Proofgold Signed Transaction

Param SNo^SNo : ι → ο
Param SNoLe^SNoLe : ι → ι → ο
Param add_SNo^add_SNo : ι → ι → ι
Param minus_SNo^minus_SNo : ι → ι
Known add_SNo_minus_R2^{add_SNo_minus_R2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo (add_SNo x0 x1) (minus_SNo x1) = x0
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 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)
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_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_minus_SNo_linv^{add_SNo_minus_SNo_linv} : ∀ x0 . SNo x0 ⟶ add_SNo (minus_SNo x0) x0 = 0
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
Theorem add_SNo_minus_Le1b^{add_SNo_minus_Le1b} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 (add_SNo x2 x1) ⟶ SNoLe (add_SNo x0 (minus_SNo x1)) x2 (proof)
Theorem add_SNo_minus_Le1b3^{add_SNo_minus_Le1b3} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNoLe (add_SNo x0 x1) (add_SNo x3 x2) ⟶ SNoLe (add_SNo x0 (add_SNo x1 (minus_SNo x2))) x3 (proof)
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 add_SNo_com_3b_1_2^{add_SNo_com_3b_1_2} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo (add_SNo x0 x1) x2 = add_SNo (add_SNo x0 x2) x1
Known add_SNo_minus_Le2b^{add_SNo_minus_Le2b} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe (add_SNo x2 x1) x0 ⟶ SNoLe x2 (add_SNo x0 (minus_SNo x1))
Theorem add_SNo_minus_Le12b3^{add_SNo_minus_Le12b3} : ∀ x0 x1 x2 x3 x4 x5 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ SNo x5 ⟶ SNoLe (add_SNo x0 (add_SNo x1 x5)) (add_SNo x3 (add_SNo x4 x2)) ⟶ SNoLe (add_SNo x0 (add_SNo x1 (minus_SNo x2))) (add_SNo x3 (add_SNo x4 (minus_SNo x5))) (proof)
Param SNoLt^SNoLt : ι → ι → ο
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 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_Lt12b3^{add_SNo_minus_Lt12b3} : ∀ x0 x1 x2 x3 x4 x5 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ SNo x4 ⟶ SNo x5 ⟶ SNoLt (add_SNo x0 (add_SNo x1 x5)) (add_SNo x3 (add_SNo x4 x2)) ⟶ SNoLt (add_SNo x0 (add_SNo x1 (minus_SNo x2))) (add_SNo x3 (add_SNo x4 (minus_SNo x5))) (proof)
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
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 binunion^binunion : ι → ι → ι
Param lam^Sigma : ι → (ι → ι) → ι
Definition setprod^setprod := λ x0 x1 . lam x0 (λ x2 . x1)
Param mul_SNo^mul_SNo : ι → ι → ι
Param ap^ap : ι → ι → ι
Param ordsucc^ordsucc : ι → ι
Param SNoLev^SNoLev : 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 and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Param SNoL^SNoL : ι → ι
Param SNoR^SNoR : 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 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 SNoL_E^SNoL_E : ∀ x0 . SNo x0 ⟶ ∀ x1 . x1 ∈ SNoL x0 ⟶ ∀ x2 : ο . (SNo x1 ⟶ SNoLev x1 ∈ SNoLev x0 ⟶ SNoLt x1 x0 ⟶ x2) ⟶ x2
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 x0)
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known SNoLt_tra^SNoLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known In_irref^In_irref : ∀ x0 . nIn 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
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Known SNoLe_tra^SNoLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLe 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 SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe 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)
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion 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 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 binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
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 . ∀ x3 : ο . (x2 ∈ x0 ⟶ x3) ⟶ (x2 ∈ x1 ⟶ x3) ⟶ x2 ∈ binunion x0 x1 ⟶ x3
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 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 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)
Param If_i^If_i : ο → ι → 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 tuple_2_1_eq^tuple_2_1_eq : ∀ x0 x1 . ap (lam 2 (λ x3 . If_i (x3 = 0) x0 x1)) 1 = x1
Known ReplI^ReplI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ prim5 x0 x1
Known tuple_2_setprod^{tuple_2_setprod} : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x1 ⟶ lam 2 (λ x4 . If_i (x4 = 0) x2 x3) ∈ setprod x0 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 ap1_Sigma^ap1_Sigma : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ ap x2 1 ∈ x1 (ap x2 0)
Known ap0_Sigma^ap0_Sigma : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ ap x2 0 ∈ x0
Theorem mul_SNoCutP_lem^{mul_SNoCutP_lem} : ∀ x0 x1 x2 x3 x4 x5 . SNoCutP x0 x1 ⟶ SNoCutP x2 x3 ⟶ x4 = SNoCut x0 x1 ⟶ x5 = SNoCut x2 x3 ⟶ and (and (SNoCutP (binunion {add_SNo (mul_SNo (ap x6 0) x5) (add_SNo (mul_SNo x4 (ap x6 1)) (minus_SNo (mul_SNo (ap x6 0) (ap x6 1))))|x6 ∈ setprod x0 x2} {add_SNo (mul_SNo (ap x6 0) x5) (add_SNo (mul_SNo x4 (ap x6 1)) (minus_SNo (mul_SNo (ap x6 0) (ap x6 1))))|x6 ∈ setprod x1 x3}) (binunion {add_SNo (mul_SNo (ap x6 0) x5) (add_SNo (mul_SNo x4 (ap x6 1)) (minus_SNo (mul_SNo (ap x6 0) (ap x6 1))))|x6 ∈ setprod x0 x3} {add_SNo (mul_SNo (ap x6 0) x5) (add_SNo (mul_SNo x4 (ap x6 1)) (minus_SNo (mul_SNo (ap x6 0) (ap x6 1))))|x6 ∈ setprod x1 x2})) (mul_SNo x4 x5 = SNoCut (binunion {add_SNo (mul_SNo (ap x7 0) x5) (add_SNo (mul_SNo x4 (ap x7 1)) (minus_SNo (mul_SNo (ap x7 0) (ap x7 1))))|x7 ∈ setprod x0 x2} {add_SNo (mul_SNo (ap x7 0) x5) (add_SNo (mul_SNo x4 (ap x7 1)) (minus_SNo (mul_SNo (ap x7 0) (ap x7 1))))|x7 ∈ setprod x1 x3}) (binunion {add_SNo (mul_SNo (ap x7 0) x5) (add_SNo (mul_SNo x4 (ap x7 1)) (minus_SNo (mul_SNo (ap x7 0) (ap x7 1))))|x7 ∈ setprod x0 x3} {add_SNo (mul_SNo (ap x7 0) x5) (add_SNo (mul_SNo x4 (ap x7 1)) (minus_SNo (mul_SNo (ap x7 0) (ap x7 1))))|x7 ∈ setprod x1 x2}))) (∀ 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) (proof)
Theorem mul_SNoCutP_gen^{mul_SNoCutP_gen} : ∀ x0 x1 x2 x3 x4 x5 . SNoCutP x0 x1 ⟶ SNoCutP x2 x3 ⟶ x4 = SNoCut x0 x1 ⟶ x5 = SNoCut x2 x3 ⟶ SNoCutP (binunion {add_SNo (mul_SNo (ap x6 0) x5) (add_SNo (mul_SNo x4 (ap x6 1)) (minus_SNo (mul_SNo (ap x6 0) (ap x6 1))))|x6 ∈ setprod x0 x2} {add_SNo (mul_SNo (ap x6 0) x5) (add_SNo (mul_SNo x4 (ap x6 1)) (minus_SNo (mul_SNo (ap x6 0) (ap x6 1))))|x6 ∈ setprod x1 x3}) (binunion {add_SNo (mul_SNo (ap x6 0) x5) (add_SNo (mul_SNo x4 (ap x6 1)) (minus_SNo (mul_SNo (ap x6 0) (ap x6 1))))|x6 ∈ setprod x0 x3} {add_SNo (mul_SNo (ap x6 0) x5) (add_SNo (mul_SNo x4 (ap x6 1)) (minus_SNo (mul_SNo (ap x6 0) (ap x6 1))))|x6 ∈ setprod x1 x2}) (proof)
Theorem mul_SNoCut_eq^{mul_SNoCut_eq} : ∀ x0 x1 x2 x3 x4 x5 . SNoCutP x0 x1 ⟶ SNoCutP x2 x3 ⟶ x4 = SNoCut x0 x1 ⟶ x5 = SNoCut x2 x3 ⟶ mul_SNo x4 x5 = SNoCut (binunion {add_SNo (mul_SNo (ap x7 0) x5) (add_SNo (mul_SNo x4 (ap x7 1)) (minus_SNo (mul_SNo (ap x7 0) (ap x7 1))))|x7 ∈ setprod x0 x2} {add_SNo (mul_SNo (ap x7 0) x5) (add_SNo (mul_SNo x4 (ap x7 1)) (minus_SNo (mul_SNo (ap x7 0) (ap x7 1))))|x7 ∈ setprod x1 x3}) (binunion {add_SNo (mul_SNo (ap x7 0) x5) (add_SNo (mul_SNo x4 (ap x7 1)) (minus_SNo (mul_SNo (ap x7 0) (ap x7 1))))|x7 ∈ setprod x0 x3} {add_SNo (mul_SNo (ap x7 0) x5) (add_SNo (mul_SNo x4 (ap x7 1)) (minus_SNo (mul_SNo (ap x7 0) (ap x7 1))))|x7 ∈ setprod x1 x2}) (proof)
Theorem 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 (proof)