Proofgold Address

Param pack_b^pack_b : ι → CT2 ι
Definition struct_b^struct_b := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ι . (∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x2 ⟶ x3 x4 x5 ∈ x2) ⟶ x1 (pack_b x2 x3)) ⟶ x1 x0
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition MetaCat_initial_p^initial_p := λ x0 : ι → ο . λ x1 : ι → ι → ι → ο . λ x2 : ι → ι . λ x3 : ι → ι → ι → ι → ι → ι . λ x4 . λ x5 : ι → ι . and (x0 x4) (∀ x6 . x0 x6 ⟶ and (x1 x4 x6 (x5 x6)) (∀ x7 . x1 x4 x6 x7 ⟶ x7 = x5 x6))
Param MagmaHom^Hom_struct_b : ι → ι → ι → ο
Param struct_id^struct_id : ι → ι
Param lam^Sigma : ι → (ι → ι) → ι
Param ap^ap : ι → ι → ι
Definition lam_comp^lam_comp := λ x0 x1 x2 . lam x0 (λ x3 . ap x1 (ap x2 x3))
Definition struct_comp^struct_comp := λ x0 x1 x2 . lam_comp (ap x0 0)
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Param Pi^Pi : ι → (ι → ι) → ι
Definition setexp^setexp := λ x0 x1 . Pi x1 (λ x2 . x0)
Known 2cd8d..^{Hom_struct_b_pack} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 . MagmaHom (pack_b x0 x2) (pack_b x1 x3) x4 = and (x4 ∈ setexp x1 x0) (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ap x4 (x2 x6 x7) = x3 (ap x4 x6) (ap x4 x7))
Known lam_Pi^lam_Pi : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1 x3) ⟶ lam x0 x2 ∈ Pi x0 x1
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 Pi_eta^Pi_eta : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ Pi x0 x1 ⟶ lam x0 (ap x2) = x2
Known encode_u_ext^encode_u_ext : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x1 x3 = x2 x3) ⟶ lam x0 x1 = lam x0 x2
Theorem 93a0d.. : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ struct_b x1) ⟶ x0 (pack_b 0 (λ x1 x2 . x1)) ⟶ MetaCat_initial_p x0 MagmaHom struct_id struct_comp (pack_b 0 (λ x1 x2 . x1)) (λ x1 . lam 0 (λ x2 . x2)) (proof)
Known pack_struct_b_I^{pack_struct_b_I} : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ∈ x0) ⟶ struct_b (pack_b x0 x1)
Theorem e9785..^{MetaCat_struct_b_initial} : ∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . MetaCat_initial_p struct_b MagmaHom struct_id struct_comp x1 x3 ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Param ordsucc^ordsucc : ι → ι
Definition MetaCat_terminal_p^terminal_p := λ x0 : ι → ο . λ x1 : ι → ι → ι → ο . λ x2 : ι → ι . λ x3 : ι → ι → ι → ι → ι → ι . λ x4 . λ x5 : ι → ι . and (x0 x4) (∀ x6 . x0 x6 ⟶ and (x1 x6 x4 (x5 x6)) (∀ x7 . x1 x6 x4 x7 ⟶ x7 = x5 x6))
Known pack_b_0_eq2^pack_b_0_eq2 : ∀ x0 . ∀ x1 : ι → ι → ι . x0 = ap (pack_b x0 x1) 0
Known In_0_1^In_0_1 : 0 ∈ 1
Known beta^beta : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ ap (lam x0 x1) x2 = x1 x2
Param Sing^Sing : ι → ι
Known SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Known eq_1_Sing0^eq_1_Sing0 : 1 = Sing 0
Known ap_Pi^ap_Pi : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ Pi x0 x1 ⟶ x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3
Theorem dfc93.. : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ struct_b x1) ⟶ x0 (pack_b 1 (λ x1 x2 . x1)) ⟶ MetaCat_terminal_p x0 MagmaHom struct_id struct_comp (pack_b 1 (λ x1 x2 . x1)) (λ x1 . lam (ap x1 0) (λ x2 . 0)) (proof)
Theorem 021fd..^{MetaCat_struct_b_terminal} : ∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . MetaCat_terminal_p struct_b MagmaHom struct_id struct_comp x1 x3 ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Param unpack_b_i^unpack_b_i : ι → (ι → CT2 ι) → ι
Definition setprod^setprod := λ x0 x1 . lam x0 (λ x2 . x1)
Param If_i^If_i : ο → ι → ι → ι
Definition 3d151.. := λ x0 x1 . unpack_b_i x0 (λ x2 . λ x3 : ι → ι → ι . unpack_b_i x1 (λ x4 . λ x5 : ι → ι → ι . pack_b (setprod x2 x4) (λ x6 x7 . lam 2 (λ x8 . If_i (x8 = 0) (x3 (ap x6 0) (ap x7 0)) (x5 (ap x6 1) (ap x7 1))))))
Known unpack_b_i_eq^{unpack_b_i_eq} : ∀ x0 : ι → (ι → ι → ι) → ι . ∀ x1 . ∀ x2 : ι → ι → ι . (∀ x3 : ι → ι → ι . (∀ x4 . x4 ∈ x1 ⟶ ∀ x5 . x5 ∈ x1 ⟶ x2 x4 x5 = x3 x4 x5) ⟶ x0 x1 x3 = x0 x1 x2) ⟶ unpack_b_i (pack_b x1 x2) x0 = x0 x1 x2
Known pack_b_ext^pack_b_ext : ∀ x0 . ∀ x1 x2 : ι → ι → ι . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = x2 x3 x4) ⟶ pack_b x0 x1 = pack_b x0 x2
Known ap0_Sigma^ap0_Sigma : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ ap x2 0 ∈ x0
Known ap1_Sigma^ap1_Sigma : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ ap x2 1 ∈ x1 (ap x2 0)
Theorem fe106.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . ∀ x3 : ι → ι → ι . 3d151.. (pack_b x0 x1) (pack_b x2 x3) = pack_b (setprod x0 x2) (λ x5 x6 . lam 2 (λ x7 . If_i (x7 = 0) (x1 (ap x5 0) (ap x6 0)) (x3 (ap x5 1) (ap x6 1)))) (proof)
Known tuple_2_Sigma^{tuple_2_Sigma} : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x1 x2 ⟶ lam 2 (λ x4 . If_i (x4 = 0) x2 x3) ∈ lam x0 x1
Theorem 2e4a2.. : ∀ x0 x1 . struct_b x0 ⟶ struct_b x1 ⟶ struct_b (3d151.. x0 x1) (proof)
Definition MetaCat_product_p^product_p := λ x0 : ι → ο . λ x1 : ι → ι → ι → ο . λ x2 : ι → ι . λ x3 : ι → ι → ι → ι → ι → ι . λ x4 x5 x6 x7 x8 . λ x9 : ι → ι → ι → ι . and (and (and (and (and (x0 x4) (x0 x5)) (x0 x6)) (x1 x6 x4 x7)) (x1 x6 x5 x8)) (∀ x10 . x0 x10 ⟶ ∀ x11 x12 . x1 x10 x4 x11 ⟶ x1 x10 x5 x12 ⟶ and (and (and (x1 x10 x6 (x9 x10 x11 x12)) (x3 x10 x6 x4 x7 (x9 x10 x11 x12) = x11)) (x3 x10 x6 x5 x8 (x9 x10 x11 x12) = x12)) (∀ x13 . x1 x10 x6 x13 ⟶ x3 x10 x6 x4 x7 x13 = x11 ⟶ x3 x10 x6 x5 x8 x13 = x12 ⟶ x13 = x9 x10 x11 x12))
Definition MetaCat_product_constr_p^{product_constr_p} := λ x0 : ι → ο . λ x1 : ι → ι → ι → ο . λ x2 : ι → ι . λ x3 : ι → ι → ι → ι → ι → ι . λ x4 x5 x6 : ι → ι → ι . λ x7 : ι → ι → ι → ι → ι → ι . ∀ x8 x9 . x0 x8 ⟶ x0 x9 ⟶ MetaCat_product_p x0 x1 x2 x3 x8 x9 (x4 x8 x9) (x5 x8 x9) (x6 x8 x9) (x7 x8 x9)
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
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 and4I^and4I : ∀ x0 x1 x2 x3 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ and (and (and x0 x1) x2) x3
Known tuple_Sigma_eta^{tuple_Sigma_eta} : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ lam 2 (λ x4 . If_i (x4 = 0) (ap x2 0) (ap x2 1)) = x2
Known and6I^and6I : ∀ x0 x1 x2 x3 x4 x5 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ x5 ⟶ and (and (and (and (and x0 x1) x2) x3) x4) x5
Theorem 91c36.. : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ struct_b x1) ⟶ (∀ x1 x2 . x0 x1 ⟶ x0 x2 ⟶ x0 (3d151.. x1 x2)) ⟶ MetaCat_product_constr_p x0 MagmaHom struct_id struct_comp 3d151.. (λ x1 x2 . lam (setprod (ap x1 0) (ap x2 0)) (λ x3 . ap x3 0)) (λ x1 x2 . lam (setprod (ap x1 0) (ap x2 0)) (λ x3 . ap x3 1)) (λ x1 x2 x3 x4 x5 . lam (ap x3 0) (λ x6 . lam 2 (λ x7 . If_i (x7 = 0) (ap x4 x6) (ap x5 x6)))) (proof)
Theorem f7929.. : MetaCat_product_constr_p struct_b MagmaHom struct_id struct_comp 3d151.. (λ x0 x1 . lam (setprod (ap x0 0) (ap x1 0)) (λ x2 . ap x2 0)) (λ x0 x1 . lam (setprod (ap x0 0) (ap x1 0)) (λ x2 . ap x2 1)) (λ x0 x1 x2 x3 x4 . lam (ap x2 0) (λ x5 . lam 2 (λ x6 . If_i (x6 = 0) (ap x3 x5) (ap x4 x5)))) (proof)
Theorem 150e3..^{MetaCat_struct_b_product_constr} : ∀ x0 : ο . (∀ x1 : ι → ι → ι . (∀ x2 : ο . (∀ x3 : ι → ι → ι . (∀ x4 : ο . (∀ x5 : ι → ι → ι . (∀ x6 : ο . (∀ x7 : ι → ι → ι → ι → ι → ι . MetaCat_product_constr_p struct_b MagmaHom struct_id struct_comp x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Param Sep^Sep : ι → (ι → ο) → ι
Definition 32592.. := λ x0 x1 x2 x3 . unpack_b_i x0 (λ x4 . pack_b {x5 ∈ x4|ap x2 x5 = ap x3 x5})
Known SepE1^SepE1 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x2 ∈ x0
Theorem e123b.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 x3 x4 . 32592.. (pack_b x0 x1) x2 x3 x4 = pack_b {x6 ∈ x0|ap x3 x6 = ap x4 x6} x1 (proof)
Definition MetaCat_equalizer_p^equalizer_p := λ x0 : ι → ο . λ x1 : ι → ι → ι → ο . λ x2 : ι → ι . λ x3 : ι → ι → ι → ι → ι → ι . λ x4 x5 x6 x7 x8 x9 . λ x10 : ι → ι → ι . and (and (and (and (and (and (and (x0 x4) (x0 x5)) (x1 x4 x5 x6)) (x1 x4 x5 x7)) (x0 x8)) (x1 x8 x4 x9)) (x3 x8 x4 x5 x6 x9 = x3 x8 x4 x5 x7 x9)) (∀ x11 . x0 x11 ⟶ ∀ x12 . x1 x11 x4 x12 ⟶ x3 x11 x4 x5 x6 x12 = x3 x11 x4 x5 x7 x12 ⟶ and (and (x1 x11 x8 (x10 x11 x12)) (x3 x11 x8 x4 x9 (x10 x11 x12) = x12)) (∀ x13 . x1 x11 x8 x13 ⟶ x3 x11 x8 x4 x9 x13 = x12 ⟶ x13 = x10 x11 x12))
Definition MetaCat_equalizer_struct_p^{equalizer_constr_p} := λ x0 : ι → ο . λ x1 : ι → ι → ι → ο . λ x2 : ι → ι . λ x3 : ι → ι → ι → ι → ι → ι . λ x4 x5 : ι → ι → ι → ι → ι . λ x6 : ι → ι → ι → ι → ι → ι → ι . ∀ x7 x8 . x0 x7 ⟶ x0 x8 ⟶ ∀ x9 x10 . x1 x7 x8 x9 ⟶ x1 x7 x8 x10 ⟶ MetaCat_equalizer_p x0 x1 x2 x3 x7 x8 x9 x10 (x4 x7 x8 x9 x10) (x5 x7 x8 x9 x10) (x6 x7 x8 x9 x10)
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known SepE2^SepE2 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x1 x2
Known 41253..^and8I : ∀ x0 x1 x2 x3 x4 x5 x6 x7 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ x5 ⟶ x6 ⟶ x7 ⟶ and (and (and (and (and (and (and x0 x1) x2) x3) x4) x5) x6) x7
Theorem d19d6.. : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ struct_b x1) ⟶ (∀ x1 x2 x3 x4 . x0 x1 ⟶ x0 x2 ⟶ MagmaHom x1 x2 x3 ⟶ MagmaHom x1 x2 x4 ⟶ x0 (32592.. x1 x2 x3 x4)) ⟶ ∀ x1 : ο . (∀ x2 : ι → ι → ι → ι → ι . (∀ x3 : ο . (∀ x4 : ι → ι → ι → ι → ι . (∀ x5 : ο . (∀ x6 : ι → ι → ι → ι → ι → ι → ι . MetaCat_equalizer_struct_p x0 MagmaHom struct_id struct_comp x2 x4 x6 ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1 (proof)
Theorem c368b..^{MetaCat_struct_b_equalizer_constr} : ∀ x0 : ο . (∀ x1 : ι → ι → ι → ι → ι . (∀ x2 : ο . (∀ x3 : ι → ι → ι → ι → ι . (∀ x4 : ο . (∀ x5 : ι → ι → ι → ι → ι → ι → ι . MetaCat_equalizer_struct_p struct_b MagmaHom struct_id struct_comp x1 x3 x5 ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Param MetaCat^MetaCat : (ι → ο) → (ι → ι → ι → ο) → (ι → ι) → (ι → ι → ι → ι → ι → ι) → ο
Known c55c1..^{MetaCat_struct_b} : MetaCat struct_b MagmaHom struct_id struct_comp
Param MetaCat_pullback_struct_p^{pullback_constr_p} : (ι → ο) → (ι → ι → ι → ο) → (ι → ι) → (ι → ι → ι → ι → ι → ι) → (ι → ι → ι → ι → ι → ι) → (ι → ι → ι → ι → ι → ι) → (ι → ι → ι → ι → ι → ι) → (ι → ι → ι → ι → ι → ι → ι → ι → ι) → ο
Known ed2b0..^{product_equalizer_pullback_constr_ex} : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ι → ι → ι → ι . MetaCat x0 x1 x2 x3 ⟶ (∀ x4 : ο . (∀ x5 : ι → ι → ι → ι → ι . (∀ x6 : ο . (∀ x7 : ι → ι → ι → ι → ι . (∀ x8 : ο . (∀ x9 : ι → ι → ι → ι → ι → ι → ι . MetaCat_equalizer_struct_p x0 x1 x2 x3 x5 x7 x9 ⟶ x8) ⟶ x8) ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ (∀ x4 : ο . (∀ x5 : ι → ι → ι . (∀ x6 : ο . (∀ x7 : ι → ι → ι . (∀ x8 : ο . (∀ x9 : ι → ι → ι . (∀ x10 : ο . (∀ x11 : ι → ι → ι → ι → ι → ι . MetaCat_product_constr_p x0 x1 x2 x3 x5 x7 x9 x11 ⟶ x10) ⟶ x10) ⟶ x8) ⟶ x8) ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ ∀ x4 : ο . (∀ x5 : ι → ι → ι → ι → ι → ι . (∀ x6 : ο . (∀ x7 : ι → ι → ι → ι → ι → ι . (∀ x8 : ο . (∀ x9 : ι → ι → ι → ι → ι → ι . (∀ x10 : ο . (∀ x11 : ι → ι → ι → ι → ι → ι → ι → ι → ι . MetaCat_pullback_struct_p x0 x1 x2 x3 x5 x7 x9 x11 ⟶ x10) ⟶ x10) ⟶ x8) ⟶ x8) ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4
Theorem a238a..^{MetaCat_struct_b_pullback_constr} : ∀ x0 : ο . (∀ x1 : ι → ι → ι → ι → ι → ι . (∀ x2 : ο . (∀ x3 : ι → ι → ι → ι → ι → ι . (∀ x4 : ο . (∀ x5 : ι → ι → ι → ι → ι → ι . (∀ x6 : ο . (∀ x7 : ι → ι → ι → ι → ι → ι → ι → ι → ι . MetaCat_pullback_struct_p struct_b MagmaHom struct_id struct_comp x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)