Proofgold Signed Transaction

Param add_SNo^add_SNo : ι → ι → ι
Param mul_SNo^mul_SNo : ι → ι → ι
Param minus_SNo^minus_SNo : ι → ι
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known mul_SNo_Subq_lem^{mul_SNo_Subq_lem} : ∀ x0 x1 x2 x3 x4 x5 x6 x7 . (∀ x8 . x8 ∈ x6 ⟶ ∀ x9 : ο . (∀ x10 . x10 ∈ x2 ⟶ ∀ x11 . x11 ∈ x3 ⟶ x8 = add_SNo (mul_SNo x10 x1) (add_SNo (mul_SNo x0 x11) (minus_SNo (mul_SNo x10 x11))) ⟶ x9) ⟶ (∀ x10 . x10 ∈ x4 ⟶ ∀ x11 . x11 ∈ x5 ⟶ x8 = add_SNo (mul_SNo x10 x1) (add_SNo (mul_SNo x0 x11) (minus_SNo (mul_SNo x10 x11))) ⟶ x9) ⟶ x9) ⟶ (∀ x8 . x8 ∈ x2 ⟶ ∀ x9 . x9 ∈ x3 ⟶ add_SNo (mul_SNo x8 x1) (add_SNo (mul_SNo x0 x9) (minus_SNo (mul_SNo x8 x9))) ∈ x7) ⟶ (∀ x8 . x8 ∈ x4 ⟶ ∀ x9 . x9 ∈ x5 ⟶ add_SNo (mul_SNo x8 x1) (add_SNo (mul_SNo x0 x9) (minus_SNo (mul_SNo x8 x9))) ∈ x7) ⟶ x6 ⊆ x7
Param SNo^SNo : ι → ο
Param SNoLt^SNoLt : ι → ι → ο
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
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
Theorem add_SNo_Lt2_cancel^{add_SNo_Lt2_cancel} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x0 x2) ⟶ SNoLt x1 x2 (proof)
Param SNoL^SNoL : ι → ι
Param SNoR^SNoR : ι → ι
Known mul_SNo_prop_1^{mul_SNo_prop_1} : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ ∀ x2 : ο . (SNo (mul_SNo x0 x1) ⟶ (∀ x3 . x3 ∈ SNoL x0 ⟶ ∀ x4 . x4 ∈ SNoL x1 ⟶ SNoLt (add_SNo (mul_SNo x3 x1) (mul_SNo x0 x4)) (add_SNo (mul_SNo x0 x1) (mul_SNo x3 x4))) ⟶ (∀ x3 . x3 ∈ SNoR x0 ⟶ ∀ x4 . x4 ∈ SNoR x1 ⟶ SNoLt (add_SNo (mul_SNo x3 x1) (mul_SNo x0 x4)) (add_SNo (mul_SNo x0 x1) (mul_SNo x3 x4))) ⟶ (∀ x3 . x3 ∈ SNoL x0 ⟶ ∀ x4 . x4 ∈ SNoR x1 ⟶ SNoLt (add_SNo (mul_SNo x0 x1) (mul_SNo x3 x4)) (add_SNo (mul_SNo x3 x1) (mul_SNo x0 x4))) ⟶ (∀ x3 . x3 ∈ SNoR x0 ⟶ ∀ x4 . x4 ∈ SNoL x1 ⟶ SNoLt (add_SNo (mul_SNo x0 x1) (mul_SNo x3 x4)) (add_SNo (mul_SNo x3 x1) (mul_SNo x0 x4))) ⟶ x2) ⟶ x2
Param SNoLev^SNoLev : ι → ι
Param binintersect^binintersect : ι → ι → ι
Param SNoEq_^SNoEq_ : ι → ι → ι → ο
Param nIn^nIn : ι → ι → ο
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
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known binintersectE^{binintersectE} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ and (x2 ∈ x0) (x2 ∈ x1)
Known SNoLt_tra^SNoLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
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_Lt3^add_SNo_Lt3 : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLt x0 x2 ⟶ SNoLt x1 x3 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x3)
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Known SNoR_I^SNoR_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x0 x1 ⟶ x1 ∈ SNoR x0
Known SNoL_I^SNoL_I : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x1 x0 ⟶ x1 ∈ SNoL x0
Known add_SNo_com_3_0_1^{add_SNo_com_3_0_1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 (add_SNo x1 x2) = add_SNo x1 (add_SNo x0 x2)
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 add_SNo_assoc^{add_SNo_assoc} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 (add_SNo x1 x2) = add_SNo (add_SNo x0 x1) x2
Known add_SNo_Lt1^add_SNo_Lt1 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x2 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x1)
Known add_SNo_rotate_3_1^{add_SNo_rotate_3_1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 (add_SNo x1 x2) = add_SNo x2 (add_SNo x0 x1)
Theorem mul_SNo_Lt^mul_SNo_Lt : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLt x2 x0 ⟶ SNoLt x3 x1 ⟶ SNoLt (add_SNo (mul_SNo x2 x1) (mul_SNo x0 x3)) (add_SNo (mul_SNo x0 x1) (mul_SNo x2 x3)) (proof)
Param SNoLe^SNoLe : ι → ι → ο
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known SNoLeE^SNoLeE : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ or (SNoLt x0 x1) (x0 = x1)
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Known SNoLe_ref^SNoLe_ref : ∀ x0 . SNoLe x0 x0
Theorem mul_SNo_Le^mul_SNo_Le : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLe x2 x0 ⟶ SNoLe x3 x1 ⟶ SNoLe (add_SNo (mul_SNo x2 x1) (mul_SNo x0 x3)) (add_SNo (mul_SNo x0 x1) (mul_SNo x2 x3)) (proof)
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
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 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 (proof)
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))
Theorem 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))) (proof)
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 SNoCutP^SNoCutP : ι → ι → ο
Param SNoCut^SNoCut : ι → ι → ι
Known mul_SNo_eq_3^mul_SNo_eq_3 : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 : ο . (∀ x3 x4 . SNoCutP x3 x4 ⟶ (∀ x5 . x5 ∈ x3 ⟶ ∀ x6 : ο . (∀ x7 . x7 ∈ SNoL x0 ⟶ ∀ x8 . x8 ∈ SNoL x1 ⟶ x5 = add_SNo (mul_SNo x7 x1) (add_SNo (mul_SNo x0 x8) (minus_SNo (mul_SNo x7 x8))) ⟶ x6) ⟶ (∀ x7 . x7 ∈ SNoR x0 ⟶ ∀ x8 . x8 ∈ SNoR x1 ⟶ x5 = add_SNo (mul_SNo x7 x1) (add_SNo (mul_SNo x0 x8) (minus_SNo (mul_SNo x7 x8))) ⟶ x6) ⟶ x6) ⟶ (∀ x5 . x5 ∈ SNoL x0 ⟶ ∀ x6 . x6 ∈ SNoL x1 ⟶ add_SNo (mul_SNo x5 x1) (add_SNo (mul_SNo x0 x6) (minus_SNo (mul_SNo x5 x6))) ∈ x3) ⟶ (∀ x5 . x5 ∈ SNoR x0 ⟶ ∀ x6 . x6 ∈ SNoR x1 ⟶ add_SNo (mul_SNo x5 x1) (add_SNo (mul_SNo x0 x6) (minus_SNo (mul_SNo x5 x6))) ∈ x3) ⟶ (∀ x5 . x5 ∈ x4 ⟶ ∀ x6 : ο . (∀ x7 . x7 ∈ SNoL x0 ⟶ ∀ x8 . x8 ∈ SNoR x1 ⟶ x5 = add_SNo (mul_SNo x7 x1) (add_SNo (mul_SNo x0 x8) (minus_SNo (mul_SNo x7 x8))) ⟶ x6) ⟶ (∀ x7 . x7 ∈ SNoR x0 ⟶ ∀ x8 . x8 ∈ SNoL x1 ⟶ x5 = add_SNo (mul_SNo x7 x1) (add_SNo (mul_SNo x0 x8) (minus_SNo (mul_SNo x7 x8))) ⟶ x6) ⟶ x6) ⟶ (∀ x5 . x5 ∈ SNoL x0 ⟶ ∀ x6 . x6 ∈ SNoR x1 ⟶ add_SNo (mul_SNo x5 x1) (add_SNo (mul_SNo x0 x6) (minus_SNo (mul_SNo x5 x6))) ∈ x4) ⟶ (∀ x5 . x5 ∈ SNoR x0 ⟶ ∀ x6 . x6 ∈ SNoL x1 ⟶ add_SNo (mul_SNo x5 x1) (add_SNo (mul_SNo x0 x6) (minus_SNo (mul_SNo x5 x6))) ∈ x4) ⟶ mul_SNo x0 x1 = SNoCut x3 x4 ⟶ x2) ⟶ x2
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 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 SNoLeLt_tra^SNoLeLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 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 In_no2cycle^In_no2cycle : ∀ x0 x1 . x0 ∈ x1 ⟶ x1 ∈ x0 ⟶ False
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 x0)
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Theorem mul_SNo_SNoL_interpolate^{mul_SNo_SNoL_interpolate} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 . x2 ∈ SNoL (mul_SNo x0 x1) ⟶ or (∀ x3 : ο . (∀ x4 . and (x4 ∈ SNoL x0) (∀ x5 : ο . (∀ x6 . and (x6 ∈ SNoL x1) (SNoLe (add_SNo x2 (mul_SNo x4 x6)) (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x6))) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) (∀ x3 : ο . (∀ x4 . and (x4 ∈ SNoR x0) (∀ x5 : ο . (∀ x6 . and (x6 ∈ SNoR x1) (SNoLe (add_SNo x2 (mul_SNo x4 x6)) (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x6))) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) (proof)
Theorem mul_SNo_SNoL_interpolate_impred^{mul_SNo_SNoL_interpolate_impred} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 . x2 ∈ SNoL (mul_SNo x0 x1) ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ SNoL x0 ⟶ ∀ x5 . x5 ∈ SNoL x1 ⟶ SNoLe (add_SNo x2 (mul_SNo x4 x5)) (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x5)) ⟶ x3) ⟶ (∀ x4 . x4 ∈ SNoR x0 ⟶ ∀ x5 . x5 ∈ SNoR x1 ⟶ SNoLe (add_SNo x2 (mul_SNo x4 x5)) (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x5)) ⟶ x3) ⟶ x3 (proof)
Theorem mul_SNo_SNoR_interpolate^{mul_SNo_SNoR_interpolate} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 . x2 ∈ SNoR (mul_SNo x0 x1) ⟶ or (∀ x3 : ο . (∀ x4 . and (x4 ∈ SNoL x0) (∀ x5 : ο . (∀ x6 . and (x6 ∈ SNoR x1) (SNoLe (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x6)) (add_SNo x2 (mul_SNo x4 x6))) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) (∀ x3 : ο . (∀ x4 . and (x4 ∈ SNoR x0) (∀ x5 : ο . (∀ x6 . and (x6 ∈ SNoL x1) (SNoLe (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x6)) (add_SNo x2 (mul_SNo x4 x6))) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) (proof)
Theorem mul_SNo_SNoR_interpolate_impred^{mul_SNo_SNoR_interpolate_impred} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 . x2 ∈ SNoR (mul_SNo x0 x1) ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ SNoL x0 ⟶ ∀ x5 . x5 ∈ SNoR x1 ⟶ SNoLe (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x5)) (add_SNo x2 (mul_SNo x4 x5)) ⟶ x3) ⟶ (∀ x4 . x4 ∈ SNoR x0 ⟶ ∀ x5 . x5 ∈ SNoL x1 ⟶ SNoLe (add_SNo (mul_SNo x4 x1) (mul_SNo x0 x5)) (add_SNo x2 (mul_SNo x4 x5)) ⟶ x3) ⟶ x3 (proof)