Proofgold Signed Transaction

Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
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 famunionI^famunionI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ x0 ⟶ x3 ∈ x1 x2 ⟶ x3 ∈ famunion x0 x1
Theorem famunion_Subq^{famunion_Subq} : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x1 x3 ⊆ x2 x3) ⟶ famunion x0 x1 ⊆ famunion x0 x2 (proof)
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Known Subq_ref^Subq_ref : ∀ x0 . x0 ⊆ x0
Theorem famunion_ext^famunion_ext : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x1 x3 = x2 x3) ⟶ famunion x0 x1 = famunion x0 x2 (proof)
Param nat_p^nat_p : ι → ο
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param add_nat^add_nat : ι → ι → ι
Param ordsucc^ordsucc : ι → ι
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 andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known add_nat_0R^add_nat_0R : ∀ x0 . add_nat x0 0 = x0
Known nat_inv_impred^{nat_inv_impred} : ∀ x0 : ι → ο . x0 0 ⟶ (∀ x1 . nat_p x1 ⟶ x0 (ordsucc x1)) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Param ordinal^ordinal : ι → ο
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known ordinal_In_Or_Subq^{ordinal_In_Or_Subq} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ or (x0 ∈ x1) (x1 ⊆ x0)
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Known nat_ordsucc_in_ordsucc^{nat_ordsucc_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ∈ ordsucc x0
Known add_nat_SR^add_nat_SR : ∀ x0 x1 . nat_p x1 ⟶ add_nat x0 (ordsucc x1) = ordsucc (add_nat x0 x1)
Theorem nat_Subq_add_ex^{nat_Subq_add_ex} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ x0 ⊆ x1 ⟶ ∀ x2 : ο . (∀ x3 . and (nat_p x3) (x1 = add_nat x3 x0) ⟶ x2) ⟶ x2 (proof)
Param SNo^SNo : ι → ο
Param SNoLe^SNoLe : ι → ι → ο
Param SNoLt^SNoLt : ι → ι → ο
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 orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Known SNoLeLt_tra^SNoLeLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Theorem SNoLeE^SNoLeE : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ or (SNoLt x0 x1) (x0 = x1) (proof)
Param mul_SNo^mul_SNo : ι → ι → ι
Known SNo_0^SNo_0 : SNo 0
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 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 mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Known SNoLe_ref^SNoLe_ref : ∀ x0 . SNoLe x0 x0
Known mul_SNo_zeroL^{mul_SNo_zeroL} : ∀ x0 . SNo x0 ⟶ mul_SNo 0 x0 = 0
Theorem 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) (proof)
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)
Theorem SNo_pos_sqr_uniq^{SNo_pos_sqr_uniq} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt 0 x0 ⟶ SNoLt 0 x1 ⟶ mul_SNo x0 x0 = mul_SNo x1 x1 ⟶ x0 = x1 (proof)
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_nonneg_sqr_uniq^{SNo_nonneg_sqr_uniq} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe 0 x0 ⟶ SNoLe 0 x1 ⟶ mul_SNo x0 x0 = mul_SNo x1 x1 ⟶ x0 = x1 (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 SNoL^SNoL : ι → ι
Param SNoCut^SNoCut : ι → ι → ι
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Param SNoLev^SNoLev : ι → ι
Param binunion^binunion : ι → ι → ι
Param iff^iff : ο → ο → ο
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)
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 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 andEL^andEL : ∀ x0 x1 : ο . and x0 x1 ⟶ x0
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 x0)
Known SNoLt_tra^SNoLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Theorem SNoL_SNoCutP_ex^{SNoL_SNoCutP_ex} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ ∀ x2 . x2 ∈ SNoL (SNoCut x0 x1) ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ x0) (SNoLe x2 x4) ⟶ x3) ⟶ x3 (proof)
Param SNoR^SNoR : ι → ι
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
Theorem SNoR_SNoCutP_ex^{SNoR_SNoCutP_ex} : ∀ x0 x1 . SNoCutP x0 x1 ⟶ ∀ x2 . x2 ∈ SNoR (SNoCut x0 x1) ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ x1) (SNoLe x4 x2) ⟶ x3) ⟶ x3 (proof)
Param binintersect^binintersect : ι → ι → ι
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 ordinal_SNoLev^{ordinal_SNoLev} : ∀ x0 . ordinal x0 ⟶ SNoLev x0 = x0
Known ordinal_Empty^{ordinal_Empty} : ordinal 0
Known binintersectE1^{binintersectE1} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ x2 ∈ x0
Known SNo_eq^SNo_eq : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x0 = SNoLev x1 ⟶ SNoEq_ (SNoLev x0) x0 x1 ⟶ x0 = x1
Known SNo_1^SNo_1 : SNo 1
Known nat_1^nat_1 : nat_p 1
Known cases_1^cases_1 : ∀ x0 . x0 ∈ 1 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 x0
Known iffI^iffI : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ iff x0 x1
Known In_0_1^In_0_1 : 0 ∈ 1
Theorem pos_low_eq_one^{pos_low_eq_one} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ SNoLev x0 ⊆ 1 ⟶ x0 = 1 (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
Known SNoLe_antisym^{SNoLe_antisym} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x0 ⟶ x0 = x1
Theorem mul_SNo_nonpos_pos^{mul_SNo_nonpos_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 0 ⟶ SNoLt 0 x1 ⟶ SNoLe (mul_SNo x0 x1) 0 (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 mul_SNo_nonpos_neg^{mul_SNo_nonpos_neg} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 0 ⟶ SNoLt x1 0 ⟶ SNoLe 0 (mul_SNo x0 x1) (proof)
Known 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)
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 Sep^Sep : ι → (ι → ο) → ι
Definition SNoL_pos^SNoL_pos := λ x0 . Sep (SNoL x0) (SNoLt 0)
Known neq_0_1^neq_0_1 : 0 = 1 ⟶ ∀ x0 : ο . x0
Known SNoLt_0_1^SNoLt_0_1 : SNoLt 0 1
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
Theorem SNo_recip_pos_pos^{SNo_recip_pos_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt 0 x0 ⟶ mul_SNo x0 x1 = 1 ⟶ SNoLt 0 x1 (proof)
Param add_SNo^add_SNo : ι → ι → ι
Param minus_SNo^minus_SNo : ι → ι
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
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 SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Known add_SNo_minus_Lt2b^{add_SNo_minus_Lt2b} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt (add_SNo x2 x1) x0 ⟶ SNoLt x2 (add_SNo x0 (minus_SNo 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 SNo_recip_lem1^{SNo_recip_lem1} : ∀ x0 x1 x2 x3 x4 . SNo x0 ⟶ SNoLt 0 x0 ⟶ x1 ∈ SNoL_pos x0 ⟶ SNo x2 ⟶ mul_SNo x1 x2 = 1 ⟶ SNo x3 ⟶ SNoLt (mul_SNo x0 x3) 1 ⟶ SNo x4 ⟶ add_SNo 1 (minus_SNo (mul_SNo x0 x4)) = mul_SNo (add_SNo 1 (minus_SNo (mul_SNo x0 x3))) (mul_SNo (add_SNo x1 (minus_SNo x0)) x2) ⟶ SNoLt 1 (mul_SNo x0 x4) (proof)
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
Theorem SNo_recip_lem2^{SNo_recip_lem2} : ∀ x0 x1 x2 x3 x4 . SNo x0 ⟶ SNoLt 0 x0 ⟶ x1 ∈ SNoL_pos x0 ⟶ SNo x2 ⟶ mul_SNo x1 x2 = 1 ⟶ SNo x3 ⟶ SNoLt 1 (mul_SNo x0 x3) ⟶ SNo x4 ⟶ add_SNo 1 (minus_SNo (mul_SNo x0 x4)) = mul_SNo (add_SNo 1 (minus_SNo (mul_SNo x0 x3))) (mul_SNo (add_SNo x1 (minus_SNo x0)) x2) ⟶ SNoLt (mul_SNo x0 x4) 1 (proof)
Theorem SNo_recip_lem3^{SNo_recip_lem3} : ∀ x0 x1 x2 x3 x4 . SNo x0 ⟶ SNoLt 0 x0 ⟶ x1 ∈ SNoR x0 ⟶ SNo x2 ⟶ mul_SNo x1 x2 = 1 ⟶ SNo x3 ⟶ SNoLt (mul_SNo x0 x3) 1 ⟶ SNo x4 ⟶ add_SNo 1 (minus_SNo (mul_SNo x0 x4)) = mul_SNo (add_SNo 1 (minus_SNo (mul_SNo x0 x3))) (mul_SNo (add_SNo x1 (minus_SNo x0)) x2) ⟶ SNoLt (mul_SNo x0 x4) 1 (proof)
Theorem SNo_recip_lem4^{SNo_recip_lem4} : ∀ x0 x1 x2 x3 x4 . SNo x0 ⟶ SNoLt 0 x0 ⟶ x1 ∈ SNoR x0 ⟶ SNo x2 ⟶ mul_SNo x1 x2 = 1 ⟶ SNo x3 ⟶ SNoLt 1 (mul_SNo x0 x3) ⟶ SNo x4 ⟶ add_SNo 1 (minus_SNo (mul_SNo x0 x4)) = mul_SNo (add_SNo 1 (minus_SNo (mul_SNo x0 x3))) (mul_SNo (add_SNo x1 (minus_SNo x0)) x2) ⟶ SNoLt 1 (mul_SNo x0 x4) (proof)
Definition SNo_recipauxset^{SNo_recipauxset} := λ x0 x1 x2 . λ x3 : ι → ι . famunion x0 (λ x4 . {mul_SNo (add_SNo 1 (mul_SNo (add_SNo x5 (minus_SNo x1)) x4)) (x3 x5)|x5 ∈ x2})
Known ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Theorem SNo_recipauxset_I^{SNo_recipauxset_I} : ∀ x0 x1 x2 . ∀ x3 : ι → ι . ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x2 ⟶ mul_SNo (add_SNo 1 (mul_SNo (add_SNo x5 (minus_SNo x1)) x4)) (x3 x5) ∈ SNo_recipauxset x0 x1 x2 x3 (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
Theorem SNo_recipauxset_E^{SNo_recipauxset_E} : ∀ x0 x1 x2 . ∀ x3 : ι → ι . ∀ x4 . x4 ∈ SNo_recipauxset x0 x1 x2 x3 ⟶ ∀ x5 : ο . (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x2 ⟶ x4 = mul_SNo (add_SNo 1 (mul_SNo (add_SNo x7 (minus_SNo x1)) x6)) (x3 x7) ⟶ x5) ⟶ x5 (proof)
Known ReplEq_ext^ReplEq_ext : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x1 x3 = x2 x3) ⟶ prim5 x0 x1 = prim5 x0 x2
Theorem SNo_recipauxset_ext^{SNo_recipauxset_ext} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι . (∀ x5 . x5 ∈ x2 ⟶ x3 x5 = x4 x5) ⟶ SNo_recipauxset x0 x1 x2 x3 = SNo_recipauxset x0 x1 x2 x4 (proof)
Param nat_primrec^nat_primrec : ι → (ι → ι → ι) → ι → ι
Param lam^Sigma : ι → (ι → ι) → ι
Param If_i^If_i : ο → ι → ι → ι
Param Sing^Sing : ι → ι
Param ap^ap : ι → ι → ι
Definition SNo_recipaux^SNo_recipaux := λ x0 . λ x1 : ι → ι . nat_primrec (lam 2 (λ x2 . If_i (x2 = 0) (Sing 0) 0)) (λ x2 x3 . lam 2 (λ x4 . If_i (x4 = 0) (binunion (binunion (ap x3 0) (SNo_recipauxset (ap x3 0) x0 (SNoR x0) x1)) (SNo_recipauxset (ap x3 1) x0 (SNoL_pos x0) x1)) (binunion (binunion (ap x3 1) (SNo_recipauxset (ap x3 0) x0 (SNoL_pos x0) x1)) (SNo_recipauxset (ap x3 1) x0 (SNoR x0) x1))))
Known nat_primrec_0^{nat_primrec_0} : ∀ x0 . ∀ x1 : ι → ι → ι . nat_primrec x0 x1 0 = x0
Theorem SNo_recipaux_0^{SNo_recipaux_0} : ∀ x0 . ∀ x1 : ι → ι . SNo_recipaux x0 x1 0 = lam 2 (λ x3 . If_i (x3 = 0) (Sing 0) 0) (proof)
Known nat_primrec_S^{nat_primrec_S} : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . nat_p x2 ⟶ nat_primrec x0 x1 (ordsucc x2) = x1 x2 (nat_primrec x0 x1 x2)
Theorem SNo_recipaux_S^{SNo_recipaux_S} : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . nat_p x2 ⟶ SNo_recipaux x0 x1 (ordsucc x2) = lam 2 (λ x4 . If_i (x4 = 0) (binunion (binunion (ap (SNo_recipaux x0 x1 x2) 0) (SNo_recipauxset (ap (SNo_recipaux x0 x1 x2) 0) x0 (SNoR x0) x1)) (SNo_recipauxset (ap (SNo_recipaux x0 x1 x2) 1) x0 (SNoL_pos x0) x1)) (binunion (binunion (ap (SNo_recipaux x0 x1 x2) 1) (SNo_recipauxset (ap (SNo_recipaux x0 x1 x2) 0) x0 (SNoL_pos x0) x1)) (SNo_recipauxset (ap (SNo_recipaux x0 x1 x2) 1) x0 (SNoR x0) x1))) (proof)
Param SNoS_^SNoS_ : ι → ι
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 SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = 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 binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Known SNoS_I2^SNoS_I2 : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x0 ∈ SNoLev x1 ⟶ x0 ∈ SNoS_ (SNoLev x1)
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 SNoLev_ordinal^{SNoLev_ordinal} : ∀ x0 . SNo x0 ⟶ ordinal (SNoLev 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 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 mul_SNo_oneL^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 x0 = x0
Known mul_SNo_oneR^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = 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 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 mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
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 SNo_recipaux_lem1^{SNo_recipaux_lem1} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ ∀ x1 : ι → ι . (∀ x2 . x2 ∈ SNoS_ (SNoLev x0) ⟶ SNoLt 0 x2 ⟶ and (SNo (x1 x2)) (mul_SNo x2 (x1 x2) = 1)) ⟶ ∀ x2 . nat_p x2 ⟶ and (∀ x3 . x3 ∈ ap (SNo_recipaux x0 x1 x2) 0 ⟶ and (SNo x3) (SNoLt (mul_SNo x0 x3) 1)) (∀ x3 . x3 ∈ ap (SNo_recipaux x0 x1 x2) 1 ⟶ and (SNo x3) (SNoLt 1 (mul_SNo x0 x3))) (proof)
Param omega^omega : ι
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
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)
Theorem SNo_recipaux_lem2^{SNo_recipaux_lem2} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ ∀ x1 : ι → ι . (∀ x2 . x2 ∈ SNoS_ (SNoLev x0) ⟶ SNoLt 0 x2 ⟶ and (SNo (x1 x2)) (mul_SNo x2 (x1 x2) = 1)) ⟶ SNoCutP (famunion omega (λ x2 . ap (SNo_recipaux x0 x1 x2) 0)) (famunion omega (λ x2 . ap (SNo_recipaux x0 x1 x2) 1)) (proof)
Known SepE1^SepE1 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x2 ∈ x0
Theorem SNo_recipaux_ext^{SNo_recipaux_ext} : ∀ x0 . SNo x0 ⟶ ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ SNoS_ (SNoLev x0) ⟶ x1 x3 = x2 x3) ⟶ ∀ x3 . nat_p x3 ⟶ SNo_recipaux x0 x1 x3 = SNo_recipaux x0 x2 x3 (proof)
Param SNo_rec_i^SNo_rec_i : (ι → (ι → ι) → ι) → ι → ι
Definition recip_SNo_pos^{recip_SNo_pos} := SNo_rec_i (λ x0 . λ x1 : ι → ι . SNoCut (famunion omega (λ x2 . ap (SNo_recipaux x0 x1 x2) 0)) (famunion omega (λ x2 . ap (SNo_recipaux x0 x1 x2) 1)))
Known SNo_rec_i_eq^SNo_rec_i_eq : ∀ x0 : ι → (ι → ι) → ι . (∀ x1 . SNo x1 ⟶ ∀ x2 x3 : ι → ι . (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ x2 x4 = x3 x4) ⟶ x0 x1 x2 = x0 x1 x3) ⟶ ∀ x1 . SNo x1 ⟶ SNo_rec_i x0 x1 = x0 x1 (SNo_rec_i x0)
Theorem recip_SNo_pos_eq^{recip_SNo_pos_eq} : ∀ x0 . SNo x0 ⟶ recip_SNo_pos x0 = SNoCut (famunion omega (λ x2 . ap (SNo_recipaux x0 recip_SNo_pos x2) 0)) (famunion omega (λ x2 . ap (SNo_recipaux x0 recip_SNo_pos x2) 1)) (proof)
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)
Definition div_SNo^div_SNo := λ x0 x1 . mul_SNo x0 (recip_SNo x1)
Definition SNoL_nonneg^SNoL_nonneg := λ x0 . Sep (SNoL x0) (SNoLe 0)
Known Empty_eq^Empty_eq : ∀ x0 . (∀ x1 . nIn x1 x0) ⟶ x0 = 0
Theorem Sep_Empty^Sep_Empty : ∀ x0 : ι → ο . Sep 0 x0 = 0 (proof)
Known SNoL_0^SNoL_0 : SNoL 0 = 0
Theorem SNoL_nonneg_0^{SNoL_nonneg_0} : SNoL_nonneg 0 = 0 (proof)
Known SNoL_1^SNoL_1 : SNoL 1 = 1
Known Sep_Subq^Sep_Subq : ∀ x0 . ∀ x1 : ι → ο . Sep x0 x1 ⊆ x0
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Theorem SNoL_nonneg_1^{SNoL_nonneg_1} : SNoL_nonneg 1 = 1 (proof)
Param ReplSep^ReplSep : ι → (ι → ο) → (ι → ι) → ι
Definition SNo_sqrtauxset^{SNo_sqrtauxset} := λ x0 x1 x2 . famunion x0 (λ x3 . {div_SNo (add_SNo x2 (mul_SNo x3 x4)) (add_SNo x3 x4)|x4 ∈ x1,SNoLt 0 (add_SNo x3 x4)})
Known ReplSepI^ReplSepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 : ι → ι . ∀ x3 . x3 ∈ x0 ⟶ x1 x3 ⟶ x2 x3 ∈ ReplSep x0 x1 x2
Theorem SNo_sqrtauxset_I^{SNo_sqrtauxset_I} : ∀ x0 x1 x2 x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x1 ⟶ SNoLt 0 (add_SNo x3 x4) ⟶ div_SNo (add_SNo x2 (mul_SNo x3 x4)) (add_SNo x3 x4) ∈ SNo_sqrtauxset x0 x1 x2 (proof)
Known ReplSepE_impred^{ReplSepE_impred} : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 : ι → ι . ∀ x3 . x3 ∈ ReplSep x0 x1 x2 ⟶ ∀ x4 : ο . (∀ x5 . x5 ∈ x0 ⟶ x1 x5 ⟶ x3 = x2 x5 ⟶ x4) ⟶ x4
Theorem 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 (proof)
Theorem SNo_sqrtauxset_0^{SNo_sqrtauxset_0} : ∀ x0 x1 . SNo_sqrtauxset 0 x0 x1 = 0 (proof)
Theorem SNo_sqrtauxset_0'^{SNo_sqrtauxset_0} : ∀ x0 x1 . SNo_sqrtauxset x0 0 x1 = 0 (proof)
Definition SNo_sqrtaux^SNo_sqrtaux := λ x0 . λ x1 : ι → ι . nat_primrec (lam 2 (λ x2 . If_i (x2 = 0) (prim5 (SNoL_nonneg x0) x1) (prim5 (SNoR x0) x1))) (λ x2 x3 . lam 2 (λ x4 . If_i (x4 = 0) (binunion (ap x3 0) (SNo_sqrtauxset (ap x3 0) (ap x3 1) x0)) (binunion (binunion (ap x3 1) (SNo_sqrtauxset (ap x3 0) (ap x3 0) x0)) (SNo_sqrtauxset (ap x3 1) (ap x3 1) x0))))
Theorem 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)) (proof)
Theorem 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))) (proof)
Known add_nat_p^add_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (add_nat x0 x1)
Known Subq_tra^Subq_tra : ∀ x0 x1 x2 . x0 ⊆ x1 ⟶ x1 ⊆ x2 ⟶ x0 ⊆ x2
Known binunion_Subq_1^{binunion_Subq_1} : ∀ x0 x1 . x0 ⊆ binunion x0 x1
Theorem SNo_sqrtaux_mon_lem^{SNo_sqrtaux_mon_lem} : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . nat_p x2 ⟶ ∀ x3 . nat_p x3 ⟶ and (ap (SNo_sqrtaux x0 x1 x2) 0 ⊆ ap (SNo_sqrtaux x0 x1 (add_nat x2 x3)) 0) (ap (SNo_sqrtaux x0 x1 x2) 1 ⊆ ap (SNo_sqrtaux x0 x1 (add_nat x2 x3)) 1) (proof)
Known add_nat_com^add_nat_com : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ add_nat x0 x1 = add_nat x1 x0
Theorem SNo_sqrtaux_mon^{SNo_sqrtaux_mon} : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . nat_p x2 ⟶ ∀ x3 . nat_p x3 ⟶ x2 ⊆ x3 ⟶ and (ap (SNo_sqrtaux x0 x1 x2) 0 ⊆ ap (SNo_sqrtaux x0 x1 x3) 0) (ap (SNo_sqrtaux x0 x1 x2) 1 ⊆ ap (SNo_sqrtaux x0 x1 x3) 1) (proof)
Theorem SNo_sqrtaux_ext^{SNo_sqrtaux_ext} : ∀ x0 . SNo x0 ⟶ ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ SNoS_ (SNoLev x0) ⟶ x1 x3 = x2 x3) ⟶ ∀ x3 . nat_p x3 ⟶ SNo_sqrtaux x0 x1 x3 = SNo_sqrtaux x0 x2 x3 (proof)
Definition sqrt_SNo_nonneg^{sqrt_SNo_nonneg} := SNo_rec_i (λ x0 . λ x1 : ι → ι . SNoCut (famunion omega (λ x2 . ap (SNo_sqrtaux x0 x1 x2) 0)) (famunion omega (λ x2 . ap (SNo_sqrtaux x0 x1 x2) 1)))
Theorem sqrt_SNo_nonneg_eq^{sqrt_SNo_nonneg_eq} : ∀ x0 . SNo x0 ⟶ sqrt_SNo_nonneg x0 = SNoCut (famunion omega (λ x2 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x2) 0)) (famunion omega (λ x2 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x2) 1)) (proof)