Proofgold Asset

Param SNo^SNo : ι → ο
Param nIn^nIn : ι → ι → ο
Param Sing^Sing : ι → ι
Param ordsucc^ordsucc : ι → ι
Known complex_tag_fresh^{complex_tag_fresh} : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ nIn (Sing 2) x1
Param pair_tag^pair_tag : ι → ι → ι → ι
Definition SNo_pair^SNo_pair := pair_tag (Sing 2)
Known SNo_pair_0^SNo_pair_0 : ∀ x0 . SNo_pair x0 0 = x0
Known SNo_pair_prop_1^{SNo_pair_prop_1} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x2 ⟶ SNo_pair x0 x1 = SNo_pair x2 x3 ⟶ x0 = x2
Known SNo_pair_prop_2^{SNo_pair_prop_2} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo_pair x0 x1 = SNo_pair x2 x3 ⟶ x1 = x3
Param CD_carr^CD_carr : ι → (ι → ο) → ι → ο
Definition CSNo^CSNo := CD_carr (Sing 2) SNo
Known CSNo_I^CSNo_I : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ CSNo (SNo_pair x0 x1)
Known CSNo_E^CSNo_E : ∀ x0 . CSNo x0 ⟶ ∀ x1 : ι → ο . (∀ x2 x3 . SNo x2 ⟶ SNo x3 ⟶ x0 = SNo_pair x2 x3 ⟶ x1 (SNo_pair x2 x3)) ⟶ x1 x0
Known SNo_CSNo^SNo_CSNo : ∀ x0 . SNo x0 ⟶ CSNo x0
Known CSNo_0^CSNo_0 : CSNo 0
Known CSNo_1^CSNo_1 : CSNo 1
Param ExtendedSNoElt_^{ExtendedSNoElt_} : ι → ι → ο
Known CSNo_ExtendedSNoElt_3^{CSNo_ExtendedSNoElt_3} : ∀ x0 . CSNo x0 ⟶ ExtendedSNoElt_ 3 x0
Definition Complex_i^Complex_i := SNo_pair 0 1
Param CD_proj0^CD_proj0 : ι → (ι → ο) → ι → ι
Definition CSNo_Re^CSNo_Re := CD_proj0 (Sing 2) SNo
Param CD_proj1^CD_proj1 : ι → (ι → ο) → ι → ι
Definition CSNo_Im^CSNo_Im := CD_proj1 (Sing 2) SNo
Known CSNo_Re2^CSNo_Re2 : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ CSNo_Re (SNo_pair x0 x1) = x0
Known CSNo_Im2^CSNo_Im2 : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ CSNo_Im (SNo_pair x0 x1) = x1
Known CSNo_ReR^CSNo_ReR : ∀ x0 . CSNo x0 ⟶ SNo (CSNo_Re x0)
Known CSNo_ImR^CSNo_ImR : ∀ x0 . CSNo x0 ⟶ SNo (CSNo_Im x0)
Known CSNo_ReIm_split^{CSNo_ReIm_split} : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo_Re x0 = CSNo_Re x1 ⟶ CSNo_Im x0 = CSNo_Im x1 ⟶ x0 = x1
Definition CD_minus^CD_minus := λ x0 . λ x1 : ι → ο . λ x2 : ι → ι . λ x3 . pair_tag x0 (x2 (CD_proj0 x0 x1 x3)) (x2 (CD_proj1 x0 x1 x3))
Param minus_SNo^minus_SNo : ι → ι
Definition minus_CSNo^minus_CSNo := CD_minus (Sing 2) SNo minus_SNo
Param CD_conj^CD_conj : ι → (ι → ο) → (ι → ι) → (ι → ι) → ι → ι
Definition conj_CSNo^conj_CSNo := CD_conj (Sing 2) SNo minus_SNo (λ x0 . x0)
Definition CD_add^CD_add := λ x0 . λ x1 : ι → ο . λ x2 : ι → ι → ι . λ x3 x4 . pair_tag x0 (x2 (CD_proj0 x0 x1 x3) (CD_proj0 x0 x1 x4)) (x2 (CD_proj1 x0 x1 x3) (CD_proj1 x0 x1 x4))
Param add_SNo^add_SNo : ι → ι → ι
Definition add_CSNo^add_CSNo := CD_add (Sing 2) SNo add_SNo
Definition CD_mul^CD_mul := λ x0 . λ x1 : ι → ο . λ x2 x3 : ι → ι . λ x4 x5 : ι → ι → ι . λ x6 x7 . pair_tag x0 (x4 (x5 (CD_proj0 x0 x1 x6) (CD_proj0 x0 x1 x7)) (x2 (x5 (x3 (CD_proj1 x0 x1 x7)) (CD_proj1 x0 x1 x6)))) (x4 (x5 (CD_proj1 x0 x1 x7) (CD_proj0 x0 x1 x6)) (x5 (CD_proj1 x0 x1 x6) (x3 (CD_proj0 x0 x1 x7))))
Param mul_SNo^mul_SNo : ι → ι → ι
Definition mul_CSNo^mul_CSNo := CD_mul (Sing 2) SNo minus_SNo (λ x0 . x0) add_SNo mul_SNo
Param CD_exp_nat^CD_exp_nat : ι → (ι → ο) → (ι → ι) → (ι → ι) → (ι → ι → ι) → (ι → ι → ι) → ι → ι → ι
Definition exp_CSNo_nat^exp_CSNo_nat := CD_exp_nat (Sing 2) SNo minus_SNo (λ x0 . x0) add_SNo mul_SNo
Param exp_SNo_nat^exp_SNo_nat : ι → ι → ι
Definition abs_sqr_CSNo^abs_sqr_CSNo := λ x0 . add_SNo (exp_SNo_nat (CSNo_Re x0) 2) (exp_SNo_nat (CSNo_Im x0) 2)
Param recip_SNo^recip_SNo : ι → ι
Definition div_SNo^div_SNo := λ x0 x1 . mul_SNo x0 (recip_SNo x1)
Definition recip_CSNo^recip_CSNo := λ x0 . SNo_pair (div_SNo (CSNo_Re x0) (abs_sqr_CSNo x0)) (minus_SNo (div_SNo (CSNo_Im x0) (abs_sqr_CSNo x0)))
Definition div_CSNo^div_CSNo := λ x0 x1 . mul_CSNo x0 (recip_CSNo x1)
Known CSNo_Complex_i^{CSNo_Complex_i} : CSNo Complex_i
Known CSNo_minus_CSNo^{CSNo_minus_CSNo} : ∀ x0 . CSNo x0 ⟶ CSNo (minus_CSNo x0)
Known minus_CSNo_CRe^{minus_CSNo_CRe} : ∀ x0 . CSNo x0 ⟶ CSNo_Re (minus_CSNo x0) = minus_SNo (CSNo_Re x0)
Known minus_CSNo_CIm^{minus_CSNo_CIm} : ∀ x0 . CSNo x0 ⟶ CSNo_Im (minus_CSNo x0) = minus_SNo (CSNo_Im x0)
Known CSNo_add_CSNo^{CSNo_add_CSNo} : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo (add_CSNo x0 x1)
Known add_CSNo_CRe^add_CSNo_CRe : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo_Re (add_CSNo x0 x1) = add_SNo (CSNo_Re x0) (CSNo_Re x1)
Known add_CSNo_CIm^add_CSNo_CIm : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo_Im (add_CSNo x0 x1) = add_SNo (CSNo_Im x0) (CSNo_Im x1)
Known CSNo_mul_CSNo^{CSNo_mul_CSNo} : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo (mul_CSNo x0 x1)
Known mul_CSNo_CRe^mul_CSNo_CRe : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo_Re (mul_CSNo x0 x1) = add_SNo (mul_SNo (CSNo_Re x0) (CSNo_Re x1)) (minus_SNo (mul_SNo (CSNo_Im x1) (CSNo_Im x0)))
Known mul_CSNo_CIm^mul_CSNo_CIm : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo_Im (mul_CSNo x0 x1) = add_SNo (mul_SNo (CSNo_Im x1) (CSNo_Re x0)) (mul_SNo (CSNo_Im x0) (CSNo_Re x1))
Known SNo_Re^SNo_Re : ∀ x0 . SNo x0 ⟶ CSNo_Re x0 = x0
Known SNo_Im^SNo_Im : ∀ x0 . SNo x0 ⟶ CSNo_Im x0 = 0
Known Re_0^Re_0 : CSNo_Re 0 = 0
Known Im_0^Im_0 : CSNo_Im 0 = 0
Known Re_1^Re_1 : CSNo_Re 1 = 1
Known Im_1^Im_1 : CSNo_Im 1 = 0
Known Re_i^Re_i : CSNo_Re Complex_i = 0
Known Im_i^Im_i : CSNo_Im Complex_i = 1
Known mul_CSNo_mul_SNo^{mul_CSNo_mul_SNo} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_CSNo x0 x1 = mul_SNo x0 x1
Known mul_CSNo_1R^mul_CSNo_1R : ∀ x0 . CSNo x0 ⟶ mul_CSNo x0 1 = x0
Known exp_CSNo_nat_2^{exp_CSNo_nat_2} : ∀ x0 . CSNo x0 ⟶ exp_CSNo_nat x0 2 = mul_CSNo x0 x0
Param nat_p^nat_p : ι → ο
Known CSNo_exp_CSNo_nat^{CSNo_exp_CSNo_nat} : ∀ x0 . CSNo x0 ⟶ ∀ x1 . nat_p x1 ⟶ CSNo (exp_CSNo_nat x0 x1)
Known mul_CSNo_com^mul_CSNo_com : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ mul_CSNo x0 x1 = mul_CSNo x1 x0
Known mul_CSNo_assoc^{mul_CSNo_assoc} : ∀ x0 x1 x2 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo x2 ⟶ mul_CSNo x0 (mul_CSNo x1 x2) = mul_CSNo (mul_CSNo x0 x1) x2
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Known SNo_exp_SNo_nat^{SNo_exp_SNo_nat} : ∀ x0 . SNo x0 ⟶ ∀ x1 . nat_p x1 ⟶ SNo (exp_SNo_nat x0 x1)
Known nat_2^nat_2 : nat_p 2
Theorem SNo_abs_sqr_CSNo^{SNo_abs_sqr_CSNo} : ∀ x0 . CSNo x0 ⟶ SNo (abs_sqr_CSNo x0) (proof)
Param SNoLe^SNoLe : ι → ι → ο
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Known SNo_0^SNo_0 : SNo 0
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 exp_SNo_nat_2^{exp_SNo_nat_2} : ∀ x0 . SNo x0 ⟶ exp_SNo_nat x0 2 = mul_SNo x0 x0
Known SNo_sqr_nonneg^{SNo_sqr_nonneg} : ∀ x0 . SNo x0 ⟶ SNoLe 0 (mul_SNo x0 x0)
Theorem abs_sqr_CSNo_nonneg^{abs_sqr_CSNo_nonneg} : ∀ x0 . CSNo x0 ⟶ SNoLe 0 (abs_sqr_CSNo x0) (proof)
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Param SNoLt^SNoLt : ι → ι → ο
Known SNo_zero_or_sqr_pos'^{SNo_zero_or_sqr_pos} : ∀ x0 . SNo x0 ⟶ or (x0 = 0) (SNoLt 0 (exp_SNo_nat x0 2))
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 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_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_sqr_nonneg'^{SNo_sqr_nonneg} : ∀ x0 . SNo x0 ⟶ SNoLe 0 (exp_SNo_nat x0 2)
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 abs_sqr_CSNo_zero^{abs_sqr_CSNo_zero} : ∀ x0 . CSNo x0 ⟶ abs_sqr_CSNo x0 = 0 ⟶ x0 = 0 (proof)
Known SNo_div_SNo^SNo_div_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (div_SNo x0 x1)
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Theorem CSNo_recip_CSNo^{CSNo_recip_CSNo} : ∀ x0 . CSNo x0 ⟶ CSNo (recip_CSNo x0) (proof)
Known mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Known mul_SNo_zeroL^{mul_SNo_zeroL} : ∀ x0 . SNo x0 ⟶ mul_SNo 0 x0 = 0
Known SNo_1^SNo_1 : SNo 1
Known minus_SNo_0^minus_SNo_0 : minus_SNo 0 = 0
Known mul_SNo_oneL^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 x0 = x0
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 minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = 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 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_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
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_minus_SNo_linv^{add_SNo_minus_SNo_linv} : ∀ x0 . SNo x0 ⟶ add_SNo (minus_SNo x0) x0 = 0
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 SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Theorem CSNo_relative_recip^{CSNo_relative_recip} : ∀ x0 . CSNo x0 ⟶ ∀ x1 . SNo x1 ⟶ mul_SNo (add_SNo (exp_SNo_nat (CSNo_Re x0) 2) (exp_SNo_nat (CSNo_Im x0) 2)) x1 = 1 ⟶ mul_CSNo x0 (add_CSNo (mul_CSNo x1 (CSNo_Re x0)) (minus_CSNo (mul_CSNo Complex_i (mul_CSNo x1 (CSNo_Im x0))))) = 1 (proof)
Known recip_SNo_invR^{recip_SNo_invR} : ∀ x0 . SNo x0 ⟶ (x0 = 0 ⟶ ∀ x1 : ο . x1) ⟶ mul_SNo x0 (recip_SNo x0) = 1
Known mul_SNo_oneR^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = x0
Known SNo_recip_SNo^{SNo_recip_SNo} : ∀ x0 . SNo x0 ⟶ SNo (recip_SNo x0)
Theorem recip_CSNo_invR^{recip_CSNo_invR} : ∀ x0 . CSNo x0 ⟶ (x0 = 0 ⟶ ∀ x1 : ο . x1) ⟶ mul_CSNo x0 (recip_CSNo x0) = 1 (proof)
Theorem recip_CSNo_invL^{recip_CSNo_invL} : ∀ x0 . CSNo x0 ⟶ (x0 = 0 ⟶ ∀ x1 : ο . x1) ⟶ mul_CSNo (recip_CSNo x0) x0 = 1 (proof)
Theorem CSNo_div_CSNo^{CSNo_div_CSNo} : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo (div_CSNo x0 x1) (proof)
Theorem mul_div_CSNo_invL^{mul_div_CSNo_invL} : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ (x1 = 0 ⟶ ∀ x2 : ο . x2) ⟶ mul_CSNo (div_CSNo x0 x1) x1 = x0 (proof)
Theorem mul_div_CSNo_invR^{mul_div_CSNo_invR} : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ (x1 = 0 ⟶ ∀ x2 : ο . x2) ⟶ mul_CSNo x1 (div_CSNo x0 x1) = x0 (proof)
Param sqrt_SNo_nonneg^{sqrt_SNo_nonneg} : ι → ι
Known SNoLeE^SNoLeE : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ or (SNoLt x0 x1) (x0 = x1)
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 pos_mul_SNo_Lt2^{pos_mul_SNo_Lt2} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLt 0 x0 ⟶ SNoLt 0 x1 ⟶ SNoLt x0 x2 ⟶ SNoLt x1 x3 ⟶ SNoLt (mul_SNo x0 x1) (mul_SNo x2 x3)
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 sqrt_SNo_nonneg_0^{sqrt_SNo_nonneg_0} : sqrt_SNo_nonneg 0 = 0
Known sqrt_SNo_nonneg_sqr^{sqrt_SNo_nonneg_sqr} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ mul_SNo (sqrt_SNo_nonneg x0) (sqrt_SNo_nonneg x0) = x0
Known sqrt_SNo_nonneg_nonneg^{sqrt_SNo_nonneg_nonneg} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ SNoLe 0 (sqrt_SNo_nonneg x0)
Known SNo_sqrt_SNo_nonneg^{SNo_sqrt_SNo_nonneg} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ SNo (sqrt_SNo_nonneg x0)
Theorem sqrt_SNo_nonneg_sqr_id^{sqrt_SNo_nonneg_sqr_id} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ sqrt_SNo_nonneg (exp_SNo_nat x0 2) = x0 (proof)
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 x0)
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)
Known SNoLe_tra^SNoLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLe x0 x2
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Theorem sqrt_SNo_nonneg_mon_strict^{sqrt_SNo_nonneg_mon_strict} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe 0 x0 ⟶ SNoLt x0 x1 ⟶ SNoLt (sqrt_SNo_nonneg x0) (sqrt_SNo_nonneg x1) (proof)
Known SNoLe_ref^SNoLe_ref : ∀ x0 . SNoLe x0 x0
Theorem sqrt_SNo_nonneg_mon^{sqrt_SNo_nonneg_mon} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe 0 x0 ⟶ SNoLe x0 x1 ⟶ SNoLe (sqrt_SNo_nonneg x0) (sqrt_SNo_nonneg x1) (proof)
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 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)
Theorem sqrt_SNo_nonneg_mul_SNo^{sqrt_SNo_nonneg_mul_SNo} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe 0 x0 ⟶ SNoLe 0 x1 ⟶ sqrt_SNo_nonneg (mul_SNo x0 x1) = mul_SNo (sqrt_SNo_nonneg x0) (sqrt_SNo_nonneg x1) (proof)
Definition modulus_CSNo^modulus_CSNo := λ x0 . sqrt_SNo_nonneg (abs_sqr_CSNo x0)
Theorem SNo_modulus_CSNo^{SNo_modulus_CSNo} : ∀ x0 . CSNo x0 ⟶ SNo (modulus_CSNo x0) (proof)
Theorem modulus_CSNo_nonneg^{modulus_CSNo_nonneg} : ∀ x0 . CSNo x0 ⟶ SNoLe 0 (modulus_CSNo x0) (proof)
Param If_i^If_i : ο → ι → ι → ι
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param eps_^eps_ : ι → ι
Definition sqrt_CSNo^sqrt_CSNo := λ x0 . If_i (or (SNoLt (CSNo_Im x0) 0) (and (CSNo_Im x0 = 0) (SNoLt (CSNo_Re x0) 0))) (SNo_pair (sqrt_SNo_nonneg (mul_SNo (eps_ 1) (add_SNo (CSNo_Re x0) (modulus_CSNo x0)))) (minus_SNo (sqrt_SNo_nonneg (mul_SNo (eps_ 1) (add_SNo (minus_SNo (CSNo_Re x0)) (modulus_CSNo x0)))))) (SNo_pair (sqrt_SNo_nonneg (mul_SNo (eps_ 1) (add_SNo (CSNo_Re x0) (modulus_CSNo x0)))) (sqrt_SNo_nonneg (mul_SNo (eps_ 1) (add_SNo (minus_SNo (CSNo_Re x0)) (modulus_CSNo x0)))))
Known If_i_1^If_i_1 : ∀ x0 : ο . ∀ x1 x2 . x0 ⟶ If_i x0 x1 x2 = 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
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 eps_1_half_eq1^{eps_1_half_eq1} : add_SNo (eps_ 1) (eps_ 1) = 1
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 orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known If_i_0^If_i_0 : ∀ x0 : ο . ∀ x1 x2 . not x0 ⟶ If_i x0 x1 x2 = x2
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_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_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 add_SNo_com_4_inner_mid^{add_SNo_com_4_inner_mid} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ add_SNo (add_SNo x0 x1) (add_SNo x2 x3) = add_SNo (add_SNo x0 x2) (add_SNo x1 x3)
Known add_SNo_minus_SNo_rinv^{add_SNo_minus_SNo_rinv} : ∀ x0 . SNo x0 ⟶ add_SNo x0 (minus_SNo x0) = 0
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_minus_L2^{add_SNo_minus_L2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo (minus_SNo x0) (add_SNo x0 x1) = x1
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Known add_SNo_minus_Le2b^{add_SNo_minus_Le2b} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe (add_SNo x2 x1) x0 ⟶ SNoLe x2 (add_SNo x0 (minus_SNo x1))
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 nat_1^nat_1 : nat_p 1
Known SNo_eps_^SNo_eps_ : ∀ x0 . x0 ∈ omega ⟶ SNo (eps_ x0)
Theorem sqrt_CSNo_sqrt^{sqrt_CSNo_sqrt} : ∀ x0 . CSNo x0 ⟶ exp_CSNo_nat (sqrt_CSNo x0) 2 = x0 (proof)
Param lam^Sigma : ι → (ι → ι) → ι
Definition setprod^setprod := λ x0 x1 . lam x0 (λ x2 . x1)
Param real^real : ι
Param ap^ap : ι → ι → ι
Definition complex^complex := {SNo_pair (ap x0 0) (ap x0 1)|x0 ∈ setprod real real}
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 tuple_2_1_eq^tuple_2_1_eq : ∀ x0 x1 . ap (lam 2 (λ x3 . If_i (x3 = 0) x0 x1)) 1 = x1
Known ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Known tuple_2_setprod^{tuple_2_setprod} : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x1 ⟶ lam 2 (λ x4 . If_i (x4 = 0) x2 x3) ∈ setprod x0 x1
Theorem complex_I^complex_I : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ SNo_pair x0 x1 ∈ complex (proof)
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 ap0_Sigma^ap0_Sigma : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ ap x2 0 ∈ x0
Known ap1_Sigma^ap1_Sigma : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ ap x2 1 ∈ x1 (ap x2 0)
Theorem complex_E^complex_E : ∀ x0 . x0 ∈ complex ⟶ ∀ x1 : ο . (∀ x2 . x2 ∈ real ⟶ ∀ x3 . x3 ∈ real ⟶ x0 = SNo_pair x2 x3 ⟶ x1) ⟶ x1 (proof)
Known real_SNo^real_SNo : ∀ x0 . x0 ∈ real ⟶ SNo x0
Theorem complex_CSNo^complex_CSNo : ∀ x0 . x0 ∈ complex ⟶ CSNo x0 (proof)
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known real_0^real_0 : 0 ∈ real
Theorem real_complex^real_complex : real ⊆ complex (proof)
Theorem complex_0^complex_0 : 0 ∈ complex (proof)
Known real_1^real_1 : 1 ∈ real
Theorem complex_1^complex_1 : 1 ∈ complex (proof)
Theorem complex_i^complex_i : Complex_i ∈ complex (proof)
Theorem complex_Re_eq^{complex_Re_eq} : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ CSNo_Re (SNo_pair x0 x1) = x0 (proof)
Theorem complex_Im_eq^{complex_Im_eq} : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ CSNo_Im (SNo_pair x0 x1) = x1 (proof)
Theorem complex_Re_real^{complex_Re_real} : ∀ x0 . x0 ∈ complex ⟶ CSNo_Re x0 ∈ real (proof)
Theorem complex_Im_real^{complex_Im_real} : ∀ x0 . x0 ∈ complex ⟶ CSNo_Im x0 ∈ real (proof)
Theorem complex_ReIm_split^{complex_ReIm_split} : ∀ x0 . x0 ∈ complex ⟶ ∀ x1 . x1 ∈ complex ⟶ CSNo_Re x0 = CSNo_Re x1 ⟶ CSNo_Im x0 = CSNo_Im x1 ⟶ x0 = x1 (proof)
Known real_minus_SNo^{real_minus_SNo} : ∀ x0 . x0 ∈ real ⟶ minus_SNo x0 ∈ real
Theorem complex_minus_CSNo^{complex_minus_CSNo} : ∀ x0 . x0 ∈ complex ⟶ minus_CSNo x0 ∈ complex (proof)
Known real_add_SNo^real_add_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ add_SNo x0 x1 ∈ real
Theorem complex_add_CSNo^{complex_add_CSNo} : ∀ x0 . x0 ∈ complex ⟶ ∀ x1 . x1 ∈ complex ⟶ add_CSNo x0 x1 ∈ complex (proof)
Known real_mul_SNo^real_mul_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ mul_SNo x0 x1 ∈ real
Theorem complex_mul_CSNo^{complex_mul_CSNo} : ∀ x0 . x0 ∈ complex ⟶ ∀ x1 . x1 ∈ complex ⟶ mul_CSNo x0 x1 ∈ complex (proof)
Theorem real_Re_eq^real_Re_eq : ∀ x0 . x0 ∈ real ⟶ CSNo_Re x0 = x0 (proof)
Theorem real_Im_eq^real_Im_eq : ∀ x0 . x0 ∈ real ⟶ CSNo_Im x0 = 0 (proof)
Theorem mul_i_real_eq^{mul_i_real_eq} : ∀ x0 . x0 ∈ real ⟶ mul_CSNo Complex_i x0 = SNo_pair 0 x0 (proof)
Theorem real_Re_i_eq^real_Re_i_eq : ∀ x0 . x0 ∈ real ⟶ CSNo_Re (mul_CSNo Complex_i x0) = 0 (proof)
Theorem real_Im_i_eq^real_Im_i_eq : ∀ x0 . x0 ∈ real ⟶ CSNo_Im (mul_CSNo Complex_i x0) = x0 (proof)
Theorem complex_eta^complex_eta : ∀ x0 . x0 ∈ complex ⟶ x0 = add_CSNo (CSNo_Re x0) (mul_CSNo Complex_i (CSNo_Im x0)) (proof)
Known real_div_SNo^real_div_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ div_SNo x0 x1 ∈ real
Theorem complex_recip_CSNo^{complex_recip_CSNo} : ∀ x0 . x0 ∈ complex ⟶ recip_CSNo x0 ∈ complex (proof)
Theorem complex_div_CSNo^{complex_div_CSNo} : ∀ x0 . x0 ∈ complex ⟶ ∀ x1 . x1 ∈ complex ⟶ div_CSNo x0 x1 ∈ complex (proof)
Param Sep^Sep : ι → (ι → ο) → ι
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 SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Theorem complex_real_set_eq^{complex_real_set_eq} : real = {x1 ∈ complex|CSNo_Re x1 = x1} (proof)