Proofgold Address

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 lam^Sigma : ι → (ι → ι) → ι
Definition setprod^setprod := λ x0 x1 . lam x0 (λ x2 . x1)
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param ap^ap : ι → ι → ι
Param ordsucc^ordsucc : ι → ι
Definition BinReln_product := λ x0 x1 . unpack_r_i x0 (λ x2 . λ x3 : ι → ι → ο . unpack_r_i x1 (λ x4 . λ x5 : ι → ι → ο . pack_r (setprod x2 x4) (λ x6 x7 . and (x3 (ap x6 0) (ap x7 0)) (x5 (ap x6 1) (ap x7 1)))))
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 iffI^iffI : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ iff x0 x1
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
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 433d4.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 . ∀ x3 x4 : ι → ι → ο . x4 (BinReln_product (pack_r x0 x1) (pack_r x2 x3)) (pack_r (setprod x0 x2) (λ x5 x6 . and (x1 (ap x5 0) (ap x6 0)) (x3 (ap x5 1) (ap x6 1)))) ⟶ x4 (pack_r (setprod x0 x2) (λ x5 x6 . and (x1 (ap x5 0) (ap x6 0)) (x3 (ap x5 1) (ap x6 1)))) (BinReln_product (pack_r x0 x1) (pack_r x2 x3)) (proof)
Definition struct_r^struct_r := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ο . x1 (pack_r x2 x3)) ⟶ x1 x0
Known pack_struct_r_I^{pack_struct_r_I} : ∀ x0 . ∀ x1 : ι → ι → ο . struct_r (pack_r x0 x1)
Theorem 08ab7.. : ∀ x0 x1 . struct_r x0 ⟶ struct_r x1 ⟶ struct_r (BinReln_product 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)
Param BinRelnHom^Hom_struct_r : ι → ι → ι → ο
Param struct_id^struct_id : 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)
Param If_i^If_i : ο → ι → ι → ι
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
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 and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
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 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 and4I^and4I : ∀ x0 x1 x2 x3 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ and (and (and x0 x1) x2) x3
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 ap_Pi^ap_Pi : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ Pi x0 x1 ⟶ x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3
Known Pi_eta^Pi_eta : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ Pi x0 x1 ⟶ lam x0 (ap x2) = x2
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 tuple_2_1_eq^tuple_2_1_eq : ∀ x0 x1 . ap (lam 2 (λ x3 . If_i (x3 = 0) x0 x1)) 1 = x1
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_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 10126.. : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ struct_r x1) ⟶ (∀ x1 x2 . x0 x1 ⟶ x0 x2 ⟶ x0 (BinReln_product x1 x2)) ⟶ MetaCat_product_constr_p x0 BinRelnHom struct_id struct_comp BinReln_product (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 060a4.. : MetaCat_product_constr_p struct_r BinRelnHom struct_id struct_comp BinReln_product (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 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 (proof)
Definition BinReln_exp := λ x0 x1 . unpack_r_i x0 (λ x2 . λ x3 : ι → ι → ο . unpack_r_i x1 (λ x4 . λ x5 : ι → ι → ο . pack_r (setexp x4 x2) (λ x6 x7 . ∀ x8 . x8 ∈ x2 ⟶ ∀ x9 . x9 ∈ x2 ⟶ x3 x8 x9 ⟶ x5 (ap x6 x8) (ap x7 x9))))
Theorem 5ade7.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 . ∀ x3 x4 : ι → ι → ο . x4 (BinReln_exp (pack_r x0 x1) (pack_r x2 x3)) (pack_r (setexp x2 x0) (λ x5 x6 . ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x1 x7 x8 ⟶ x3 (ap x5 x7) (ap x6 x8))) ⟶ x4 (pack_r (setexp x2 x0) (λ x5 x6 . ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x1 x7 x8 ⟶ x3 (ap x5 x7) (ap x6 x8))) (BinReln_exp (pack_r x0 x1) (pack_r x2 x3)) (proof)
Theorem 89edf.. : ∀ x0 x1 . struct_r x0 ⟶ struct_r x1 ⟶ struct_r (BinReln_exp x0 x1) (proof)
Definition MetaCat_exp_p^exponent_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 . λ x8 x9 x10 x11 . λ x12 : ι → ι → ι . and (and (and (and (x0 x8) (x0 x9)) (x0 x10)) (x1 (x4 x10 x8) x9 x11)) (∀ x13 . x0 x13 ⟶ ∀ x14 . x1 (x4 x13 x8) x9 x14 ⟶ and (and (x1 x13 x10 (x12 x13 x14)) (x3 (x4 x13 x8) (x4 x10 x8) x9 x11 (x7 x10 x8 (x4 x13 x8) (x3 (x4 x13 x8) x13 x10 (x12 x13 x14) (x5 x13 x8)) (x6 x13 x8)) = x14)) (∀ x15 . x1 x13 x10 x15 ⟶ x3 (x4 x13 x8) (x4 x10 x8) x9 x11 (x7 x10 x8 (x4 x13 x8) (x3 (x4 x13 x8) x13 x10 x15 (x5 x13 x8)) (x6 x13 x8)) = x14 ⟶ x15 = x12 x13 x14))
Definition MetaCat_exp_constr_p^{product_exponent_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 . λ x8 x9 : ι → ι → ι . λ x10 : ι → ι → ι → ι → ι . and (MetaCat_product_constr_p x0 x1 x2 x3 x4 x5 x6 x7) (∀ x11 x12 . x0 x11 ⟶ x0 x12 ⟶ MetaCat_exp_p x0 x1 x2 x3 x4 x5 x6 x7 x11 x12 (x8 x11 x12) (x9 x11 x12) (x10 x11 x12))
Known and5I^and5I : ∀ x0 x1 x2 x3 x4 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ and (and (and (and x0 x1) x2) x3) x4
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
Theorem 2450d.. : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ struct_r x1) ⟶ (∀ x1 x2 . x0 x1 ⟶ x0 x2 ⟶ x0 (BinReln_product x1 x2)) ⟶ (∀ x1 x2 . x0 x1 ⟶ x0 x2 ⟶ x0 (BinReln_exp x1 x2)) ⟶ MetaCat_exp_constr_p x0 BinRelnHom struct_id struct_comp BinReln_product (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)))) BinReln_exp (λ x1 x2 . lam (setprod (setexp (ap x2 0) (ap x1 0)) (ap x1 0)) (λ x3 . ap (ap x3 0) (ap x3 1))) (λ x1 x2 x3 x4 . lam (ap x3 0) (λ x5 . lam (ap x1 0) (λ x6 . ap x4 (lam 2 (λ x7 . If_i (x7 = 0) x5 x6))))) (proof)
Theorem 4f1fe.. : MetaCat_exp_constr_p struct_r BinRelnHom struct_id struct_comp BinReln_product (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)))) BinReln_exp (λ x0 x1 . lam (setprod (setexp (ap x1 0) (ap x0 0)) (ap x0 0)) (λ x2 . ap (ap x2 0) (ap x2 1))) (λ x0 x1 x2 x3 . lam (ap x2 0) (λ x4 . lam (ap x0 0) (λ x5 . ap x3 (lam 2 (λ x6 . If_i (x6 = 0) x4 x5))))) (proof)
Theorem 7f417..^{MetaCat_struct_r_product_exponent} : ∀ 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 . (∀ x8 : ο . (∀ x9 : ι → ι → ι . (∀ x10 : ο . (∀ x11 : ι → ι → ι . (∀ x12 : ο . (∀ x13 : ι → ι → ι → ι → ι . MetaCat_exp_constr_p struct_r BinRelnHom struct_id struct_comp x1 x3 x5 x7 x9 x11 x13 ⟶ x12) ⟶ x12) ⟶ x10) ⟶ x10) ⟶ x8) ⟶ x8) ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Param unpack_r_o^unpack_r_o : ι → (ι → (ι → ι → ο) → ο) → ο
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
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 5344f.. : ∀ x0 . ∀ x1 : ι → ι → ο . unpack_r_o (pack_r x0 x1) (λ x3 . λ x4 : ι → ι → ο . and (∀ x5 . x5 ∈ x3 ⟶ not (x4 x5 x5)) (∀ x5 . x5 ∈ x3 ⟶ ∀ x6 . x6 ∈ x3 ⟶ x4 x5 x6 ⟶ x4 x6 x5)) = and (∀ x3 . x3 ∈ x0 ⟶ not (x1 x3 x3)) (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 ⟶ x1 x4 x3)
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 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 (proof)
Theorem e7d0d.. : ∀ x0 . IrreflexiveSymmetricReln x0 ⟶ struct_r x0 (proof)
Theorem 49c4f.. : ∀ x0 x1 . IrreflexiveSymmetricReln x0 ⟶ IrreflexiveSymmetricReln x1 ⟶ IrreflexiveSymmetricReln (BinReln_product x0 x1) (proof)
Theorem 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 (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 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)
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
Theorem 1b5dd.. : ∀ x0 . PER x0 ⟶ struct_r x0 (proof)
Theorem a9f70.. : ∀ x0 x1 . PER x0 ⟶ PER x1 ⟶ PER (BinReln_product x0 x1) (proof)
Theorem 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 (proof)
Theorem 310d5.. : ∀ x0 x1 . PER x0 ⟶ PER x1 ⟶ PER (BinReln_exp x0 x1) (proof)
Theorem 97ee8..^{MetaCat_struct_r_per_product_exponent} : ∀ 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 . (∀ x8 : ο . (∀ x9 : ι → ι → ι . (∀ x10 : ο . (∀ x11 : ι → ι → ι . (∀ x12 : ο . (∀ x13 : ι → ι → ι → ι → ι . MetaCat_exp_constr_p PER BinRelnHom struct_id struct_comp x1 x3 x5 x7 x9 x11 x13 ⟶ x12) ⟶ x12) ⟶ x10) ⟶ x10) ⟶ x8) ⟶ x8) ⟶ 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 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)
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
Theorem 2effc.. : ∀ x0 . EquivReln x0 ⟶ struct_r x0 (proof)
Theorem 7da0d.. : ∀ x0 x1 . EquivReln x0 ⟶ EquivReln x1 ⟶ EquivReln (BinReln_product x0 x1) (proof)
Theorem 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 (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 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)
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
Theorem 294e6.. : ∀ x0 . IrreflexiveTransitiveReln x0 ⟶ struct_r x0 (proof)
Theorem e27aa.. : ∀ x0 x1 . IrreflexiveTransitiveReln x0 ⟶ IrreflexiveTransitiveReln x1 ⟶ IrreflexiveTransitiveReln (BinReln_product x0 x1) (proof)
Theorem 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 (proof)