Proofgold Address

Param SNo^SNo : ι → ο
Param add_SNo^add_SNo : ι → ι → ι
Known SNo_add_SNo_3^{SNo_add_SNo_3} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo (add_SNo x0 (add_SNo x1 x2))
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Theorem SNo_add_SNo_4^{SNo_add_SNo_4} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo (add_SNo x0 (add_SNo x1 (add_SNo x2 x3))) (proof)
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)
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)
Theorem add_SNo_rotate_4_1^{add_SNo_rotate_4_1} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ add_SNo x0 (add_SNo x1 (add_SNo x2 x3)) = add_SNo x3 (add_SNo x0 (add_SNo x1 x2)) (proof)
Param SNoLe^SNoLe : ι → ι → ο
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Param SNoLt^SNoLt : ι → ι → ο
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 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 add_SNo_cancel_R^{add_SNo_cancel_R} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 x1 = add_SNo x2 x1 ⟶ x0 = x2
Known SNoLe_ref^SNoLe_ref : ∀ x0 . SNoLe x0 x0
Theorem add_SNo_Le1_cancel^{add_SNo_Le1_cancel} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe (add_SNo x0 x1) (add_SNo x2 x1) ⟶ SNoLe x0 x2 (proof)
Known 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
Known add_SNo_cancel_L^{add_SNo_cancel_L} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 x1 = add_SNo x0 x2 ⟶ x1 = x2
Theorem 8ebd3.. : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe (add_SNo x0 x1) (add_SNo x0 x2) ⟶ SNoLe x1 x2 (proof)
Param mul_SNo^mul_SNo : ι → ι → ι
Param SNoS_^SNoS_ : ι → ι
Param SNoLev^SNoLev : ι → ι
Known SNoLev_ind3^SNoLev_ind3 : ∀ x0 : ι → ι → ι → ο . (∀ x1 x2 x3 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ x0 x4 x2 x3) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x2) ⟶ x0 x1 x4 x3) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x3) ⟶ x0 x1 x2 x4) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ ∀ x5 . x5 ∈ SNoS_ (SNoLev x2) ⟶ x0 x4 x5 x3) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ ∀ x5 . x5 ∈ SNoS_ (SNoLev x3) ⟶ x0 x4 x2 x5) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x2) ⟶ ∀ x5 . x5 ∈ SNoS_ (SNoLev x3) ⟶ x0 x1 x4 x5) ⟶ (∀ x4 . x4 ∈ SNoS_ (SNoLev x1) ⟶ ∀ x5 . x5 ∈ SNoS_ (SNoLev x2) ⟶ ∀ x6 . x6 ∈ SNoS_ (SNoLev x3) ⟶ x0 x4 x5 x6) ⟶ x0 x1 x2 x3) ⟶ ∀ x1 x2 x3 . SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ x0 x1 x2 x3
Param SNoCutP^SNoCutP : ι → ι → ο
Param SNoL^SNoL : ι → ι
Param minus_SNo^minus_SNo : ι → ι
Param SNoR^SNoR : ι → ι
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
Param binunion^binunion : ι → ι → ι
Known SNoCut_ext^SNoCut_ext : ∀ x0 x1 x2 x3 . SNoCutP x0 x1 ⟶ SNoCutP x2 x3 ⟶ (∀ x4 . x4 ∈ x0 ⟶ SNoLt x4 (SNoCut x2 x3)) ⟶ (∀ x4 . x4 ∈ x1 ⟶ SNoLt (SNoCut x2 x3) x4) ⟶ (∀ x4 . x4 ∈ x2 ⟶ SNoLt x4 (SNoCut x0 x1)) ⟶ (∀ x4 . x4 ∈ x3 ⟶ SNoLt (SNoCut x0 x1) x4) ⟶ SNoCut x0 x1 = SNoCut x2 x3
Known add_SNo_SNoCutP^{add_SNo_SNoCutP} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoCutP (binunion {add_SNo x2 x1|x2 ∈ SNoL x0} (prim5 (SNoL x1) (add_SNo x0))) (binunion {add_SNo x2 x1|x2 ∈ SNoR x0} (prim5 (SNoR x1) (add_SNo x0)))
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
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 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 SNoL_SNoS^SNoL_SNoS : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoL x0 ⟶ x1 ∈ SNoS_ (SNoLev x0)
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
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known add_SNo_SNoL_interpolate^{add_SNo_SNoL_interpolate} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 . x2 ∈ SNoL (add_SNo x0 x1) ⟶ or (∀ x3 : ο . (∀ x4 . and (x4 ∈ SNoL x0) (SNoLe x2 (add_SNo x4 x1)) ⟶ x3) ⟶ x3) (∀ x3 : ο . (∀ x4 . and (x4 ∈ SNoL x1) (SNoLe x2 (add_SNo x0 x4)) ⟶ x3) ⟶ x3)
Known add_SNo_assoc_4^{add_SNo_assoc_4} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ add_SNo x0 (add_SNo x1 (add_SNo x2 x3)) = add_SNo (add_SNo x0 (add_SNo x1 x2)) x3
Known SNoLeLt_tra^SNoLeLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt 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 add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Known 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))
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_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))
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 SNoR_SNoS^SNoR_SNoS : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoR x0 ⟶ x1 ∈ SNoS_ (SNoLev x0)
Known add_SNo_SNoR_interpolate^{add_SNo_SNoR_interpolate} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 . x2 ∈ SNoR (add_SNo x0 x1) ⟶ or (∀ x3 : ο . (∀ x4 . and (x4 ∈ SNoR x0) (SNoLe (add_SNo x4 x1) x2) ⟶ x3) ⟶ x3) (∀ x3 : ο . (∀ x4 . and (x4 ∈ SNoR x1) (SNoLe (add_SNo x0 x4) x2) ⟶ x3) ⟶ x3)
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)))
Known SNoLtLe_tra^SNoLtLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLt x0 x2
Known binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Known ReplE_impred^ReplE_impred : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ prim5 x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ x0 ⟶ x2 = x1 x4 ⟶ x3) ⟶ x3
Known 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
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_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)
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 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
Known add_SNo_eq^add_SNo_eq : ∀ x0 . SNo x0 ⟶ ∀ x1 . SNo x1 ⟶ add_SNo x0 x1 = SNoCut (binunion {add_SNo x3 x1|x3 ∈ SNoL x0} (prim5 (SNoL x1) (add_SNo x0))) (binunion {add_SNo x3 x1|x3 ∈ SNoR x0} (prim5 (SNoR x1) (add_SNo x0)))
Theorem 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) (proof)
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
Theorem 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) (proof)