Proofgold Signed Transaction

Definition False^False := ∀ x0 : ο . x0
Known In_ind^In_ind : ∀ x0 : ι → ο . (∀ x1 . (∀ x2 . x2 ∈ x1 ⟶ x0 x2) ⟶ x0 x1) ⟶ ∀ x1 . x0 x1
Theorem In_no4cycle^In_no4cycle : ∀ x0 x1 x2 x3 . x0 ∈ x1 ⟶ x1 ∈ x2 ⟶ x2 ∈ x3 ⟶ x3 ∈ x0 ⟶ False (proof)
Param SNo^SNo : ι → ο
Param mul_SNo^mul_SNo : ι → ι → ι
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
Known SNo_0^SNo_0 : SNo 0
Known mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Theorem mul_SNo_zeroL^{mul_SNo_zeroL} : ∀ x0 . SNo x0 ⟶ mul_SNo 0 x0 = 0 (proof)
Param ordsucc^ordsucc : ι → ι
Known SNo_1^SNo_1 : SNo 1
Known mul_SNo_oneR^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = x0
Theorem mul_SNo_oneL^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 x0 = x0 (proof)
Param SNoLt^SNoLt : ι → ι → ο
Param add_SNo^add_SNo : ι → ι → ι
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 add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Theorem 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) (proof)
Param SNoLe^SNoLe : ι → ι → ο
Known mul_SNo_Le^mul_SNo_Le : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLe x2 x0 ⟶ SNoLe x3 x1 ⟶ SNoLe (add_SNo (mul_SNo x2 x1) (mul_SNo x0 x3)) (add_SNo (mul_SNo x0 x1) (mul_SNo x2 x3))
Theorem 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) (proof)
Theorem neg_mul_SNo_Lt^{neg_mul_SNo_Lt} : ∀ x0 x1 x2 . SNo x0 ⟶ SNoLt x0 0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x2 x1 ⟶ SNoLt (mul_SNo x0 x1) (mul_SNo x0 x2) (proof)
Theorem 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) (proof)
Param If_i^If_i : ο → ι → ι → ι
Param minus_SNo^minus_SNo : ι → ι
Definition abs_SNo^abs_SNo := λ x0 . If_i (SNoLe 0 x0) x0 (minus_SNo x0)
Known If_i_1^If_i_1 : ∀ x0 : ο . ∀ x1 x2 . x0 ⟶ If_i x0 x1 x2 = x1
Theorem nonneg_abs_SNo^{nonneg_abs_SNo} : ∀ x0 . SNoLe 0 x0 ⟶ abs_SNo x0 = x0 (proof)
Definition not^not := λ x0 : ο . x0 ⟶ False
Known If_i_0^If_i_0 : ∀ x0 : ο . ∀ x1 x2 . not x0 ⟶ If_i x0 x1 x2 = x2
Theorem not_nonneg_abs_SNo^{not_nonneg_abs_SNo} : ∀ x0 . not (SNoLe 0 x0) ⟶ abs_SNo x0 = minus_SNo x0 (proof)
Known SNoLe_ref^SNoLe_ref : ∀ x0 . SNoLe x0 x0
Theorem abs_SNo_0^abs_SNo_0 : abs_SNo 0 = 0 (proof)
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Theorem pos_abs_SNo^pos_abs_SNo : ∀ x0 . SNoLt 0 x0 ⟶ abs_SNo x0 = x0 (proof)
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
Theorem neg_abs_SNo^neg_abs_SNo : ∀ x0 . SNo x0 ⟶ SNoLt x0 0 ⟶ abs_SNo x0 = minus_SNo x0 (proof)
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Theorem SNo_abs_SNo^SNo_abs_SNo : ∀ x0 . SNo x0 ⟶ SNo (abs_SNo x0) (proof)
Param SNoLev^SNoLev : ι → ι
Known minus_SNo_Lev^{minus_SNo_Lev} : ∀ x0 . SNo x0 ⟶ SNoLev (minus_SNo x0) = SNoLev x0
Theorem abs_SNo_Lev^abs_SNo_Lev : ∀ x0 . SNo x0 ⟶ SNoLev (abs_SNo x0) = SNoLev x0 (proof)
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 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 minus_SNo_0^minus_SNo_0 : minus_SNo 0 = 0
Known minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
Known SNoLe_antisym^{SNoLe_antisym} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x0 ⟶ 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)
Theorem abs_SNo_minus^{abs_SNo_minus} : ∀ x0 . SNo x0 ⟶ abs_SNo (minus_SNo x0) = abs_SNo x0 (proof)
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_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Theorem 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)) (proof)
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 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 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_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 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_Le3^add_SNo_Le3 : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLe x0 x2 ⟶ SNoLe x1 x3 ⟶ SNoLe (add_SNo x0 x1) (add_SNo x2 x3)
Theorem SNo_triangle^SNo_triangle : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe (abs_SNo (add_SNo x0 x1)) (add_SNo (abs_SNo x0) (abs_SNo x1)) (proof)
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 add_SNo_minus_L2^{add_SNo_minus_L2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo (minus_SNo x0) (add_SNo x0 x1) = x1
Theorem 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)))) (proof)
Param nat_p^nat_p : ι → ο
Param SNoS_^SNoS_ : ι → ι
Param binunion^binunion : ι → ι → ι
Param Sing^Sing : ι → ι
Param SetAdjoin^SetAdjoin : ι → ι → ι
Definition eps_^eps_ := λ x0 . binunion (Sing 0) {SetAdjoin (ordsucc x1) (Sing 1)|x1 ∈ x0}
Param omega^omega : ι
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 nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known nat_ordsucc^nat_ordsucc : ∀ x0 . nat_p x0 ⟶ nat_p (ordsucc 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_eps_^SNo_eps_ : ∀ x0 . x0 ∈ omega ⟶ SNo (eps_ x0)
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Known SNoLev_eps_^SNoLev_eps_ : ∀ x0 . x0 ∈ omega ⟶ SNoLev (eps_ x0) = ordsucc x0
Param binintersect^binintersect : ι → ι → ι
Param SNoEq_^SNoEq_ : ι → ι → ι → ο
Known SNoLtE^SNoLtE : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 x1 ⟶ ∀ x2 : ο . (∀ x3 . SNo x3 ⟶ SNoLev x3 ∈ binintersect (SNoLev x0) (SNoLev x1) ⟶ SNoEq_ (SNoLev x3) x3 x0 ⟶ SNoEq_ (SNoLev x3) x3 x1 ⟶ SNoLt x0 x3 ⟶ SNoLt x3 x1 ⟶ nIn (SNoLev x3) x0 ⟶ SNoLev x3 ∈ x1 ⟶ x2) ⟶ (SNoLev x0 ∈ SNoLev x1 ⟶ SNoEq_ (SNoLev x0) x0 x1 ⟶ SNoLev x0 ∈ x1 ⟶ x2) ⟶ (SNoLev x1 ∈ SNoLev x0 ⟶ SNoEq_ (SNoLev x1) x0 x1 ⟶ nIn (SNoLev x1) x0 ⟶ x2) ⟶ x2
Known SNoLev_0_eq_0^{SNoLev_0_eq_0} : ∀ x0 . SNo x0 ⟶ SNoLev x0 = 0 ⟶ x0 = 0
Known eps_ordinal_In_eq_0^{eps_ordinal_In_eq_0} : ∀ x0 x1 . ordinal x1 ⟶ x1 ∈ eps_ x0 ⟶ x1 = 0
Known SNoLev_ordinal^{SNoLev_ordinal} : ∀ x0 . SNo x0 ⟶ ordinal (SNoLev x0)
Known In_no2cycle^In_no2cycle : ∀ x0 x1 . x0 ∈ x1 ⟶ x1 ∈ x0 ⟶ False
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Theorem SNo_pos_eps_Lt^{SNo_pos_eps_Lt} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ SNoS_ (ordsucc x0) ⟶ SNoLt 0 x1 ⟶ SNoLt (eps_ x0) x1 (proof)
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Theorem SNo_pos_eps_Le^{SNo_pos_eps_Le} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ SNoS_ (ordsucc (ordsucc x0)) ⟶ SNoLt 0 x1 ⟶ SNoLe (eps_ x0) x1 (proof)
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ 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 andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known nat_ordsucc_trans^{nat_ordsucc_trans} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ ordsucc x0 ⟶ x1 ⊆ x0
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Known SNo_eq^SNo_eq : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x0 = SNoLev x1 ⟶ SNoEq_ (SNoLev x0) x0 x1 ⟶ x0 = x1
Known SNoEq_tra_^SNoEq_tra_ : ∀ x0 x1 x2 x3 . SNoEq_ x0 x1 x2 ⟶ SNoEq_ x0 x2 x3 ⟶ SNoEq_ x0 x1 x3
Known SNoEq_sym_^SNoEq_sym_ : ∀ x0 x1 x2 . SNoEq_ x0 x1 x2 ⟶ SNoEq_ x0 x2 x1
Param iff^iff : ο → ο → ο
Known SNoEq_I^SNoEq_I : ∀ x0 x1 x2 . (∀ x3 . x3 ∈ x0 ⟶ iff (x3 ∈ x1) (x3 ∈ x2)) ⟶ SNoEq_ x0 x1 x2
Known iffI^iffI : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ iff x0 x1
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Known nat_p_trans^nat_p_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ nat_p x1
Theorem eps_SNo_eq^eps_SNo_eq : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ SNoS_ (ordsucc x0) ⟶ SNoLt 0 x1 ⟶ SNoEq_ (SNoLev x1) (eps_ x0) x1 ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ x0) (x1 = eps_ x3) ⟶ x2) ⟶ x2 (proof)
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 and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = 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 SNo_eps_pos^SNo_eps_pos : ∀ x0 . x0 ∈ omega ⟶ SNoLt 0 (eps_ x0)
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Theorem eps_SNoCutP^eps_SNoCutP : ∀ x0 . x0 ∈ omega ⟶ SNoCutP (Sing 0) (prim5 x0 eps_) (proof)
Param SNoCut^SNoCut : ι → ι → ι
Param famunion^famunion : ι → (ι → ι) → ι
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 binintersectE2^{binintersectE2} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ x2 ∈ x1
Known ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Known SNoS_I^SNoS_I : ∀ x0 . ordinal x0 ⟶ ∀ x1 x2 . x2 ∈ x0 ⟶ SNo_ x2 x1 ⟶ x1 ∈ SNoS_ x0
Known SNoLev_^SNoLev_ : ∀ x0 . SNo x0 ⟶ SNo_ (SNoLev x0) x0
Known SNo_eps_decr^SNo_eps_decr : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ x0 ⟶ SNoLt (eps_ x0) (eps_ x1)
Known famunion_Empty^{famunion_Empty} : ∀ x0 : ι → ι . famunion 0 x0 = 0
Known binunion_idr^binunion_idr : ∀ x0 . binunion x0 0 = x0
Known Empty_eq^Empty_eq : ∀ x0 . (∀ x1 . nIn x1 x0) ⟶ x0 = 0
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Known Subq_binunion_eq^{Subq_binunion_eq} : ∀ x0 x1 . x0 ⊆ x1 = (binunion x0 x1 = x1)
Known cases_1^cases_1 : ∀ x0 . x0 ∈ 1 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 x0
Known nat_0_in_ordsucc^{nat_0_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ 0 ∈ ordsucc x0
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ 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 nat_trans^nat_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Known ordinal_ordsucc_In^{ordinal_ordsucc_In} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ∈ ordsucc x0
Known famunionI^famunionI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ x0 ⟶ x3 ∈ x1 x2 ⟶ x3 ∈ famunion x0 x1
Known SNoLev_0^SNoLev_0 : SNoLev 0 = 0
Known In_0_1^In_0_1 : 0 ∈ 1
Theorem eps_SNoCut^eps_SNoCut : ∀ x0 . x0 ∈ omega ⟶ eps_ x0 = SNoCut (Sing 0) (prim5 x0 eps_) (proof)
Known nat_complete_ind^{nat_complete_ind} : ∀ x0 : ι → ο . (∀ x1 . nat_p x1 ⟶ (∀ x2 . x2 ∈ x1 ⟶ x0 x2) ⟶ x0 x1) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1
Param SNoL^SNoL : ι → ι
Param SNoR^SNoR : ι → ι
Known add_SNo_eq^add_SNo_eq : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ add_SNo x0 x1 = SNoCut (binunion {add_SNo x3 x1|x3 ∈ SNoL x0} (prim5 (SNoL x1) (add_SNo x0))) (binunion {add_SNo x3 x1|x3 ∈ SNoR x0} (prim5 (SNoR x1) (add_SNo x0)))
Known SNoCut_ext^SNoCut_ext : ∀ x0 x1 x2 x3 . SNoCutP x0 x1 ⟶ SNoCutP x2 x3 ⟶ (∀ x4 . x4 ∈ x0 ⟶ SNoLt x4 (SNoCut x2 x3)) ⟶ (∀ x4 . x4 ∈ x1 ⟶ SNoLt (SNoCut x2 x3) x4) ⟶ (∀ x4 . x4 ∈ x2 ⟶ SNoLt x4 (SNoCut x0 x1)) ⟶ (∀ x4 . x4 ∈ x3 ⟶ SNoLt (SNoCut x0 x1) x4) ⟶ SNoCut x0 x1 = SNoCut x2 x3
Known add_SNo_SNoCutP^{add_SNo_SNoCutP} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoCutP (binunion {add_SNo x2 x1|x2 ∈ SNoL x0} (prim5 (SNoL x1) (add_SNo x0))) (binunion {add_SNo x2 x1|x2 ∈ SNoR x0} (prim5 (SNoR x1) (add_SNo x0)))
Known binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Known andEL^andEL : ∀ x0 x1 : ο . and x0 x1 ⟶ x0
Known andER^andER : ∀ x0 x1 : ο . and x0 x1 ⟶ 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 omega_ordsucc^{omega_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ ordsucc x0 ∈ omega
Known SNoLeLt_tra^SNoLeLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 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 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)
Theorem eps_ordsucc_half_add^{eps_ordsucc_half_add} : ∀ x0 . nat_p x0 ⟶ add_SNo (eps_ (ordsucc x0)) (eps_ (ordsucc x0)) = eps_ x0 (proof)
Known nat_1^nat_1 : nat_p 1
Theorem SNo_eps_1^SNo_eps_1 : SNo (eps_ 1) (proof)
Known nat_0^nat_0 : nat_p 0
Known eps_0_1^eps_0_1 : eps_ 0 = 1
Theorem eps_1_half_eq1^{eps_1_half_eq1} : add_SNo (eps_ 1) (eps_ 1) = 1 (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 add_nat_1_1_2^{add_nat_1_1_2} : add_nat 1 1 = 2
Theorem add_SNo_1_1_2^{add_SNo_1_1_2} : add_SNo 1 1 = 2 (proof)
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)
Theorem eps_1_half_eq2^{eps_1_half_eq2} : mul_SNo 2 (eps_ 1) = 1 (proof)
Known SNo_2^SNo_2 : SNo 2
Theorem eps_1_half_eq3^{eps_1_half_eq3} : mul_SNo (eps_ 1) 2 = 1 (proof)
Theorem eps_1_split_eq^{eps_1_split_eq} : ∀ x0 . SNo x0 ⟶ add_SNo (mul_SNo (eps_ 1) x0) (mul_SNo (eps_ 1) x0) = x0 (proof)
Theorem eps_1_split_Le_bd^{eps_1_split_Le_bd} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x1 (mul_SNo (eps_ 1) x0) ⟶ SNoLe x2 (mul_SNo (eps_ 1) x0) ⟶ SNoLe (add_SNo x1 x2) x0 (proof)
Theorem eps_1_split_LeLt_bd^{eps_1_split_LeLt_bd} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x1 (mul_SNo (eps_ 1) x0) ⟶ SNoLt x2 (mul_SNo (eps_ 1) x0) ⟶ SNoLt (add_SNo x1 x2) x0 (proof)
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)
Theorem eps_1_split_LtLe_bd^{eps_1_split_LtLe_bd} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x1 (mul_SNo (eps_ 1) x0) ⟶ SNoLe x2 (mul_SNo (eps_ 1) x0) ⟶ SNoLt (add_SNo x1 x2) x0 (proof)
Theorem eps_1_split_Lt_bd^{eps_1_split_Lt_bd} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x1 (mul_SNo (eps_ 1) x0) ⟶ SNoLt x2 (mul_SNo (eps_ 1) x0) ⟶ SNoLt (add_SNo x1 x2) x0 (proof)
Definition SNo_ord_seq^SNo_ord_seq := λ x0 . λ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ SNo (x1 x2)
Param setminus^setminus : ι → ι → ι
Definition 2be79.. := λ x0 . λ x1 : ι → ι . λ x2 . ∀ x3 . SNo x3 ⟶ SNoLt 0 x3 ⟶ ∀ x4 : ο . (∀ x5 . and (x5 ∈ x0) (∀ x6 . x6 ∈ setminus x0 x5 ⟶ SNoLt (abs_SNo (add_SNo (x1 x6) (minus_SNo x2))) x3) ⟶ x4) ⟶ x4
Definition c59b5.. := λ x0 . λ x1 : ι → ι . ∀ x2 . SNo x2 ⟶ SNoLt 0 x2 ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ x0) (∀ x5 . x5 ∈ setminus x0 x4 ⟶ ∀ x6 . x6 ∈ setminus x0 x4 ⟶ SNoLt (abs_SNo (add_SNo (x1 x5) (minus_SNo (x1 x6)))) x2) ⟶ x3) ⟶ x3
Known ordinal_linear^{ordinal_linear} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ or (x0 ⊆ x1) (x1 ⊆ x0)
Known setminusI^setminusI : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ nIn x2 x1 ⟶ x2 ∈ setminus x0 x1
Known ordinal_Hered^{ordinal_Hered} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordinal 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 add_SNo_minus_SNo_rinv^{add_SNo_minus_SNo_rinv} : ∀ x0 . SNo x0 ⟶ add_SNo x0 (minus_SNo x0) = 0
Theorem 5bedf.. : ∀ x0 . ordinal x0 ⟶ ∀ x1 : ι → ι . SNo_ord_seq x0 x1 ⟶ ∀ x2 . SNo x2 ⟶ 2be79.. x0 x1 x2 ⟶ ∀ x3 . SNo x3 ⟶ 2be79.. x0 x1 x3 ⟶ not (SNoLt x2 x3) (proof)
Theorem 5eef1.. : ∀ x0 . ordinal x0 ⟶ ∀ x1 : ι → ι . SNo_ord_seq x0 x1 ⟶ ∀ x2 . SNo x2 ⟶ 2be79.. x0 x1 x2 ⟶ ∀ x3 . SNo x3 ⟶ 2be79.. x0 x1 x3 ⟶ x2 = x3 (proof)