Proofgold Address

Param SNo^SNo : ι → ο
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Param PNoLt^PNoLt : ι → (ι → ο) → ι → (ι → ο) → ο
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param PNoEq_^PNoEq_ : ι → (ι → ο) → (ι → ο) → ο
Definition PNoLe^PNoLe := λ x0 . λ x1 : ι → ο . λ x2 . λ x3 : ι → ο . or (PNoLt x0 x1 x2 x3) (and (x0 = x2) (PNoEq_ x0 x1 x3))
Param SNoLev^SNoLev : ι → ι
Definition SNoLe^SNoLe := λ x0 x1 . PNoLe (SNoLev x0) (λ x2 . x2 ∈ x0) (SNoLev x1) (λ x2 . x2 ∈ x1)
Param SNoLt^SNoLt : ι → ι → ο
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
Param omega^omega : ι
Param ap^ap : ι → ι → ι
Param SNo_sqrtaux^SNo_sqrtaux : ι → (ι → ι) → ι → ι
Param sqrt_SNo_nonneg^{sqrt_SNo_nonneg} : ι → ι
Param ordsucc^ordsucc : ι → ι
Param mul_SNo^mul_SNo : ι → ι → ι
Param SNoCut^SNoCut : ι → ι → ι
Known SNoCut_0_0^SNoCut_0_0 : SNoCut 0 0 = 0
Known SNoCut_Le^SNoCut_Le : ∀ x0 x1 x2 x3 . SNoCutP x0 x1 ⟶ SNoCutP x2 x3 ⟶ (∀ x4 . x4 ∈ x0 ⟶ SNoLt x4 (SNoCut x2 x3)) ⟶ (∀ x4 . x4 ∈ x3 ⟶ SNoLt (SNoCut x0 x1) x4) ⟶ SNoLe (SNoCut x0 x1) (SNoCut x2 x3)
Known SNoCutP_0_0^SNoCutP_0_0 : SNoCutP 0 0
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 SNoLeE^SNoLeE : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ or (SNoLt x0 x1) (x0 = x1)
Known SNo_0^SNo_0 : SNo 0
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 mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Theorem sqrt_SNo_nonneg_prop1c^{sqrt_SNo_nonneg_prop1c} : ∀ x0 . SNo x0 ⟶ SNoLe 0 x0 ⟶ 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)) ⟶ (∀ x1 . x1 ∈ famunion omega (λ x2 . ap (SNo_sqrtaux x0 sqrt_SNo_nonneg x2) 1) ⟶ ∀ x2 : ο . (SNo x1 ⟶ SNoLe 0 x1 ⟶ SNoLt x0 (mul_SNo x1 x1) ⟶ x2) ⟶ x2) ⟶ SNoLe 0 (SNoCut (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)
Param SNoS_^SNoS_ : ι → ι
Param binunion^binunion : ι → ι → ι
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param SNoEq_^SNoEq_ : ι → ι → ι → ο
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
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 SNoS_I2^SNoS_I2 : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x0 ∈ SNoLev x1 ⟶ x0 ∈ SNoS_ (SNoLev x1)
Known binintersectE2^{binintersectE2} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ x2 ∈ x1
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 SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Known famunionI^famunionI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ x0 ⟶ x3 ∈ x1 x2 ⟶ x3 ∈ famunion x0 x1
Param nat_p^nat_p : ι → ο
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known nat_0^nat_0 : nat_p 0
Param lam^Sigma : ι → (ι → ι) → ι
Param If_i^If_i : ο → ι → ι → ι
Param Sep^Sep : ι → (ι → ο) → ι
Param SNoL^SNoL : 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 ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
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 SNoLeLt_tra^SNoLeLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known 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
Param add_SNo^add_SNo : ι → ι → ι
Known mul_SNo_SNoCut_SNoR_interpolate_impred^{mul_SNo_SNoCut_SNoR_interpolate_impred} : ∀ x0 x1 x2 x3 . SNoCutP x0 x1 ⟶ SNoCutP x2 x3 ⟶ ∀ x4 x5 . x4 = SNoCut x0 x1 ⟶ x5 = SNoCut x2 x3 ⟶ ∀ x6 . x6 ∈ SNoR (mul_SNo x4 x5) ⟶ ∀ x7 : ο . (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x3 ⟶ SNoLe (add_SNo (mul_SNo x8 x5) (mul_SNo x4 x9)) (add_SNo x6 (mul_SNo x8 x9)) ⟶ x7) ⟶ (∀ x8 . x8 ∈ x1 ⟶ ∀ x9 . x9 ∈ x2 ⟶ SNoLe (add_SNo (mul_SNo x8 x5) (mul_SNo x4 x9)) (add_SNo x6 (mul_SNo x8 x9)) ⟶ x7) ⟶ x7
Known SNoR_I^SNoR_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x0 x1 ⟶ x1 ∈ SNoR x0
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
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 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)))
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Known famunionE^famunionE : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ famunion x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ x0) (x2 ∈ x1 x4) ⟶ x3) ⟶ x3
Param ordinal^ordinal : ι → ο
Known ordinal_linear^{ordinal_linear} : ∀ 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 and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known 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)
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_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)
Param div_SNo^div_SNo : ι → ι → ι
Known 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
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)
Known SNo_div_SNo^SNo_div_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (div_SNo x0 x1)
Known nat_ordsucc^nat_ordsucc : ∀ x0 . nat_p x0 ⟶ nat_p (ordsucc x0)
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 binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
Known 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
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
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 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)
Theorem sqrt_SNo_nonneg_prop1d^{sqrt_SNo_nonneg_prop1d} : ∀ 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)) ⟶ 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)) ⟶ SNoLe 0 (SNoCut (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))) ⟶ SNoLt (mul_SNo (SNoCut (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))) (SNoCut (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)))) x0 ⟶ False (proof)