Proofgold Signed Transaction

Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param SNo^SNo : ι → ο
Param SNoLe^SNoLe : ι → ι → ο
Definition SNo_max_of^SNo_max_of := λ x0 x1 . and (and (x1 ∈ x0) (SNo x1)) (∀ x2 . x2 ∈ x0 ⟶ SNo x2 ⟶ SNoLe x2 x1)
Definition SNo_min_of^SNo_min_of := λ x0 x1 . and (and (x1 ∈ x0) (SNo x1)) (∀ x2 . x2 ∈ x0 ⟶ SNo x2 ⟶ SNoLe x1 x2)
Known SNoLe_antisym^{SNoLe_antisym} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x0 ⟶ x0 = x1
Theorem 75b8f.. : ∀ x0 x1 x2 . SNo_max_of x0 x1 ⟶ SNo_max_of x0 x2 ⟶ x1 = x2 (proof)
Theorem 64b6a.. : ∀ x0 x1 x2 . SNo_min_of x0 x1 ⟶ SNo_min_of x0 x2 ⟶ x1 = x2 (proof)
Param mul_SNo^mul_SNo : ι → ι → ι
Param eps_^eps_ : ι → ι
Param ordsucc^ordsucc : ι → ι
Param add_SNo^add_SNo : ι → ι → ι
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)
Param omega^omega : ι
Known SNo_eps_^SNo_eps_ : ∀ x0 . x0 ∈ omega ⟶ SNo (eps_ x0)
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 eps_1_split_eq^{eps_1_split_eq} : ∀ x0 . SNo x0 ⟶ add_SNo (mul_SNo (eps_ 1) x0) (mul_SNo (eps_ 1) x0) = x0
Theorem 421ce.. : ∀ x0 . SNo x0 ⟶ x0 = mul_SNo (eps_ 1) (add_SNo x0 x0) (proof)
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
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 ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Theorem Repl_inv_eq^Repl_inv_eq : ∀ x0 : ι → ο . ∀ x1 x2 : ι → ι . (∀ x3 . x0 x3 ⟶ x2 (x1 x3) = x3) ⟶ ∀ x3 . (∀ x4 . x4 ∈ x3 ⟶ x0 x4) ⟶ prim5 (prim5 x3 x1) x2 = x3 (proof)
Theorem Repl_invol_eq^{Repl_invol_eq} : ∀ x0 : ι → ο . ∀ x1 : ι → ι . (∀ x2 . x0 x2 ⟶ x1 (x1 x2) = x2) ⟶ ∀ x2 . (∀ x3 . x3 ∈ x2 ⟶ x0 x3) ⟶ prim5 (prim5 x2 x1) x1 = x2 (proof)
Param minus_SNo^minus_SNo : ι → ι
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
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 minus_SNo_max_min^{minus_SNo_max_min} : ∀ x0 x1 . (∀ x2 . x2 ∈ x0 ⟶ SNo x2) ⟶ SNo_max_of x0 x1 ⟶ SNo_min_of (prim5 x0 minus_SNo) (minus_SNo x1) (proof)
Known minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
Theorem minus_SNo_max_min'^{minus_SNo_max_min} : ∀ x0 x1 . (∀ x2 . x2 ∈ x0 ⟶ SNo x2) ⟶ SNo_max_of (prim5 x0 minus_SNo) x1 ⟶ SNo_min_of x0 (minus_SNo x1) (proof)
Theorem minus_SNo_min_max^{minus_SNo_min_max} : ∀ x0 x1 . (∀ x2 . x2 ∈ x0 ⟶ SNo x2) ⟶ SNo_min_of x0 x1 ⟶ SNo_max_of (prim5 x0 minus_SNo) (minus_SNo x1) (proof)
Theorem f6de3.. : ∀ x0 x1 . (∀ x2 . x2 ∈ x0 ⟶ SNo x2) ⟶ SNo_min_of (prim5 x0 minus_SNo) x1 ⟶ SNo_max_of x0 (minus_SNo x1) (proof)
Param SNoL^SNoL : ι → ι
Param SNoLt^SNoLt : ι → ι → ο
Param SNoR^SNoR : ι → ι
Param SNoLev^SNoLev : ι → ι
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 SNoS_^SNoS_ : 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
Definition False^False := ∀ x0 : ο . x0
Known SNoLt_SNoL_or_SNoR_impred^{SNoLt_SNoL_or_SNoR_impred} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 x1 ⟶ ∀ x2 : ο . (∀ x3 . x3 ∈ SNoL x1 ⟶ x3 ∈ SNoR x0 ⟶ x2) ⟶ (x0 ∈ SNoL x1 ⟶ x2) ⟶ (x1 ∈ SNoR x0 ⟶ x2) ⟶ x2
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known add_SNo_SNoR_interpolate^{add_SNo_SNoR_interpolate} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 . x2 ∈ SNoR (add_SNo x0 x1) ⟶ or (∀ x3 : ο . (∀ x4 . and (x4 ∈ SNoR x0) (SNoLe (add_SNo x4 x1) x2) ⟶ x3) ⟶ x3) (∀ x3 : ο . (∀ x4 . and (x4 ∈ SNoR x1) (SNoLe (add_SNo x0 x4) x2) ⟶ x3) ⟶ x3)
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 FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 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 add_SNo_SNoL_interpolate^{add_SNo_SNoL_interpolate} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 . x2 ∈ SNoL (add_SNo x0 x1) ⟶ or (∀ x3 : ο . (∀ x4 . and (x4 ∈ SNoL x0) (SNoLe x2 (add_SNo x4 x1)) ⟶ x3) ⟶ x3) (∀ x3 : ο . (∀ x4 . and (x4 ∈ SNoL x1) (SNoLe x2 (add_SNo x0 x4)) ⟶ x3) ⟶ x3)
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 x0)
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known SNoR_SNoS_^SNoR_SNoS_ : ∀ x0 . SNoR x0 ⊆ SNoS_ (SNoLev x0)
Known SNoLt_tra^SNoLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
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
Definition TransSet^TransSet := λ x0 . ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Known ordinal_TransSet^{ordinal_TransSet} : ∀ x0 . ordinal x0 ⟶ TransSet x0
Known SNoLev_ordinal^{SNoLev_ordinal} : ∀ x0 . SNo x0 ⟶ ordinal (SNoLev x0)
Definition not^not := λ x0 : ο . x0 ⟶ False
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 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 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 add_SNo_Lt3b^add_SNo_Lt3b : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLe x0 x2 ⟶ SNoLt x1 x3 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x3)
Known SNoL_I^SNoL_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x1 x0 ⟶ x1 ∈ SNoL x0
Theorem double_SNo_max_1^{double_SNo_max_1} : ∀ x0 x1 . SNo x0 ⟶ SNo_max_of (SNoL x0) x1 ⟶ ∀ x2 . SNo x2 ⟶ SNoLt x0 x2 ⟶ SNoLt (add_SNo x1 x2) (add_SNo x0 x0) ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ SNoR x2) (add_SNo x1 x4 = add_SNo x0 x0) ⟶ x3) ⟶ x3 (proof)
Known minus_SNo_Lev^{minus_SNo_Lev} : ∀ x0 . SNo x0 ⟶ SNoLev (minus_SNo x0) = SNoLev 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_Lt_contra^{minus_SNo_Lt_contra} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 x1 ⟶ SNoLt (minus_SNo x1) (minus_SNo x0)
Theorem SNoL_minus_SNoR^{SNoL_minus_SNoR} : ∀ x0 . SNo x0 ⟶ SNoL (minus_SNo x0) = prim5 (SNoR x0) minus_SNo (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)
Theorem double_SNo_min_1^{double_SNo_min_1} : ∀ x0 x1 . SNo x0 ⟶ SNo_min_of (SNoR x0) x1 ⟶ ∀ x2 . SNo x2 ⟶ SNoLt x2 x0 ⟶ SNoLt (add_SNo x0 x0) (add_SNo x1 x2) ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ SNoL x2) (add_SNo x1 x4 = add_SNo x0 x0) ⟶ x3) ⟶ x3 (proof)
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_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Known omega_SNoS_omega^{omega_SNoS_omega} : omega ⊆ SNoS_ omega
Known nat_0^nat_0 : nat_p 0
Known add_SNo_1_ordsucc^{add_SNo_1_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ add_SNo x0 1 = ordsucc x0
Known omega_SNo^omega_SNo : ∀ x0 . x0 ∈ omega ⟶ SNo x0
Known mul_SNo_oneR^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = 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 SNo_eps_SNoS_omega^{SNo_eps_SNoS_omega} : ∀ x0 . x0 ∈ omega ⟶ eps_ x0 ∈ SNoS_ omega
Theorem nonneg_diadic_rational_p_SNoS_omega^{nonneg_diadic_rational_p_SNoS_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . nat_p x1 ⟶ mul_SNo (eps_ x0) x1 ∈ SNoS_ omega (proof)
Definition 368eb.. := λ x0 . ∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (∀ x3 : ο . (∀ x4 . and (x4 ∈ omega) (or (x0 = mul_SNo (eps_ x2) x4) (x0 = minus_SNo (mul_SNo (eps_ x2) x4))) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Known minus_SNo_SNoS_omega^{minus_SNo_SNoS_omega} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ minus_SNo x0 ∈ SNoS_ omega
Theorem 19a1e.. : ∀ x0 . 368eb.. x0 ⟶ x0 ∈ SNoS_ omega (proof)
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
Theorem 752a1.. : ∀ x0 . 368eb.. x0 ⟶ SNo x0 (proof)
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known eps_0_1^eps_0_1 : eps_ 0 = 1
Known mul_SNo_oneL^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 x0 = x0
Theorem bb191.. : ∀ x0 . x0 ∈ omega ⟶ 368eb.. x0 (proof)
Theorem 12a08.. : ∀ x0 . x0 ∈ omega ⟶ 368eb.. (eps_ x0) (proof)
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Theorem 81797.. : ∀ x0 . 368eb.. x0 ⟶ 368eb.. (minus_SNo x0) (proof)
Known add_SNo_In_omega^{add_SNo_In_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ add_SNo x0 x1 ∈ omega
Known mul_SNo_In_omega^{mul_SNo_In_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_SNo x0 x1 ∈ omega
Known mul_SNo_eps_eps_add_SNo^{mul_SNo_eps_eps_add_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_SNo (eps_ x0) (eps_ x1) = eps_ (add_SNo x0 x1)
Known mul_SNo_com_4_inner_mid^{mul_SNo_com_4_inner_mid} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ mul_SNo (mul_SNo x0 x1) (mul_SNo x2 x3) = mul_SNo (mul_SNo x0 x2) (mul_SNo x1 x3)
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_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_minus^{mul_SNo_minus_minus} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo (minus_SNo x0) (minus_SNo x1) = mul_SNo x0 x1
Theorem 43fd3.. : ∀ x0 x1 . 368eb.. x0 ⟶ 368eb.. x1 ⟶ 368eb.. (mul_SNo x0 x1) (proof)
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 nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known Subq_Empty^Subq_Empty : ∀ x0 . 0 ⊆ x0
Theorem omega_nonneg^omega_nonneg : ∀ x0 . x0 ∈ omega ⟶ SNoLe 0 x0 (proof)
Param exp_SNo_nat^exp_SNo_nat : ι → ι → ι
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_eps_power_2^{mul_SNo_eps_power_2} : ∀ x0 . nat_p x0 ⟶ mul_SNo (eps_ x0) (exp_SNo_nat 2 x0) = 1
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 SNo_0^SNo_0 : SNo 0
Known minus_SNo_0^minus_SNo_0 : minus_SNo 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_eps_pos^SNo_eps_pos : ∀ x0 . x0 ∈ omega ⟶ SNoLt 0 (eps_ x0)
Theorem 4e3c5.. : ∀ x0 x1 . 368eb.. x0 ⟶ 368eb.. x1 ⟶ SNoLt 0 x0 ⟶ SNoLt 0 x1 ⟶ 368eb.. (add_SNo x0 x1) (proof)
Known mul_SNo_nonzero_cancel^{mul_SNo_nonzero_cancel_L} : ∀ x0 x1 x2 . SNo x0 ⟶ (x0 = 0 ⟶ ∀ x3 : ο . x3) ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 x1 = mul_SNo x0 x2 ⟶ x1 = x2
Known SNo_2^SNo_2 : SNo 2
Known neq_2_0^neq_2_0 : 2 = 0 ⟶ ∀ x0 : ο . x0
Known eps_1_half_eq2^{eps_1_half_eq2} : mul_SNo 2 (eps_ 1) = 1
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 SNo_1^SNo_1 : SNo 1
Theorem double_eps_1^double_eps_1 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 x0 = add_SNo x1 x2 ⟶ x0 = mul_SNo (eps_ 1) (add_SNo x1 x2) (proof)