Proofgold Asset

Param unpack_r_i^unpack_r_i : ι → (ι → (ι → ι → ο) → ι) → ι
Param pack_r^pack_r : ι → (ι → ι → ο) → ι
Param Sep^Sep : ι → (ι → ο) → ι
Param ap^ap : ι → ι → ι
Definition 05907.. := λ x0 x1 x2 x3 . unpack_r_i x0 (λ x4 . pack_r {x5 ∈ x4|ap x2 x5 = ap x3 x5})
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition iff^iff := λ x0 x1 : ο . and (x0 ⟶ x1) (x1 ⟶ x0)
Known unpack_r_i_eq^{unpack_r_i_eq} : ∀ x0 : ι → (ι → ι → ο) → ι . ∀ x1 . ∀ x2 : ι → ι → ο . (∀ x3 : ι → ι → ο . (∀ x4 . x4 ∈ x1 ⟶ ∀ x5 . x5 ∈ x1 ⟶ iff (x2 x4 x5) (x3 x4 x5)) ⟶ x0 x1 x3 = x0 x1 x2) ⟶ unpack_r_i (pack_r x1 x2) x0 = x0 x1 x2
Known pack_r_ext^pack_r_ext : ∀ x0 . ∀ x1 x2 : ι → ι → ο . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ iff (x1 x3 x4) (x2 x3 x4)) ⟶ pack_r x0 x1 = pack_r x0 x2
Known SepE1^SepE1 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x2 ∈ x0
Known iffI^iffI : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ iff x0 x1
Theorem 08d58.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 x3 x4 . 05907.. (pack_r x0 x1) x2 x3 x4 = pack_r {x6 ∈ x0|ap x3 x6 = ap x4 x6} x1 (proof)
Definition struct_r^struct_r := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ο . x1 (pack_r x2 x3)) ⟶ x1 x0
Param BinRelnHom^Hom_struct_r : ι → ι → ι → ο
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)
Param struct_id^struct_id : ι → ι
Param lam^Sigma : ι → (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 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
Param Pi^Pi : ι → (ι → ι) → ι
Definition setexp^setexp := λ x0 x1 . Pi x1 (λ x2 . x0)
Known c84ab..^{Hom_struct_r_pack} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 . BinRelnHom (pack_r x0 x2) (pack_r x1 x3) x4 = and (x4 ∈ setexp x1 x0) (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x2 x6 x7 ⟶ x3 (ap x4 x6) (ap x4 x7))
Known pack_r_0_eq2^pack_r_0_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι → ο . x2 x0 (ap (pack_r x0 x1) 0) ⟶ x2 (ap (pack_r x0 x1) 0) x0
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known lam_Pi^lam_Pi : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1 x3) ⟶ lam x0 x2 ∈ Pi x0 x1
Known beta^beta : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ ap (lam x0 x1) x2 = x1 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
Known SepE2^SepE2 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x1 x2
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known Pi_eta^Pi_eta : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ Pi x0 x1 ⟶ lam x0 (ap x2) = x2
Known ap_Pi^ap_Pi : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ Pi x0 x1 ⟶ x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Theorem 3d2c4.. : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ struct_r x1) ⟶ (∀ x1 x2 x3 x4 . x0 x1 ⟶ x0 x2 ⟶ BinRelnHom x1 x2 x3 ⟶ BinRelnHom x1 x2 x4 ⟶ x0 (05907.. 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 BinRelnHom struct_id struct_comp x2 x4 x6 ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1 (proof)
Known pack_struct_r_I^{pack_struct_r_I} : ∀ x0 . ∀ x1 : ι → ι → ο . struct_r (pack_r x0 x1)
Theorem 1d8ba..^{MetaCat_struct_r_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_r BinRelnHom 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 6955f..^{MetaCat_struct_r} : MetaCat struct_r BinRelnHom 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
Param MetaCat_product_constr_p^{product_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) → 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
Known ece68..^{MetaCat_struct_r_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_r BinRelnHom struct_id struct_comp x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0
Theorem 97f16..^{MetaCat_struct_r_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_r BinRelnHom struct_id struct_comp x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Param unpack_r_o^unpack_r_o : ι → (ι → (ι → ι → ο) → ο) → ο
Param not^not : ο → ο
Definition IrreflexiveSymmetricReln^{struct_r_graph} := λ x0 . and (struct_r x0) (unpack_r_o x0 (λ x1 . λ x2 : ι → ι → ο . and (∀ x3 . x3 ∈ x1 ⟶ not (x2 x3 x3)) (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x3 x4 ⟶ x2 x4 x3)))
Known 96ca7.. : ∀ x0 . IrreflexiveSymmetricReln x0 ⟶ ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ο . (∀ x4 . x4 ∈ x2 ⟶ not (x3 x4 x4)) ⟶ (∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x2 ⟶ x3 x4 x5 ⟶ x3 x5 x4) ⟶ x1 (pack_r x2 x3)) ⟶ x1 x0
Known 36176.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 . x2 ∈ x0 ⟶ not (x1 x2 x2)) ⟶ (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ⟶ x1 x3 x2) ⟶ IrreflexiveSymmetricReln (pack_r x0 x1)
Theorem 06b27..^{MetaCat_struct_r_graph_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 IrreflexiveSymmetricReln BinRelnHom struct_id struct_comp x1 x3 x5 ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Known 71675..^{MetaCat_struct_r_graph} : MetaCat IrreflexiveSymmetricReln BinRelnHom struct_id struct_comp
Known 709ef..^{MetaCat_struct_r_graph_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 IrreflexiveSymmetricReln BinRelnHom struct_id struct_comp x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0
Theorem c2d12..^{MetaCat_struct_r_graph_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 IrreflexiveSymmetricReln BinRelnHom struct_id struct_comp x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Definition PER^struct_r_per := λ x0 . and (struct_r x0) (unpack_r_o x0 (λ x1 . λ x2 : ι → ι → ο . and (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x3 x4 ⟶ x2 x4 x3) (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ ∀ x5 . x5 ∈ x1 ⟶ x2 x3 x4 ⟶ x2 x4 x5 ⟶ x2 x3 x5)))
Known 0bd5c.. : ∀ x0 . PER x0 ⟶ ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ο . (∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x2 ⟶ x3 x4 x5 ⟶ x3 x5 x4) ⟶ (∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x2 ⟶ ∀ x6 . x6 ∈ x2 ⟶ x3 x4 x5 ⟶ x3 x5 x6 ⟶ x3 x4 x6) ⟶ x1 (pack_r x2 x3)) ⟶ x1 x0
Known a3466.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ⟶ x1 x3 x2) ⟶ (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x2 x3 ⟶ x1 x3 x4 ⟶ x1 x2 x4) ⟶ PER (pack_r x0 x1)
Theorem 57f50..^{MetaCat_struct_r_per_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 PER BinRelnHom struct_id struct_comp x1 x3 x5 ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Known 259fb..^{MetaCat_struct_r_per} : MetaCat PER BinRelnHom struct_id struct_comp
Known b370d..^{MetaCat_struct_r_per_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 PER BinRelnHom struct_id struct_comp x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0
Theorem 44067..^{MetaCat_struct_r_per_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 PER BinRelnHom struct_id struct_comp x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Definition EquivReln^{struct_r_equivreln} := λ x0 . and (struct_r x0) (unpack_r_o x0 (λ x1 . λ x2 : ι → ι → ο . and (and (∀ x3 . x3 ∈ x1 ⟶ x2 x3 x3) (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x3 x4 ⟶ x2 x4 x3)) (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ ∀ x5 . x5 ∈ x1 ⟶ x2 x3 x4 ⟶ x2 x4 x5 ⟶ x2 x3 x5)))
Known 909a7.. : ∀ x0 . EquivReln x0 ⟶ ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ο . (∀ x4 . x4 ∈ x2 ⟶ x3 x4 x4) ⟶ (∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x2 ⟶ x3 x4 x5 ⟶ x3 x5 x4) ⟶ (∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x2 ⟶ ∀ x6 . x6 ∈ x2 ⟶ x3 x4 x5 ⟶ x3 x5 x6 ⟶ x3 x4 x6) ⟶ x1 (pack_r x2 x3)) ⟶ x1 x0
Known 517b3.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 . x2 ∈ x0 ⟶ x1 x2 x2) ⟶ (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ⟶ x1 x3 x2) ⟶ (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x2 x3 ⟶ x1 x3 x4 ⟶ x1 x2 x4) ⟶ EquivReln (pack_r x0 x1)
Theorem 99d06..^{MetaCat_struct_r_equivreln_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 EquivReln BinRelnHom struct_id struct_comp x1 x3 x5 ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Known ca919..^{MetaCat_struct_r_equivreln} : MetaCat EquivReln BinRelnHom struct_id struct_comp
Known 4d1df..^{MetaCat_struct_r_equivreln_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 EquivReln BinRelnHom struct_id struct_comp x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0
Theorem e7f7d..^{MetaCat_struct_r_equivreln_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 EquivReln BinRelnHom struct_id struct_comp x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Definition IrreflexiveTransitiveReln^{struct_r_partialord} := λ x0 . and (struct_r x0) (unpack_r_o x0 (λ x1 . λ x2 : ι → ι → ο . and (∀ x3 . x3 ∈ x1 ⟶ not (x2 x3 x3)) (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ ∀ x5 . x5 ∈ x1 ⟶ x2 x3 x4 ⟶ x2 x4 x5 ⟶ x2 x3 x5)))
Known af4aa.. : ∀ x0 . IrreflexiveTransitiveReln x0 ⟶ ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ο . (∀ x4 . x4 ∈ x2 ⟶ not (x3 x4 x4)) ⟶ (∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x2 ⟶ ∀ x6 . x6 ∈ x2 ⟶ x3 x4 x5 ⟶ x3 x5 x6 ⟶ x3 x4 x6) ⟶ x1 (pack_r x2 x3)) ⟶ x1 x0
Known b25e7.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 . x2 ∈ x0 ⟶ not (x1 x2 x2)) ⟶ (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x2 x3 ⟶ x1 x3 x4 ⟶ x1 x2 x4) ⟶ IrreflexiveTransitiveReln (pack_r x0 x1)
Theorem 3d0dd..^{MetaCat_struct_r_partialord_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 IrreflexiveTransitiveReln BinRelnHom struct_id struct_comp x1 x3 x5 ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Known c6620..^{MetaCat_struct_r_partialord} : MetaCat IrreflexiveTransitiveReln BinRelnHom struct_id struct_comp
Known 42715..^{MetaCat_struct_r_partialord_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 IrreflexiveTransitiveReln BinRelnHom struct_id struct_comp x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0
Theorem d1a34..^{MetaCat_struct_r_partialord_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 IrreflexiveTransitiveReln BinRelnHom struct_id struct_comp x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)