Proofgold Address

Param SNo^SNo : ι → ο
Param exp_SNo_nat^exp_SNo_nat : ι → ι → ι
Param ordsucc^ordsucc : ι → ι
Param mul_SNo^mul_SNo : ι → ι → ι
Param nat_p^nat_p : ι → ο
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_1^nat_1 : nat_p 1
Known exp_SNo_nat_1^{exp_SNo_nat_1} : ∀ x0 . SNo x0 ⟶ exp_SNo_nat x0 1 = x0
Theorem exp_SNo_nat_2^{exp_SNo_nat_2} : ∀ x0 . SNo x0 ⟶ exp_SNo_nat x0 2 = mul_SNo x0 x0 (proof)
Param SNoLe^SNoLe : ι → ι → ο
Known SNo_sqr_nonneg^{SNo_sqr_nonneg} : ∀ x0 . SNo x0 ⟶ SNoLe 0 (mul_SNo x0 x0)
Theorem SNo_sqr_nonneg'^{SNo_sqr_nonneg} : ∀ x0 . SNo x0 ⟶ SNoLe 0 (exp_SNo_nat x0 2) (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 (mul_SNo x0 x0))
Theorem SNo_zero_or_sqr_pos'^{SNo_zero_or_sqr_pos} : ∀ x0 . SNo x0 ⟶ or (x0 = 0) (SNoLt 0 (exp_SNo_nat x0 2)) (proof)
Param SNo_pair^SNo_pair : ι → ι → ι
Param recip_SNo^recip_SNo : ι → ι
Definition div_SNo^div_SNo := λ x0 x1 . mul_SNo x0 (recip_SNo x1)
Param CSNo_Re^CSNo_Re : ι → ι
Param add_SNo^add_SNo : ι → ι → ι
Param CSNo_Im^CSNo_Im : ι → ι
Param minus_SNo^minus_SNo : ι → ι
Definition cbf48.. := λ x0 . SNo_pair (div_SNo (CSNo_Re x0) (add_SNo (exp_SNo_nat (CSNo_Re x0) 2) (exp_SNo_nat (CSNo_Im x0) 2))) (minus_SNo (div_SNo (CSNo_Im x0) (add_SNo (exp_SNo_nat (CSNo_Re x0) 2) (exp_SNo_nat (CSNo_Im x0) 2))))
Param mul_CSNo^mul_CSNo : ι → ι → ι
Definition bf082.. := λ x0 x1 . mul_CSNo x0 (cbf48.. x1)
Param CSNo^CSNo : ι → ο
Known 1e9ba..^mul_CSNo_com : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ mul_CSNo x0 x1 = mul_CSNo x1 x0
Known ca69e..^CSNo_1 : CSNo 1
Known 42258..^{mul_CSNo_oneL} : ∀ x0 . CSNo x0 ⟶ mul_CSNo 1 x0 = x0
Theorem 10244.. : ∀ x0 . CSNo x0 ⟶ mul_CSNo x0 1 = x0 (proof)
Known CSNo_I^CSNo_I : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ CSNo (SNo_pair x0 x1)
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)
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
Known CSNo_ImR^CSNo_ImR : ∀ x0 . CSNo x0 ⟶ SNo (CSNo_Im x0)
Known CSNo_ReR^CSNo_ReR : ∀ x0 . CSNo x0 ⟶ SNo (CSNo_Re x0)
Theorem 435ba.. : ∀ x0 . CSNo x0 ⟶ CSNo (cbf48.. x0) (proof)
Param add_CSNo^add_CSNo : ι → ι → ι
Param minus_CSNo^minus_CSNo : ι → ι
Param Complex_i^Complex_i : ι
Known 15de6..^{mul_SNo_mul_CSNo} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_CSNo x0 x1
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
Known d8721..^{CSNo_mul_CSNo} : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo (mul_CSNo x0 x1)
Known Re_1^Re_1 : CSNo_Re 1 = 1
Known a8c42..^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 x0) (CSNo_Im x1)))
Known 15fd0..^add_CSNo_CRe : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo_Re (add_CSNo x0 x1) = add_SNo (CSNo_Re x0) (CSNo_Re x1)
Known 2fac0..^add_CSNo_CIm : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo_Im (add_CSNo x0 x1) = add_SNo (CSNo_Im x0) (CSNo_Im x1)
Known 31582..^{minus_CSNo_CRe} : ∀ x0 . CSNo x0 ⟶ CSNo_Re (minus_CSNo x0) = minus_SNo (CSNo_Re x0)
Known 4f721..^{minus_CSNo_CIm} : ∀ x0 . CSNo x0 ⟶ CSNo_Im (minus_CSNo x0) = minus_SNo (CSNo_Im x0)
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 SNo_Complex_i^{SNo_Complex_i} : CSNo Complex_i
Known 2c425..^mul_CSNo_CIm : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo_Im (mul_CSNo x0 x1) = add_SNo (mul_SNo (CSNo_Re x0) (CSNo_Im x1)) (mul_SNo (CSNo_Im x0) (CSNo_Re x1))
Known Re_i^Re_i : CSNo_Re Complex_i = 0
Known Im_i^Im_i : CSNo_Im Complex_i = 1
Known mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Known SNo_0^SNo_0 : SNo 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 add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
Known mul_SNo_oneL^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 x0 = x0
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_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_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 Im_1^Im_1 : CSNo_Im 1 = 0
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)
Known CSNo_add_CSNo^{CSNo_add_CSNo} : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo (add_CSNo x0 x1)
Known CSNo_minus_CSNo^{CSNo_minus_CSNo} : ∀ x0 . CSNo x0 ⟶ CSNo (minus_CSNo x0)
Known SNo_CSNo^SNo_CSNo : ∀ x0 . SNo x0 ⟶ CSNo x0
Theorem da734.. : ∀ 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 b5ed6..^CSNo_0 : CSNo 0
Known Re_0^Re_0 : CSNo_Re 0 = 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 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_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 Im_0^Im_0 : CSNo_Im 0 = 0
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 8a0d1.. : ∀ x0 . CSNo x0 ⟶ add_SNo (exp_SNo_nat (CSNo_Re x0) 2) (exp_SNo_nat (CSNo_Im x0) 2) = 0 ⟶ x0 = 0 (proof)
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 recip_SNo_invR^{recip_SNo_invR} : ∀ x0 . SNo x0 ⟶ (x0 = 0 ⟶ ∀ x1 : ο . x1) ⟶ mul_SNo x0 (recip_SNo x0) = 1
Known SNo_recip_SNo^{SNo_recip_SNo} : ∀ x0 . SNo x0 ⟶ SNo (recip_SNo x0)
Theorem 7e4da.. : ∀ x0 . CSNo x0 ⟶ (x0 = 0 ⟶ ∀ x1 : ο . x1) ⟶ mul_CSNo x0 (cbf48.. x0) = 1 (proof)
Theorem 5ce1e.. : ∀ x0 . CSNo x0 ⟶ (x0 = 0 ⟶ ∀ x1 : ο . x1) ⟶ mul_CSNo (cbf48.. x0) x0 = 1 (proof)
Theorem f57ff.. : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ CSNo (bf082.. x0 x1) (proof)
Known 8912c..^{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
Theorem 4f630.. : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ (x1 = 0 ⟶ ∀ x2 : ο . x2) ⟶ mul_CSNo (bf082.. x0 x1) x1 = x0 (proof)
Theorem 5eaa5.. : ∀ x0 x1 . CSNo x0 ⟶ CSNo x1 ⟶ (x1 = 0 ⟶ ∀ x2 : ο . x2) ⟶ mul_CSNo x1 (bf082.. x0 x1) = x0 (proof)
Param complex^complex : ι
Param real^real : ι
Known complex_E^complex_E : ∀ x0 . x0 ∈ complex ⟶ ∀ x1 : ο . (∀ x2 . x2 ∈ real ⟶ ∀ x3 . x3 ∈ real ⟶ x0 = SNo_pair x2 x3 ⟶ x1) ⟶ x1
Known complex_I^complex_I : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ SNo_pair x0 x1 ∈ complex
Known real_div_SNo^real_div_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ div_SNo x0 x1 ∈ real
Known real_minus_SNo^{real_minus_SNo} : ∀ x0 . x0 ∈ real ⟶ minus_SNo x0 ∈ real
Known real_add_SNo^real_add_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ add_SNo x0 x1 ∈ real
Known real_mul_SNo^real_mul_SNo : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ mul_SNo x0 x1 ∈ real
Known complex_Im_eq^{complex_Im_eq} : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ CSNo_Im (SNo_pair x0 x1) = x1
Known complex_Re_eq^{complex_Re_eq} : ∀ x0 . x0 ∈ real ⟶ ∀ x1 . x1 ∈ real ⟶ CSNo_Re (SNo_pair x0 x1) = x0
Known complex_CSNo^complex_CSNo : ∀ x0 . x0 ∈ complex ⟶ CSNo x0
Theorem 4041d.. : ∀ x0 . x0 ∈ complex ⟶ cbf48.. x0 ∈ complex (proof)
Known complex_mul_CSNo^{complex_mul_CSNo} : ∀ x0 . x0 ∈ complex ⟶ ∀ x1 . x1 ∈ complex ⟶ mul_CSNo x0 x1 ∈ complex
Theorem e1c77.. : ∀ x0 . x0 ∈ complex ⟶ ∀ x1 . x1 ∈ complex ⟶ bf082.. x0 x1 ∈ complex (proof)