Proofgold Address

Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param SNo^SNo : ι → ο
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 SNoCut^SNoCut : ι → ι → ι
Param SNoL^SNoL : ι → ι
Param mul_SNo^mul_SNo : ι → ι → ι
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Param SNoLe^SNoLe : ι → ι → ο
Param add_SNo^add_SNo : ι → ι → ι
Param SNoLev^SNoLev : ι → ι
Param ordsucc^ordsucc : ι → ι
Param binunion^binunion : ι → ι → ι
Param famunion^famunion : ι → (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
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
Param SNoS_^SNoS_ : ι → ι
Known SNoLev_ind^SNoLev_ind : ∀ x0 : ι → ο . (∀ x1 . SNo x1 ⟶ (∀ x2 . x2 ∈ SNoS_ (SNoLev x1) ⟶ x0 x2) ⟶ x0 x1) ⟶ ∀ x1 . SNo x1 ⟶ x0 x1
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
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
Param minus_SNo^minus_SNo : ι → ι
Known mul_SNoCut_abs^{mul_SNoCut_abs} : ∀ x0 x1 x2 x3 x4 x5 . SNoCutP x0 x1 ⟶ SNoCutP x2 x3 ⟶ x4 = SNoCut x0 x1 ⟶ x5 = SNoCut x2 x3 ⟶ ∀ x6 : ο . (∀ x7 x8 x9 x10 . (∀ x11 . x11 ∈ x7 ⟶ ∀ x12 : ο . (∀ x13 . x13 ∈ x0 ⟶ ∀ x14 . x14 ∈ x2 ⟶ SNo x13 ⟶ SNo x14 ⟶ SNoLt x13 x4 ⟶ SNoLt x14 x5 ⟶ x11 = add_SNo (mul_SNo x13 x5) (add_SNo (mul_SNo x4 x14) (minus_SNo (mul_SNo x13 x14))) ⟶ x12) ⟶ x12) ⟶ (∀ x11 . x11 ∈ x0 ⟶ ∀ x12 . x12 ∈ x2 ⟶ add_SNo (mul_SNo x11 x5) (add_SNo (mul_SNo x4 x12) (minus_SNo (mul_SNo x11 x12))) ∈ x7) ⟶ (∀ x11 . x11 ∈ x8 ⟶ ∀ x12 : ο . (∀ x13 . x13 ∈ x1 ⟶ ∀ x14 . x14 ∈ x3 ⟶ SNo x13 ⟶ SNo x14 ⟶ SNoLt x4 x13 ⟶ SNoLt x5 x14 ⟶ x11 = add_SNo (mul_SNo x13 x5) (add_SNo (mul_SNo x4 x14) (minus_SNo (mul_SNo x13 x14))) ⟶ x12) ⟶ x12) ⟶ (∀ x11 . x11 ∈ x1 ⟶ ∀ x12 . x12 ∈ x3 ⟶ add_SNo (mul_SNo x11 x5) (add_SNo (mul_SNo x4 x12) (minus_SNo (mul_SNo x11 x12))) ∈ x8) ⟶ (∀ x11 . x11 ∈ x9 ⟶ ∀ x12 : ο . (∀ x13 . x13 ∈ x0 ⟶ ∀ x14 . x14 ∈ x3 ⟶ SNo x13 ⟶ SNo x14 ⟶ SNoLt x13 x4 ⟶ SNoLt x5 x14 ⟶ x11 = add_SNo (mul_SNo x13 x5) (add_SNo (mul_SNo x4 x14) (minus_SNo (mul_SNo x13 x14))) ⟶ x12) ⟶ x12) ⟶ (∀ x11 . x11 ∈ x0 ⟶ ∀ x12 . x12 ∈ x3 ⟶ add_SNo (mul_SNo x11 x5) (add_SNo (mul_SNo x4 x12) (minus_SNo (mul_SNo x11 x12))) ∈ x9) ⟶ (∀ x11 . x11 ∈ x10 ⟶ ∀ x12 : ο . (∀ x13 . x13 ∈ x1 ⟶ ∀ x14 . x14 ∈ x2 ⟶ SNo x13 ⟶ SNo x14 ⟶ SNoLt x4 x13 ⟶ SNoLt x14 x5 ⟶ x11 = add_SNo (mul_SNo x13 x5) (add_SNo (mul_SNo x4 x14) (minus_SNo (mul_SNo x13 x14))) ⟶ x12) ⟶ x12) ⟶ (∀ x11 . x11 ∈ x1 ⟶ ∀ x12 . x12 ∈ x2 ⟶ add_SNo (mul_SNo x11 x5) (add_SNo (mul_SNo x4 x12) (minus_SNo (mul_SNo x11 x12))) ∈ x10) ⟶ SNoCutP (binunion x7 x8) (binunion x9 x10) ⟶ mul_SNo x4 x5 = SNoCut (binunion x7 x8) (binunion x9 x10) ⟶ x6) ⟶ x6
Param SNoR^SNoR : ι → ι
Known SNo_eta^SNo_eta : ∀ x0 . SNo x0 ⟶ x0 = SNoCut (SNoL x0) (SNoR x0)
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_SNoL_SNoR^{SNoCutP_SNoL_SNoR} : ∀ x0 . SNo x0 ⟶ SNoCutP (SNoL x0) (SNoR x0)
Known binunionE'^binunionE : ∀ x0 x1 x2 . ∀ x3 : ο . (x2 ∈ x0 ⟶ x3) ⟶ (x2 ∈ x1 ⟶ x3) ⟶ x2 ∈ binunion x0 x1 ⟶ x3
Known add_SNo_minus_Lt1b3^{add_SNo_minus_Lt1b3} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x3 x2) ⟶ SNoLt (add_SNo x0 (add_SNo x1 (minus_SNo x2))) x3
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 x0)
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 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 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 SNoS_I2^SNoS_I2 : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLev x0 ∈ SNoLev x1 ⟶ x0 ∈ SNoS_ (SNoLev x1)
Param ordinal^ordinal : ι → ο
Definition TransSet^TransSet := λ x0 . ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Known ordinal_TransSet^{ordinal_TransSet} : ∀ x0 . ordinal x0 ⟶ TransSet x0
Known SNoLev_ordinal^{SNoLev_ordinal} : ∀ x0 . SNo x0 ⟶ ordinal (SNoLev x0)
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . 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 add_SNo_Lt1_cancel^{add_SNo_Lt1_cancel} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x1) ⟶ SNoLt x0 x2
Known SNoLeLt_tra^SNoLeLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known In_no2cycle^In_no2cycle : ∀ x0 x1 . x0 ∈ x1 ⟶ x1 ∈ x0 ⟶ False
Theorem mul_SNo_SNoCut_SNoL_interpolate^{mul_SNo_SNoCut_SNoL_interpolate} : ∀ x0 x1 x2 x3 . SNoCutP x0 x1 ⟶ SNoCutP x2 x3 ⟶ ∀ x4 x5 . x4 = SNoCut x0 x1 ⟶ x5 = SNoCut x2 x3 ⟶ ∀ x6 . x6 ∈ SNoL (mul_SNo x4 x5) ⟶ or (∀ x7 : ο . (∀ x8 . and (x8 ∈ x0) (∀ x9 : ο . (∀ x10 . and (x10 ∈ x2) (SNoLe (add_SNo x6 (mul_SNo x8 x10)) (add_SNo (mul_SNo x8 x5) (mul_SNo x4 x10))) ⟶ x9) ⟶ x9) ⟶ x7) ⟶ x7) (∀ x7 : ο . (∀ x8 . and (x8 ∈ x1) (∀ x9 : ο . (∀ x10 . and (x10 ∈ x3) (SNoLe (add_SNo x6 (mul_SNo x8 x10)) (add_SNo (mul_SNo x8 x5) (mul_SNo x4 x10))) ⟶ x9) ⟶ x9) ⟶ x7) ⟶ x7) (proof)
Theorem mul_SNo_SNoCut_SNoL_interpolate_impred^{mul_SNo_SNoCut_SNoL_interpolate_impred} : ∀ x0 x1 x2 x3 . SNoCutP x0 x1 ⟶ SNoCutP x2 x3 ⟶ ∀ x4 x5 . x4 = SNoCut x0 x1 ⟶ x5 = SNoCut x2 x3 ⟶ ∀ x6 . x6 ∈ SNoL (mul_SNo x4 x5) ⟶ ∀ x7 : ο . (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x2 ⟶ SNoLe (add_SNo x6 (mul_SNo x8 x9)) (add_SNo (mul_SNo x8 x5) (mul_SNo x4 x9)) ⟶ x7) ⟶ (∀ x8 . x8 ∈ x1 ⟶ ∀ x9 . x9 ∈ x3 ⟶ SNoLe (add_SNo x6 (mul_SNo x8 x9)) (add_SNo (mul_SNo x8 x5) (mul_SNo x4 x9)) ⟶ x7) ⟶ x7 (proof)
Known add_SNo_minus_Lt2b3^{add_SNo_minus_Lt2b3} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLt (add_SNo x3 x2) (add_SNo x0 x1) ⟶ SNoLt x3 (add_SNo x0 (add_SNo x1 (minus_SNo x2)))
Theorem mul_SNo_SNoCut_SNoR_interpolate^{mul_SNo_SNoCut_SNoR_interpolate} : ∀ 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) ⟶ or (∀ x7 : ο . (∀ x8 . and (x8 ∈ x0) (∀ x9 : ο . (∀ x10 . and (x10 ∈ x3) (SNoLe (add_SNo (mul_SNo x8 x5) (mul_SNo x4 x10)) (add_SNo x6 (mul_SNo x8 x10))) ⟶ x9) ⟶ x9) ⟶ x7) ⟶ x7) (∀ x7 : ο . (∀ x8 . and (x8 ∈ x1) (∀ x9 : ο . (∀ x10 . and (x10 ∈ x2) (SNoLe (add_SNo (mul_SNo x8 x5) (mul_SNo x4 x10)) (add_SNo x6 (mul_SNo x8 x10))) ⟶ x9) ⟶ x9) ⟶ x7) ⟶ x7) (proof)
Theorem 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 (proof)