Proofgold Address

Param pack_u^pack_u : ι → (ι → ι) → ι
Definition struct_u^struct_u := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι . (∀ x4 . x4 ∈ x2 ⟶ x3 x4 ∈ x2) ⟶ x1 (pack_u 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 UnaryFuncHom^Hom_struct_u : 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 66c4c..^{Hom_struct_u_pack} : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 . UnaryFuncHom (pack_u x0 x2) (pack_u x1 x3) x4 = and (x4 ∈ setexp x1 x0) (∀ x6 . x6 ∈ x0 ⟶ ap x4 (x2 x6) = x3 (ap x4 x6))
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 35202.. : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ struct_u x1) ⟶ x0 (pack_u 0 (λ x1 . x1)) ⟶ MetaCat_initial_p x0 UnaryFuncHom struct_id struct_comp (pack_u 0 (λ x1 . x1)) (λ x1 . lam 0 (λ x2 . x2)) (proof)
Known pack_struct_u_I^{pack_struct_u_I} : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ x0) ⟶ struct_u (pack_u x0 x1)
Theorem 1e2ae..^{MetaCat_struct_u_initial} : ∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . MetaCat_initial_p struct_u UnaryFuncHom 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_u_0_eq2^pack_u_0_eq2 : ∀ x0 . ∀ x1 : ι → ι . x0 = ap (pack_u 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 a3e3e.. : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ struct_u x1) ⟶ x0 (pack_u 1 (λ x1 . x1)) ⟶ MetaCat_terminal_p x0 UnaryFuncHom struct_id struct_comp (pack_u 1 (λ x1 . x1)) (λ x1 . lam (ap x1 0) (λ x2 . 0)) (proof)
Theorem 75fea..^{MetaCat_struct_u_terminal} : ∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . MetaCat_terminal_p struct_u UnaryFuncHom struct_id struct_comp x1 x3 ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Param unpack_u_i^unpack_u_i : ι → (ι → (ι → ι) → ι) → ι
Definition setprod^setprod := λ x0 x1 . lam x0 (λ x2 . x1)
Param If_i^If_i : ο → ι → ι → ι
Definition 5a1fb.. := λ x0 x1 . unpack_u_i x0 (λ x2 . λ x3 : ι → ι . unpack_u_i x1 (λ x4 . λ x5 : ι → ι . pack_u (setprod x2 x4) (λ x6 . lam 2 (λ x7 . If_i (x7 = 0) (x3 (ap x6 0)) (x5 (ap x6 1))))))
Known unpack_u_i_eq^{unpack_u_i_eq} : ∀ x0 : ι → (ι → ι) → ι . ∀ x1 . ∀ x2 : ι → ι . (∀ x3 : ι → ι . (∀ x4 . x4 ∈ x1 ⟶ x2 x4 = x3 x4) ⟶ x0 x1 x3 = x0 x1 x2) ⟶ unpack_u_i (pack_u x1 x2) x0 = x0 x1 x2
Known pack_u_ext^pack_u_ext : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x1 x3 = x2 x3) ⟶ pack_u x0 x1 = pack_u 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 ec613.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . ∀ x3 : ι → ι . 5a1fb.. (pack_u x0 x1) (pack_u x2 x3) = pack_u (setprod x0 x2) (λ x5 . lam 2 (λ x6 . If_i (x6 = 0) (x1 (ap x5 0)) (x3 (ap x5 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 0ef6e.. : ∀ x0 x1 . struct_u x0 ⟶ struct_u x1 ⟶ struct_u (5a1fb.. 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 e49b4.. : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ struct_u x1) ⟶ (∀ x1 x2 . x0 x1 ⟶ x0 x2 ⟶ x0 (5a1fb.. x1 x2)) ⟶ MetaCat_product_constr_p x0 UnaryFuncHom struct_id struct_comp 5a1fb.. (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 e13b6.. : MetaCat_product_constr_p struct_u UnaryFuncHom struct_id struct_comp 5a1fb.. (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 a9079..^{MetaCat_struct_u_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_u UnaryFuncHom 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 9fd7a.. := λ x0 x1 x2 x3 . unpack_u_i x0 (λ x4 . pack_u {x5 ∈ x4|ap x2 x5 = ap x3 x5})
Known SepE1^SepE1 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x2 ∈ x0
Theorem 70b0f.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 x4 . 9fd7a.. (pack_u x0 x1) x2 x3 x4 = pack_u {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 1077c.. : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ struct_u x1) ⟶ (∀ x1 x2 x3 x4 . x0 x1 ⟶ x0 x2 ⟶ UnaryFuncHom x1 x2 x3 ⟶ UnaryFuncHom x1 x2 x4 ⟶ x0 (9fd7a.. 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 UnaryFuncHom struct_id struct_comp x2 x4 x6 ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1 (proof)
Theorem ee44b..^{MetaCat_struct_u_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_u UnaryFuncHom 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 73eab..^{MetaCat_struct_u} : MetaCat struct_u UnaryFuncHom 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 75a41..^{MetaCat_struct_u_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_u UnaryFuncHom struct_id struct_comp x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)