Proofgold Signed Transaction

Param SNo^SNo : ι → ο
Param SNoLt^SNoLt : ι → ι → ο
Param recip_SNo_pos^{recip_SNo_pos} : ι → ι
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param mul_SNo^mul_SNo : ι → ι → ι
Param ordsucc^ordsucc : ι → ι
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)
Theorem SNo_recip_SNo_pos^{SNo_recip_SNo_pos} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ SNo (recip_SNo_pos x0) (proof)
Theorem recip_SNo_pos_invR^{recip_SNo_pos_invR} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ mul_SNo x0 (recip_SNo_pos x0) = 1 (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_1^SNo_1 : SNo 1
Known neq_1_0^neq_1_0 : 1 = 0 ⟶ ∀ x0 : ο . x0
Known SNoLt_0_1^SNoLt_0_1 : SNoLt 0 1
Known mul_SNo_oneL^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 x0 = x0
Theorem recip_SNo_pos_1^{recip_SNo_pos_1} : recip_SNo_pos 1 = 1 (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 SNo_0^SNo_0 : SNo 0
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 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 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 mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Theorem recip_SNo_pos_pos^{recip_SNo_pos_pos} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ SNoLt 0 (recip_SNo_pos x0) (proof)
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
Theorem recip_SNo_pos_invol^{recip_SNo_pos_invol} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ recip_SNo_pos (recip_SNo_pos x0) = x0 (proof)
Param nat_p^nat_p : ι → ο
Param eps_^eps_ : ι → ι
Param exp_SNo_nat^exp_SNo_nat : ι → ι → ι
Known SNo_exp_SNo_nat^{SNo_exp_SNo_nat} : ∀ x0 . SNo x0 ⟶ ∀ x1 . nat_p x1 ⟶ SNo (exp_SNo_nat x0 x1)
Known SNo_2^SNo_2 : SNo 2
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
Param omega^omega : ι
Known SNo_eps_pos^SNo_eps_pos : ∀ x0 . x0 ∈ omega ⟶ SNoLt 0 (eps_ x0)
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known SNo_eps_^SNo_eps_ : ∀ x0 . x0 ∈ omega ⟶ SNo (eps_ x0)
Theorem recip_SNo_pos_eps_^{recip_SNo_pos_eps_} : ∀ x0 . nat_p x0 ⟶ recip_SNo_pos (eps_ x0) = exp_SNo_nat 2 x0 (proof)
Theorem recip_SNo_pos_pow_2^{recip_SNo_pos_pow_2} : ∀ x0 . nat_p x0 ⟶ recip_SNo_pos (exp_SNo_nat 2 x0) = eps_ x0 (proof)
Known exp_SNo_nat_S^{exp_SNo_nat_S} : ∀ x0 . SNo x0 ⟶ ∀ x1 . nat_p x1 ⟶ exp_SNo_nat x0 (ordsucc x1) = mul_SNo x0 (exp_SNo_nat x0 x1)
Known nat_0^nat_0 : nat_p 0
Known exp_SNo_nat_0^{exp_SNo_nat_0} : ∀ x0 . SNo x0 ⟶ exp_SNo_nat x0 0 = 1
Known mul_SNo_oneR^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = x0
Theorem exp_SNo_nat_1^{exp_SNo_nat_1} : ∀ x0 . SNo x0 ⟶ exp_SNo_nat x0 1 = x0 (proof)
Known nat_1^nat_1 : nat_p 1
Theorem recip_SNo_pos_2^{recip_SNo_pos_2} : recip_SNo_pos 2 = eps_ 1 (proof)
Param If_i^If_i : ο → ι → ι → ι
Param minus_SNo^minus_SNo : ι → ι
Definition recip_SNo^recip_SNo := λ x0 . If_i (SNoLt 0 x0) (recip_SNo_pos x0) (If_i (SNoLt x0 0) (minus_SNo (recip_SNo_pos (minus_SNo x0))) 0)
Known If_i_1^If_i_1 : ∀ x0 : ο . ∀ x1 x2 . x0 ⟶ If_i x0 x1 x2 = x1
Theorem recip_SNo_poscase^{recip_SNo_poscase} : ∀ x0 . SNoLt 0 x0 ⟶ recip_SNo x0 = recip_SNo_pos x0 (proof)
Known If_i_0^If_i_0 : ∀ x0 : ο . ∀ x1 x2 . not x0 ⟶ If_i x0 x1 x2 = x2
Theorem recip_SNo_negcase^{recip_SNo_negcase} : ∀ x0 . SNo x0 ⟶ SNoLt x0 0 ⟶ recip_SNo x0 = minus_SNo (recip_SNo_pos (minus_SNo x0)) (proof)
Theorem recip_SNo_0^recip_SNo_0 : recip_SNo 0 = 0 (proof)
Theorem recip_SNo_1^recip_SNo_1 : recip_SNo 1 = 1 (proof)
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (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 minus_SNo_0^minus_SNo_0 : minus_SNo 0 = 0
Theorem SNo_recip_SNo^{SNo_recip_SNo} : ∀ x0 . SNo x0 ⟶ SNo (recip_SNo x0) (proof)
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)
Theorem recip_SNo_invR^{recip_SNo_invR} : ∀ x0 . SNo x0 ⟶ (x0 = 0 ⟶ ∀ x1 : ο . x1) ⟶ mul_SNo x0 (recip_SNo x0) = 1 (proof)
Theorem recip_SNo_invL^{recip_SNo_invL} : ∀ x0 . SNo x0 ⟶ (x0 = 0 ⟶ ∀ x1 : ο . x1) ⟶ mul_SNo (recip_SNo x0) x0 = 1 (proof)
Theorem recip_SNo_eps_^{recip_SNo_eps_} : ∀ x0 . nat_p x0 ⟶ recip_SNo (eps_ x0) = exp_SNo_nat 2 x0 (proof)
Known exp_SNo_nat_pos^{exp_SNo_nat_pos} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ ∀ x1 . nat_p x1 ⟶ SNoLt 0 (exp_SNo_nat x0 x1)
Known SNoLt_0_2^SNoLt_0_2 : SNoLt 0 2
Theorem recip_SNo_pow_2^{recip_SNo_pow_2} : ∀ x0 . nat_p x0 ⟶ recip_SNo (exp_SNo_nat 2 x0) = eps_ x0 (proof)
Theorem recip_SNo_2^recip_SNo_2 : recip_SNo 2 = eps_ 1 (proof)
Known minus_SNo_Lt_contra1^{minus_SNo_Lt_contra1} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt (minus_SNo x0) x1 ⟶ SNoLt (minus_SNo x1) x0
Known minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
Theorem recip_SNo_invol^{recip_SNo_invol} : ∀ x0 . SNo x0 ⟶ recip_SNo (recip_SNo x0) = x0 (proof)
Theorem recip_SNo_of_pos_is_pos^{recip_SNo_of_pos_is_pos} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ SNoLt 0 (recip_SNo x0) (proof)
Theorem recip_SNo_of_neg_is_neg^{recip_SNo_of_neg_is_neg} : ∀ x0 . SNo x0 ⟶ SNoLt x0 0 ⟶ SNoLt (recip_SNo x0) 0 (proof)
Definition div_SNo^div_SNo := λ x0 x1 . mul_SNo x0 (recip_SNo x1)
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Theorem SNo_div_SNo^SNo_div_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (div_SNo x0 x1) (proof)
Known mul_SNo_zeroL^{mul_SNo_zeroL} : ∀ x0 . SNo x0 ⟶ mul_SNo 0 x0 = 0
Theorem div_SNo_0_num^{div_SNo_0_num} : ∀ x0 . SNo x0 ⟶ div_SNo 0 x0 = 0 (proof)
Theorem div_SNo_0_denum^{div_SNo_0_denum} : ∀ x0 . SNo x0 ⟶ div_SNo x0 0 = 0 (proof)
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
Theorem mul_div_SNo_invL^{mul_div_SNo_invL} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ (x1 = 0 ⟶ ∀ x2 : ο . x2) ⟶ mul_SNo (div_SNo x0 x1) x1 = x0 (proof)
Theorem mul_div_SNo_invR^{mul_div_SNo_invR} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ (x1 = 0 ⟶ ∀ x2 : ο . x2) ⟶ mul_SNo x1 (div_SNo x0 x1) = 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)
Theorem mul_div_SNo_R^{mul_div_SNo_R} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo (div_SNo x0 x1) x2 = div_SNo (mul_SNo x0 x2) x1 (proof)
Theorem mul_div_SNo_L^{mul_div_SNo_L} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x2 (div_SNo x0 x1) = div_SNo (mul_SNo x2 x0) x1 (proof)
Theorem div_mul_SNo_invL^{div_mul_SNo_invL} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ (x1 = 0 ⟶ ∀ x2 : ο . x2) ⟶ div_SNo (mul_SNo x0 x1) x1 = x0 (proof)
Theorem div_div_SNo^div_div_SNo : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ div_SNo (div_SNo x0 x1) x2 = div_SNo x0 (mul_SNo x1 x2) (proof)
Theorem mul_div_SNo_both^{mul_div_SNo_both} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ mul_SNo (div_SNo x0 x1) (div_SNo x2 x3) = div_SNo (mul_SNo x0 x2) (mul_SNo x1 x3) (proof)
Param SNoLe^SNoLe : ι → ι → ο
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 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 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 SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Theorem recip_SNo_pos_pos^{recip_SNo_pos_pos} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ SNoLt 0 (recip_SNo_pos x0) (proof)
Theorem recip_SNo_of_neg_is_neg^{recip_SNo_of_neg_is_neg} : ∀ x0 . SNo x0 ⟶ SNoLt x0 0 ⟶ SNoLt (recip_SNo x0) 0 (proof)
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)
Theorem div_SNo_pos_pos^{div_SNo_pos_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt 0 x0 ⟶ SNoLt 0 x1 ⟶ SNoLt 0 (div_SNo x0 x1) (proof)
Known mul_SNo_neg_neg^{mul_SNo_neg_neg} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 0 ⟶ SNoLt x1 0 ⟶ SNoLt 0 (mul_SNo x0 x1)
Theorem div_SNo_neg_neg^{div_SNo_neg_neg} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 0 ⟶ SNoLt x1 0 ⟶ SNoLt 0 (div_SNo x0 x1) (proof)
Theorem div_SNo_pos_neg^{div_SNo_pos_neg} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt 0 x0 ⟶ SNoLt x1 0 ⟶ SNoLt (div_SNo x0 x1) 0 (proof)
Known mul_SNo_neg_pos^{mul_SNo_neg_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 0 ⟶ SNoLt 0 x1 ⟶ SNoLt (mul_SNo x0 x1) 0
Theorem div_SNo_neg_pos^{div_SNo_neg_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 0 ⟶ SNoLt 0 x1 ⟶ SNoLt (div_SNo x0 x1) 0 (proof)
Known pos_mul_SNo_Lt'^{pos_mul_SNo_Lt} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt 0 x2 ⟶ SNoLt x0 x1 ⟶ SNoLt (mul_SNo x0 x2) (mul_SNo x1 x2)
Theorem 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 (proof)
Theorem 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) (proof)
Theorem 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) (proof)
Theorem 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 (proof)
Theorem 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 (proof)
Param SNo_^SNo_ : ι → ι → ο
Param SNoLev^SNoLev : ι → ι
Known SNoLev_0_eq_0^{SNoLev_0_eq_0} : ∀ x0 . SNo x0 ⟶ SNoLev x0 = 0 ⟶ x0 = 0
Param ordinal^ordinal : ι → ο
Known SNo_SNo^SNo_SNo : ∀ x0 . ordinal x0 ⟶ ∀ x1 . SNo_ x0 x1 ⟶ SNo x1
Known ordinal_Empty^{ordinal_Empty} : ordinal 0
Known SNoLev_uniq2^SNoLev_uniq2 : ∀ x0 . ordinal x0 ⟶ ∀ x1 . SNo_ x0 x1 ⟶ SNoLev x1 = x0
Theorem SNo_0_eq_0^SNo_0_eq_0 : ∀ x0 . SNo_ 0 x0 ⟶ x0 = 0 (proof)
Param add_SNo^add_SNo : ι → ι → ι
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_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 SNo_gt2_double_ltS^{SNo_gt2_double_ltS} : ∀ x0 . SNo x0 ⟶ SNoLt 1 x0 ⟶ SNoLt (add_SNo x0 1) (mul_SNo 2 x0) (proof)
Param real^real : ι
Param SNoS_^SNoS_ : ι → ι
Param abs_SNo^abs_SNo : 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
Known eps_0_1^eps_0_1 : eps_ 0 = 1
Known SNoS_E^SNoS_E : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ SNoS_ x0 ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ x0) (SNo_ x3 x1) ⟶ x2) ⟶ x2
Known omega_ordinal^{omega_ordinal} : ordinal omega
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 SNoLeLt_tra^SNoLeLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known nat_p_SNo^nat_p_SNo : ∀ x0 . nat_p x0 ⟶ SNo x0
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known nat_ordsucc^nat_ordsucc : ∀ x0 . nat_p x0 ⟶ nat_p (ordsucc x0)
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Known ordinal_SNoLev_max_2^{ordinal_SNoLev_max_2} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ ordsucc x0 ⟶ SNoLe x1 x0
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
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 ordsuccI1^ordsuccI1 : ∀ x0 . x0 ⊆ ordsucc x0
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Known Subq_tra^Subq_tra : ∀ x0 x1 x2 . x0 ⊆ x1 ⟶ x1 ⊆ x2 ⟶ x0 ⊆ x2
Known add_SNo_Lev_bd^{add_SNo_Lev_bd} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev (add_SNo x0 x1) ⊆ add_SNo (SNoLev x0) (SNoLev x1)
Known ordinal_SNoLev^{ordinal_SNoLev} : ∀ x0 . ordinal x0 ⟶ SNoLev x0 = x0
Known add_SNo_1_ordsucc^{add_SNo_1_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ add_SNo x0 1 = ordsucc x0
Known Subq_ref^Subq_ref : ∀ x0 . x0 ⊆ x0
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Theorem nonneg_real_nat_interval^{nonneg_real_nat_interval} : ∀ x0 . x0 ∈ real ⟶ SNoLe 0 x0 ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (and (SNoLe x2 x0) (SNoLt x0 (ordsucc x2))) ⟶ x1) ⟶ x1 (proof)
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 dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Known SNoS_I^SNoS_I : ∀ x0 . ordinal x0 ⟶ ∀ x1 x2 . x2 ∈ x0 ⟶ SNo_ x2 x1 ⟶ x1 ∈ SNoS_ x0
Known ordinal_SNo_^ordinal_SNo_ : ∀ x0 . ordinal x0 ⟶ SNo_ x0 x0
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Known abs_SNo_minus^{abs_SNo_minus} : ∀ x0 . SNo x0 ⟶ abs_SNo (minus_SNo x0) = abs_SNo x0
Known pos_abs_SNo^pos_abs_SNo : ∀ x0 . SNoLt 0 x0 ⟶ abs_SNo x0 = x0
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known SNoL_I^SNoL_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x1 x0 ⟶ x1 ∈ SNoL x0
Known pos_low_eq_one^{pos_low_eq_one} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ SNoLev x0 ⊆ 1 ⟶ x0 = 1
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 SNoLeE^SNoLeE : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ or (SNoLt x0 x1) (x0 = x1)
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known cases_2^cases_2 : ∀ x0 . x0 ∈ 2 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 1 ⟶ x1 x0
Known ordinal_SNoLt_In^{ordinal_SNoLt_In} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ SNoLt x0 x1 ⟶ x0 ∈ x1
Known nat_2^nat_2 : nat_p 2
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
Known ordinal_ordsucc_SNo_eq^{ordinal_ordsucc_SNo_eq} : ∀ x0 . ordinal x0 ⟶ ordsucc x0 = add_SNo 1 x0
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_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 ordinal_Subq_SNoLe^{ordinal_Subq_SNoLe} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ x0 ⊆ x1 ⟶ SNoLe x0 x1
Known Subq_Empty^Subq_Empty : ∀ x0 . 0 ⊆ x0
Known ordinal_In_SNoLt^{ordinal_In_SNoLt} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ SNoLt x1 x0
Known SNoLe_ref^SNoLe_ref : ∀ x0 . SNoLe x0 x0
Known ordinal_trichotomy_or_impred^{ordinal_trichotomy_or_impred} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ ∀ x2 : ο . (x0 ∈ x1 ⟶ x2) ⟶ (x0 = x1 ⟶ x2) ⟶ (x1 ∈ x0 ⟶ x2) ⟶ x2
Known ordinal_SNoLev_max^{ordinal_SNoLev_max} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ x0 ⟶ SNoLt x1 x0
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 SNoLt_1_2^SNoLt_1_2 : SNoLt 1 2
Param binintersect^binintersect : ι → ι → ι
Definition iff^iff := λ x0 x1 : ο . and (x0 ⟶ x1) (x1 ⟶ x0)
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)
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
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 eps_ordinal_In_eq_0^{eps_ordinal_In_eq_0} : ∀ x0 x1 . ordinal x1 ⟶ x1 ∈ eps_ x0 ⟶ x1 = 0
Known SNoLev_eps_^SNoLev_eps_ : ∀ x0 . x0 ∈ omega ⟶ SNoLev (eps_ x0) = ordsucc x0
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Known binintersectE1^{binintersectE1} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ x2 ∈ x0
Known In_no2cycle^In_no2cycle : ∀ x0 x1 . x0 ∈ x1 ⟶ x1 ∈ x0 ⟶ False
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Known eps_ordsucc_half_add^{eps_ordsucc_half_add} : ∀ x0 . nat_p x0 ⟶ add_SNo (eps_ (ordsucc x0)) (eps_ (ordsucc x0)) = eps_ x0
Known omega_ordsucc^{omega_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ ordsucc x0 ∈ omega
Theorem 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 (proof)
Param sqrt_SNo_nonneg^{sqrt_SNo_nonneg} : ι → ι
Param ap^ap : ι → ι → ι
Param SNo_sqrtaux^SNo_sqrtaux : ι → (ι → ι) → ι → ι
Param lam^Sigma : ι → (ι → ι) → ι
Definition SNoL_nonneg^SNoL_nonneg := λ x0 . Sep (SNoL x0) (SNoLe 0)
Param SNoR^SNoR : ι → ι
Known SNo_sqrtaux_0^{SNo_sqrtaux_0} : ∀ x0 . ∀ x1 : ι → ι . SNo_sqrtaux x0 x1 0 = lam 2 (λ x3 . If_i (x3 = 0) (prim5 (SNoL_nonneg x0) x1) (prim5 (SNoR x0) x1))
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 ReplE_impred^ReplE_impred : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ prim5 x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ x0 ⟶ x2 = x1 x4 ⟶ x3) ⟶ x3
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
Known SNoS_I2^SNoS_I2 : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x0 ∈ SNoLev x1 ⟶ x0 ∈ SNoS_ (SNoLev 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 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 SNoLe_tra^SNoLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLe x0 x2
Param binunion^binunion : ι → ι → ι
Param SNo_sqrtauxset^{SNo_sqrtauxset} : ι → ι → ι → ι
Known SNo_sqrtaux_S^{SNo_sqrtaux_S} : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . nat_p x2 ⟶ SNo_sqrtaux x0 x1 (ordsucc x2) = lam 2 (λ x4 . If_i (x4 = 0) (binunion (ap (SNo_sqrtaux x0 x1 x2) 0) (SNo_sqrtauxset (ap (SNo_sqrtaux x0 x1 x2) 0) (ap (SNo_sqrtaux x0 x1 x2) 1) x0)) (binunion (binunion (ap (SNo_sqrtaux x0 x1 x2) 1) (SNo_sqrtauxset (ap (SNo_sqrtaux x0 x1 x2) 0) (ap (SNo_sqrtaux x0 x1 x2) 0) x0)) (SNo_sqrtauxset (ap (SNo_sqrtaux x0 x1 x2) 1) (ap (SNo_sqrtaux x0 x1 x2) 1) x0)))
Known binunionE'^binunionE : ∀ x0 x1 x2 . ∀ x3 : ο . (x2 ∈ x0 ⟶ x3) ⟶ (x2 ∈ x1 ⟶ x3) ⟶ x2 ∈ binunion x0 x1 ⟶ x3
Known SNo_sqrtauxset_E^{SNo_sqrtauxset_E} : ∀ x0 x1 x2 x3 . x3 ∈ SNo_sqrtauxset x0 x1 x2 ⟶ ∀ x4 : ο . (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x1 ⟶ SNoLt 0 (add_SNo x5 x6) ⟶ x3 = div_SNo (add_SNo x2 (mul_SNo x5 x6)) (add_SNo x5 x6) ⟶ x4) ⟶ x4
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)
Known 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)
Known SNo_foil^SNo_foil : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ mul_SNo (add_SNo x0 x1) (add_SNo x2 x3) = add_SNo (mul_SNo x0 x2) (add_SNo (mul_SNo x0 x3) (add_SNo (mul_SNo x1 x2) (mul_SNo x1 x3)))
Known add_SNo_rotate_4_1^{add_SNo_rotate_4_1} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ add_SNo x0 (add_SNo x1 (add_SNo x2 x3)) = add_SNo x3 (add_SNo x0 (add_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 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_Lt2^{add_SNo_minus_Lt2} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x2 (add_SNo x0 (minus_SNo x1)) ⟶ SNoLt (add_SNo x2 x1) 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_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
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 SNo_foil_mm^SNo_foil_mm : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ mul_SNo (add_SNo x0 (minus_SNo x1)) (add_SNo x2 (minus_SNo x3)) = add_SNo (mul_SNo x0 x2) (add_SNo (minus_SNo (mul_SNo x0 x3)) (add_SNo (minus_SNo (mul_SNo x1 x2)) (mul_SNo x1 x3)))
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 SNo_mul_SNo_3^{SNo_mul_SNo_3} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo (mul_SNo x0 (mul_SNo x1 x2))
Known add_SNo_minus_Lt1^{add_SNo_minus_Lt1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt (add_SNo x0 (minus_SNo x1)) x2 ⟶ SNoLt x0 (add_SNo x2 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
Theorem sqrt_SNo_nonneg_prop1a^{sqrt_SNo_nonneg_prop1a} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ (∀ x1 . x1 ∈ SNoS_ (SNoLev x0) ⟶ SNoLe 0 x1 ⟶ and (and (SNo (sqrt_SNo_nonneg x1)) (SNoLe 0 (sqrt_SNo_nonneg x1))) (mul_SNo (sqrt_SNo_nonneg x1) (sqrt_SNo_nonneg x1) = x1)) ⟶ ∀ x1 . nat_p x1 ⟶ and (∀ x2 . x2 ∈ ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 0 ⟶ and (and (SNo x2) (SNoLe 0 x2)) (SNoLt (mul_SNo x2 x2) x0)) (∀ x2 . x2 ∈ ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 1 ⟶ and (and (SNo x2) (SNoLe 0 x2)) (SNoLt x0 (mul_SNo 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)
Param famunion^famunion : ι → (ι → ι) → ι
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 nonneg_mul_SNo_Le2^{nonneg_mul_SNo_Le2} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLe 0 x0 ⟶ SNoLe 0 x1 ⟶ SNoLe x0 x2 ⟶ SNoLe x1 x3 ⟶ SNoLe (mul_SNo x0 x1) (mul_SNo x2 x3)
Theorem sqrt_SNo_nonneg_prop1b^{sqrt_SNo_nonneg_prop1b} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ (∀ x1 . nat_p x1 ⟶ and (∀ x2 . x2 ∈ ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 0 ⟶ and (and (SNo x2) (SNoLe 0 x2)) (SNoLt (mul_SNo x2 x2) x0)) (∀ x2 . x2 ∈ ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 1 ⟶ and (and (SNo x2) (SNoLe 0 x2)) (SNoLt x0 (mul_SNo x2 x2)))) ⟶ SNoCutP (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 0)) (famunion omega (λ x1 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x1) 1)) (proof)