
Known SNoLev_^SNoLev_ : ∀ x0 . SNo x0 ⟶ SNo_ (SNoLev x0) x0
Known SNo_SNo^SNo_SNo : ∀ x0 . ordinal x0 ⟶ ∀ x1 . SNo_ x0 x1 ⟶ SNo x1
Theorem 8fb40.. : ∀ x0 . SNo_ omega x0 ⟶ ∀ x1 . x1 ∈ SNoS_ omega ⟶ (∀ x2 . x2 ∈ omega ⟶ SNoLt (abs_SNo (add_SNo x1 (minus_SNo x0))) (eps_ x2)) ⟶ or (and (nIn (SNoLev x1) x0) (∀ x2 . x2 ∈ omega ⟶ SNoLev x1 ∈ x2 ⟶ x2 ∈ x0)) (and (SNoLev x1 ∈ x0) (∀ x2 . x2 ∈ omega ⟶ SNoLev x1 ∈ x2 ⟶ nIn x2 x0)) (proof)
Known minus_SNo_In^minus_SNo_In : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoLev x0 ⟶ x1 ∈ x0 ⟶ nIn x1 (minus_SNo x0)
Known SNo_omega^SNo_omega : SNo omega
Known ordinal_SNoLev^{ordinal_SNoLev} : ∀ x0 . ordinal x0 ⟶ SNoLev x0 = x0
Theorem nat_nIn_minus_SNo_omega : ∀ x0 . x0 ∈ omega ⟶ nIn x0 (minus_SNo omega) (proof)
Param real^real : ι
Param Sep^Sep : ι → (ι → ο) → ι
Param ReplSep^ReplSep : ι → (ι → ο) → (ι → ι) → ι
Definition PSNo^PSNo := λ x0 . λ x1 : ι → ο . binunion (Sep x0 x1) {SetAdjoin x2 (Sing 1)|x2 ∈ x0,not (x1 x2)}
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 ordsucc_omega_ordinal^{ordsucc_omega_ordinal} : ordinal (ordsucc omega)
Known nat_3^nat_3 : nat_p 3
Known SNoLev_uniq2^SNoLev_uniq2 : ∀ x0 . ordinal x0 ⟶ ∀ x1 . SNo_ x0 x1 ⟶ SNoLev x1 = x0
Known SNo_PSNo^SNo_PSNo : ∀ x0 . ordinal x0 ⟶ ∀ x1 : ι → ο . SNo_ x0 (PSNo x0 x1)
Known PowerE^PowerE : ∀ x0 x1 . x1 ∈ prim4 x0 ⟶ x1 ⊆ x0
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known ReplSepI^ReplSepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 : ι → ι . ∀ x3 . x3 ∈ x0 ⟶ x1 x3 ⟶ x2 x3 ∈ ReplSep x0 x1 x2
Known SepE2^SepE2 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x1 x2
Known ReplSepE_impred^{ReplSepE_impred} : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 : ι → ι . ∀ x3 . x3 ∈ ReplSep x0 x1 x2 ⟶ ∀ x4 : ο . (∀ x5 . x5 ∈ x0 ⟶ x1 x5 ⟶ x3 = x2 x5 ⟶ x4) ⟶ x4
Known tagged_not_ordinal^{tagged_not_ordinal} : ∀ x0 . not (ordinal (SetAdjoin x0 (Sing 1)))
Known ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Theorem atleastp_Power_omega_real : atleastp (prim4 omega) real (proof)
Param equip^equip : ι → ι → ο
Known atleastp_tra^atleastp_tra : ∀ x0 x1 x2 . atleastp x0 x1 ⟶ atleastp x1 x2 ⟶ atleastp x0 x2
Known equip_atleastp^{equip_atleastp} : ∀ x0 x1 . equip x0 x1 ⟶ atleastp x0 x1
Known equip_sym^equip_sym : ∀ x0 x1 . equip x0 x1 ⟶ equip x1 x0
Theorem real_uncountable : not (equip omega real) (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_SNo^real_SNo : ∀ x0 . x0 ∈ real ⟶ SNo x0
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Known real_0^real_0 : 0 ∈ real
Known real_minus_SNo^{real_minus_SNo} : ∀ x0 . x0 ∈ real ⟶ minus_SNo x0 ∈ real
Known add_SNo_minus_SNo_rinv^{add_SNo_minus_SNo_rinv} : ∀ x0 . SNo x0 ⟶ add_SNo x0 (minus_SNo x0) = 0
Known real_mul_SNo^real_mul_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ mul_SNo x0 x1 ∈ real
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
Known real_1^real_1 : 1 ∈ real
Known 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
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
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 SNoLe_antisym^{SNoLe_antisym} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x0 ⟶ x0 = x1
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 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)
Theorem explicit_OrderedField_real : explicit_OrderedField real 0 1 add_SNo mul_SNo SNoLe (proof)
Param explicit_Reals^{explicit_Reals} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ο
Param natOfOrderedField_p^{natOfOrderedField_p} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ι → ο
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
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 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 SNoLtLe_tra^SNoLtLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLt x0 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
Known add_nat_add_SNo^{add_nat_add_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ add_nat x0 x1 = add_SNo x0 x1
Known add_nat_p^add_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (add_nat x0 x1)
Known ordinal_Hered^{ordinal_Hered} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordinal x1
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)
Theorem explicit_Reals_real : explicit_Reals real 0 1 add_SNo mul_SNo SNoLe (proof)
Definition c3146.. := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . λ x5 : ι → ι → ο . λ x6 . ∀ x7 : ο . (∀ x8 . (∀ x9 : ο . (∀ x10 . and (and (natOfOrderedField_p x0 x1 x2 x3 x4 x5 x8) (natOfOrderedField_p x0 x1 x2 x3 x4 x5 x10)) (x4 x10 x6 = x8) ⟶ x9) ⟶ x9) ⟶ x7) ⟶ x7
Known Sep_Subq^Sep_Subq : ∀ x0 . ∀ x1 : ι → ο . Sep x0 x1 ⊆ x0
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known explicit_Field_zero_multL^{explicit_Field_zero_multL} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Field x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x4 x1 x5 = x1
Known explicit_OrderedField_E^{explicit_OrderedField_E} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 : ο . (explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ explicit_Field x0 x1 x2 x3 x4 ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x5 x7 x8 ⟶ x5 x8 x9 ⟶ x5 x7 x9) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ iff (and (x5 x7 x8) (x5 x8 x7)) (x7 = x8)) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ or (x5 x7 x8) (x5 x8 x7)) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x5 x7 x8 ⟶ x5 (x3 x7 x9) (x3 x8 x9)) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x5 x1 x7 ⟶ x5 x1 x8 ⟶ x5 x1 (x4 x7 x8)) ⟶ x6) ⟶ explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ x6
Theorem ea079.. : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ Sep x0 (c3146.. x0 x1 x2 x3 x4 x5) = x0 (proof)
Theorem 426ae.. : (∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . explicit_Reals x0 x1 x2 x3 x4 x5 ⟶ equip omega (Sep x0 (c3146.. x0 x1 x2 x3 x4 x5))) ⟶ ∀ x0 : ο . x0 (proof)