Proofgold Address

Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known 7c691..^and9I : ∀ x0 x1 x2 x3 x4 x5 x6 x7 x8 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ x5 ⟶ x6 ⟶ x7 ⟶ x8 ⟶ and (and (and (and (and (and (and (and x0 x1) x2) x3) x4) x5) x6) x7) x8
Definition MetaFunctor_prop1^idT := λ x0 : ι → ο . λ x1 : ι → ι → ι → ο . λ x2 : ι → ι . λ x3 : ι → ι → ι → ι → ι → ι . ∀ x4 . x0 x4 ⟶ x1 x4 x4 (x2 x4)
Definition MetaFunctor_prop2^compT := λ x0 : ι → ο . λ x1 : ι → ι → ι → ο . λ x2 : ι → ι . λ x3 : ι → ι → ι → ι → ι → ι . ∀ x4 x5 x6 x7 x8 . x0 x4 ⟶ x0 x5 ⟶ x0 x6 ⟶ x1 x4 x5 x7 ⟶ x1 x5 x6 x8 ⟶ x1 x4 x6 (x3 x4 x5 x6 x8 x7)
Definition MetaCat_IdR_p^idL := λ x0 : ι → ο . λ x1 : ι → ι → ι → ο . λ x2 : ι → ι . λ x3 : ι → ι → ι → ι → ι → ι . ∀ x4 x5 x6 . x0 x4 ⟶ x0 x5 ⟶ x1 x4 x5 x6 ⟶ x3 x4 x4 x5 x6 (x2 x4) = x6
Definition MetaCat_IdL_p^idR := λ x0 : ι → ο . λ x1 : ι → ι → ι → ο . λ x2 : ι → ι . λ x3 : ι → ι → ι → ι → ι → ι . ∀ x4 x5 x6 . x0 x4 ⟶ x0 x5 ⟶ x1 x4 x5 x6 ⟶ x3 x4 x5 x5 (x2 x5) x6 = x6
Definition MetaCat_Comp_assoc_p^compAssoc := λ x0 : ι → ο . λ x1 : ι → ι → ι → ο . λ x2 : ι → ι . λ x3 : ι → ι → ι → ι → ι → ι . ∀ x4 x5 x6 x7 x8 x9 x10 . x0 x4 ⟶ x0 x5 ⟶ x0 x6 ⟶ x0 x7 ⟶ x1 x4 x5 x8 ⟶ x1 x5 x6 x9 ⟶ x1 x6 x7 x10 ⟶ x3 x4 x5 x7 (x3 x5 x6 x7 x10 x9) x8 = x3 x4 x6 x7 x10 (x3 x4 x5 x6 x9 x8)
Definition MetaCat^MetaCat := λ x0 : ι → ο . λ x1 : ι → ι → ι → ο . λ x2 : ι → ι . λ x3 : ι → ι → ι → ι → ι → ι . and (and (and (MetaFunctor_prop1 x0 x1 x2 x3) (MetaFunctor_prop2 x0 x1 x2 x3)) (and (MetaCat_IdR_p x0 x1 x2 x3) (MetaCat_IdL_p x0 x1 x2 x3))) (MetaCat_Comp_assoc_p x0 x1 x2 x3)
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Theorem 047c9..^MetaCat_I : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ι → ι → ι → ι . MetaFunctor_prop1 x0 x1 x2 x3 ⟶ MetaFunctor_prop2 x0 x1 x2 x3 ⟶ (∀ x4 x5 x6 . x0 x4 ⟶ x0 x5 ⟶ x1 x4 x5 x6 ⟶ x3 x4 x4 x5 x6 (x2 x4) = x6) ⟶ (∀ x4 x5 x6 . x0 x4 ⟶ x0 x5 ⟶ x1 x4 x5 x6 ⟶ x3 x4 x5 x5 (x2 x5) x6 = x6) ⟶ (∀ x4 x5 x6 x7 x8 x9 x10 . x0 x4 ⟶ x0 x5 ⟶ x0 x6 ⟶ x0 x7 ⟶ x1 x4 x5 x8 ⟶ x1 x5 x6 x9 ⟶ x1 x6 x7 x10 ⟶ x3 x4 x5 x7 (x3 x5 x6 x7 x10 x9) x8 = x3 x4 x6 x7 x10 (x3 x4 x5 x6 x9 x8)) ⟶ MetaCat x0 x1 x2 x3 (proof)
Theorem 7da4b..^MetaCat_E : ∀ 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 . (MetaFunctor_prop1 x0 x1 x2 x3 ⟶ MetaFunctor_prop2 x0 x1 x2 x3 ⟶ (∀ x5 x6 x7 . x0 x5 ⟶ x0 x6 ⟶ x1 x5 x6 x7 ⟶ x3 x5 x5 x6 x7 (x2 x5) = x7) ⟶ (∀ x5 x6 x7 . x0 x5 ⟶ x0 x6 ⟶ x1 x5 x6 x7 ⟶ x3 x5 x6 x6 (x2 x6) x7 = x7) ⟶ (∀ x5 x6 x7 x8 x9 x10 x11 . x0 x5 ⟶ x0 x6 ⟶ x0 x7 ⟶ x0 x8 ⟶ x1 x5 x6 x9 ⟶ x1 x6 x7 x10 ⟶ x1 x7 x8 x11 ⟶ x3 x5 x6 x8 (x3 x6 x7 x8 x11 x10) x9 = x3 x5 x7 x8 x11 (x3 x5 x6 x7 x10 x9)) ⟶ x4) ⟶ x4 (proof)
Theorem 6f34f..^MetaCatOp : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ι → ι → ι → ι . MetaCat x0 x1 x2 x3 ⟶ MetaCat x0 (λ x4 x5 . x1 x5 x4) x2 (λ x4 x5 x6 x7 x8 . x3 x6 x5 x4 x8 x7) (proof)
Definition MetaCat_monic_p^monic := 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 . and (and (and (x0 x4) (x0 x5)) (x1 x4 x5 x6)) (∀ x7 . x0 x7 ⟶ ∀ x8 x9 . x1 x7 x4 x8 ⟶ x1 x7 x4 x9 ⟶ x3 x7 x4 x5 x6 x8 = x3 x7 x4 x5 x6 x9 ⟶ x8 = x9)
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))
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))
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)
Definition MetaCat_coproduct_p^coproduct_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 x4 x6 x7)) (x1 x5 x6 x8)) (∀ x10 . x0 x10 ⟶ ∀ x11 x12 . x1 x4 x10 x11 ⟶ x1 x5 x10 x12 ⟶ and (and (and (x1 x6 x10 (x9 x10 x11 x12)) (x3 x4 x6 x10 (x9 x10 x11 x12) x7 = x11)) (x3 x5 x6 x10 (x9 x10 x11 x12) x8 = x12)) (∀ x13 . x1 x6 x10 x13 ⟶ x3 x4 x6 x10 x13 x7 = x11 ⟶ x3 x5 x6 x10 x13 x8 = x12 ⟶ x13 = x9 x10 x11 x12))
Definition MetaCat_coproduct_constr_p^{coproduct_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_coproduct_p x0 x1 x2 x3 x8 x9 (x4 x8 x9) (x5 x8 x9) (x6 x8 x9) (x7 x8 x9)
Definition MetaCat_equalizer_buggy_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 (x0 x4) (x0 x5)) (x1 x4 x5 x6)) (x1 x4 x5 x7)) (x0 x8)) (x1 x8 x4 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_buggy_struct_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_buggy_p x0 x1 x2 x3 x7 x8 x9 x10 (x4 x7 x8 x9 x10) (x5 x7 x8 x9 x10) (x6 x7 x8 x9 x10)
Definition MetaCat_coequalizer_buggy_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 (x0 x4) (x0 x5)) (x1 x4 x5 x6)) (x1 x4 x5 x7)) (x0 x8)) (x1 x5 x8 x9)) (∀ x11 . x0 x11 ⟶ ∀ x12 . x1 x5 x11 x12 ⟶ x3 x4 x5 x11 x12 x6 = x3 x4 x5 x11 x12 x7 ⟶ and (and (x1 x8 x11 (x10 x11 x12)) (x3 x5 x8 x11 (x10 x11 x12) x9 = x12)) (∀ x13 . x1 x8 x11 x13 ⟶ x3 x5 x8 x11 x13 x9 = x12 ⟶ x13 = x10 x11 x12))
Definition MetaCat_coequalizer_buggy_struct_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_coequalizer_buggy_p x0 x1 x2 x3 x7 x8 x9 x10 (x4 x7 x8 x9 x10) (x5 x7 x8 x9 x10) (x6 x7 x8 x9 x10)
Definition MetaCat_pullback_buggy_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 x10 x11 . λ x12 : ι → ι → ι → ι . and (and (and (and (and (and (and (and (x0 x4) (x0 x5)) (x0 x6)) (x1 x4 x6 x7)) (x1 x5 x6 x8)) (x0 x9)) (x1 x9 x4 x10)) (x1 x9 x5 x11)) (∀ x13 . x0 x13 ⟶ ∀ x14 . x1 x13 x4 x14 ⟶ ∀ x15 . x1 x13 x5 x15 ⟶ x3 x13 x4 x6 x7 x14 = x3 x13 x5 x6 x8 x15 ⟶ and (and (and (x1 x13 x9 (x12 x13 x14 x15)) (x3 x13 x9 x4 x10 (x12 x13 x14 x15) = x14)) (x3 x13 x9 x5 x11 (x12 x13 x14 x15) = x15)) (∀ x16 . x1 x13 x9 x16 ⟶ x3 x13 x9 x4 x10 x16 = x14 ⟶ x3 x13 x9 x5 x11 x16 = x15 ⟶ x16 = x12 x13 x14 x15))
Definition MetaCat_pullback_buggy_struct_p := λ x0 : ι → ο . λ x1 : ι → ι → ι → ο . λ x2 : ι → ι . λ x3 x4 x5 x6 : ι → ι → ι → ι → ι → ι . λ x7 : ι → ι → ι → ι → ι → ι → ι → ι → ι . ∀ x8 x9 x10 . x0 x8 ⟶ x0 x9 ⟶ x0 x10 ⟶ ∀ x11 x12 . x1 x8 x10 x11 ⟶ x1 x9 x10 x12 ⟶ MetaCat_pullback_buggy_p x0 x1 x2 x3 x8 x9 x10 x11 x12 (x4 x8 x9 x10 x11 x12) (x5 x8 x9 x10 x11 x12) (x6 x8 x9 x10 x11 x12) (x7 x8 x9 x10 x11 x12)
Definition MetaCat_pushout_buggy_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 x10 x11 . λ x12 : ι → ι → ι → ι . and (and (and (and (and (and (and (and (x0 x4) (x0 x5)) (x0 x6)) (x1 x6 x4 x7)) (x1 x6 x5 x8)) (x0 x9)) (x1 x4 x9 x10)) (x1 x5 x9 x11)) (∀ x13 . x0 x13 ⟶ ∀ x14 . x1 x4 x13 x14 ⟶ ∀ x15 . x1 x5 x13 x15 ⟶ x3 x6 x4 x13 x14 x7 = x3 x6 x5 x13 x15 x8 ⟶ and (and (and (x1 x9 x13 (x12 x13 x14 x15)) (x3 x4 x9 x13 (x12 x13 x14 x15) x10 = x14)) (x3 x5 x9 x13 (x12 x13 x14 x15) x11 = x15)) (∀ x16 . x1 x9 x13 x16 ⟶ x3 x4 x9 x13 x16 x10 = x14 ⟶ x3 x5 x9 x13 x16 x11 = x15 ⟶ x16 = x12 x13 x14 x15))
Definition MetaCat_pushout_buggy_constr_p := λ x0 : ι → ο . λ x1 : ι → ι → ι → ο . λ x2 : ι → ι . λ x3 x4 x5 x6 : ι → ι → ι → ι → ι → ι . λ x7 : ι → ι → ι → ι → ι → ι → ι → ι → ι . ∀ x8 x9 x10 . x0 x8 ⟶ x0 x9 ⟶ x0 x10 ⟶ ∀ x11 x12 . x1 x10 x8 x11 ⟶ x1 x10 x9 x12 ⟶ MetaCat_pushout_buggy_p x0 x1 x2 x3 x8 x9 x10 x11 x12 (x4 x8 x9 x10 x11 x12) (x5 x8 x9 x10 x11 x12) (x6 x8 x9 x10 x11 x12) (x7 x8 x9 x10 x11 x12)
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))
Definition MetaCat_subobject_classifier_buggy_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 . λ x6 x7 . λ x8 : ι → ι → ι → ι . λ x9 : ι → ι → ι → ι → ι → ι → ι . and (and (MetaCat_terminal_p x0 x1 x2 x3 x4 x5) (x1 x4 x6 x7)) (∀ x10 x11 x12 . MetaCat_monic_p x0 x1 x2 x3 x10 x11 x12 ⟶ and (x1 x11 x6 (x8 x10 x11 x12)) (MetaCat_pullback_buggy_p x0 x1 x2 x3 x4 x11 x6 x7 (x8 x10 x11 x12) x10 (x5 x10) x12 (x9 x10 x11 x12)))
Definition MetaCat_nno_p^nno_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 . λ x6 x7 x8 . λ x9 : ι → ι → ι → ι . and (and (and (and (MetaCat_terminal_p x0 x1 x2 x3 x4 x5) (x0 x6)) (x1 x4 x6 x7)) (x1 x6 x6 x8)) (∀ x10 x11 x12 . x0 x10 ⟶ x1 x4 x10 x11 ⟶ x1 x10 x10 x12 ⟶ and (and (and (x1 x6 x10 (x9 x10 x11 x12)) (x3 x4 x6 x10 (x9 x10 x11 x12) x7 = x11)) (x3 x6 x6 x10 (x9 x10 x11 x12) x8 = x3 x6 x10 x10 x12 (x9 x10 x11 x12))) (∀ x13 . x1 x6 x10 x13 ⟶ x3 x4 x6 x10 x13 x7 = x11 ⟶ x3 x6 x6 x10 x13 x8 = x3 x6 x10 x10 x12 x13 ⟶ x13 = x9 x10 x11 x12))
Theorem 9dd4b..^{product_coproduct_Op} : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ι → ι → ι → ι . ∀ x4 x5 x6 x7 x8 . ∀ x9 : ι → ι → ι → ι . MetaCat_product_p x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 ⟶ MetaCat_coproduct_p x0 (λ x10 x11 . x1 x11 x10) x2 (λ x10 x11 x12 x13 x14 . x3 x12 x11 x10 x14 x13) x4 x5 x6 x7 x8 x9 (proof)
Theorem 31c77..^{product_coproduct_constr_Op} : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ι → ι → ι → ι . ∀ x4 x5 x6 : ι → ι → ι . ∀ x7 : ι → ι → ι → ι → ι → ι . MetaCat_product_constr_p x0 x1 x2 x3 x4 x5 x6 x7 ⟶ MetaCat_coproduct_constr_p x0 (λ x8 x9 . x1 x9 x8) x2 (λ x8 x9 x10 x11 x12 . x3 x10 x9 x8 x12 x11) x4 x5 x6 x7 (proof)
Theorem df89e..^{coproduct_product_Op} : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ι → ι → ι → ι . ∀ x4 x5 x6 x7 x8 . ∀ x9 : ι → ι → ι → ι . MetaCat_coproduct_p x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 ⟶ MetaCat_product_p x0 (λ x10 x11 . x1 x11 x10) x2 (λ x10 x11 x12 x13 x14 . x3 x12 x11 x10 x14 x13) x4 x5 x6 x7 x8 x9 (proof)
Theorem 2bad6..^{coproduct_product_constr_Op} : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ι → ι → ι → ι . ∀ x4 x5 x6 : ι → ι → ι . ∀ x7 : ι → ι → ι → ι → ι → ι . MetaCat_coproduct_constr_p x0 x1 x2 x3 x4 x5 x6 x7 ⟶ MetaCat_product_constr_p x0 (λ x8 x9 . x1 x9 x8) x2 (λ x8 x9 x10 x11 x12 . x3 x10 x9 x8 x12 x11) x4 x5 x6 x7 (proof)
Known and7I^and7I : ∀ x0 x1 x2 x3 x4 x5 x6 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ x5 ⟶ x6 ⟶ and (and (and (and (and (and x0 x1) x2) x3) x4) x5) x6
Theorem d9a97.. : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ι → ι → ι → ι . ∀ x4 x5 x6 x7 x8 x9 . ∀ x10 : ι → ι → ι . MetaCat_equalizer_buggy_p x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 ⟶ MetaCat_coequalizer_buggy_p x0 (λ x11 x12 . x1 x12 x11) x2 (λ x11 x12 x13 x14 x15 . x3 x13 x12 x11 x15 x14) x5 x4 x6 x7 x8 x9 x10 (proof)
Theorem bcf11.. : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ι → ι → ι → ι . ∀ x4 x5 : ι → ι → ι → ι → ι . ∀ x6 : ι → ι → ι → ι → ι → ι → ι . MetaCat_equalizer_buggy_struct_p x0 x1 x2 x3 x4 x5 x6 ⟶ MetaCat_coequalizer_buggy_struct_p x0 (λ x7 x8 . x1 x8 x7) x2 (λ x7 x8 x9 x10 x11 . x3 x9 x8 x7 x11 x10) (λ x7 x8 . x4 x8 x7) (λ x7 x8 . x5 x8 x7) (λ x7 x8 . x6 x8 x7) (proof)
Theorem db622.. : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ι → ι → ι → ι . ∀ x4 x5 x6 x7 x8 x9 . ∀ x10 : ι → ι → ι . MetaCat_coequalizer_buggy_p x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 ⟶ MetaCat_equalizer_buggy_p x0 (λ x11 x12 . x1 x12 x11) x2 (λ x11 x12 x13 x14 x15 . x3 x13 x12 x11 x15 x14) x5 x4 x6 x7 x8 x9 x10 (proof)
Theorem e8468.. : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ι → ι → ι → ι . ∀ x4 x5 : ι → ι → ι → ι → ι . ∀ x6 : ι → ι → ι → ι → ι → ι → ι . MetaCat_coequalizer_buggy_struct_p x0 x1 x2 x3 x4 x5 x6 ⟶ MetaCat_equalizer_buggy_struct_p x0 (λ x7 x8 . x1 x8 x7) x2 (λ x7 x8 x9 x10 x11 . x3 x9 x8 x7 x11 x10) (λ x7 x8 . x4 x8 x7) (λ x7 x8 . x5 x8 x7) (λ x7 x8 . x6 x8 x7) (proof)
Theorem 8f137.. : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ι → ι → ι → ι . ∀ x4 x5 x6 x7 x8 x9 x10 x11 . ∀ x12 : ι → ι → ι → ι . MetaCat_pullback_buggy_p x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 ⟶ MetaCat_pushout_buggy_p x0 (λ x13 x14 . x1 x14 x13) x2 (λ x13 x14 x15 x16 x17 . x3 x15 x14 x13 x17 x16) x4 x5 x6 x7 x8 x9 x10 x11 x12 (proof)
Theorem 417a6.. : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι → ο . ∀ x2 : ι → ι . ∀ x3 x4 x5 x6 : ι → ι → ι → ι → ι → ι . ∀ x7 : ι → ι → ι → ι → ι → ι → ι → ι → ι . MetaCat_pullback_buggy_struct_p x0 x1 x2 x3 x4 x5 x6 x7 ⟶ MetaCat_pushout_buggy_constr_p x0 (λ x8 x9 . x1 x9 x8) x2 (λ x8 x9 x10 x11 x12 . x3 x10 x9 x8 x12 x11) x4 x5 x6 x7 (proof)
Theorem 02b99.. : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ι → ι → ι → ι . ∀ x4 x5 x6 x7 x8 x9 x10 x11 . ∀ x12 : ι → ι → ι → ι . MetaCat_pushout_buggy_p x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 ⟶ MetaCat_pullback_buggy_p x0 (λ x13 x14 . x1 x14 x13) x2 (λ x13 x14 x15 x16 x17 . x3 x15 x14 x13 x17 x16) x4 x5 x6 x7 x8 x9 x10 x11 x12 (proof)
Theorem a4533.. : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι → ο . ∀ x2 : ι → ι . ∀ x3 x4 x5 x6 : ι → ι → ι → ι → ι → ι . ∀ x7 : ι → ι → ι → ι → ι → ι → ι → ι → ι . MetaCat_pushout_buggy_constr_p x0 x1 x2 x3 x4 x5 x6 x7 ⟶ MetaCat_pullback_buggy_struct_p x0 (λ x8 x9 . x1 x9 x8) x2 (λ x8 x9 x10 x11 x12 . x3 x10 x9 x8 x12 x11) x4 x5 x6 x7 (proof)
Known and4I^and4I : ∀ x0 x1 x2 x3 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ and (and (and x0 x1) x2) x3
Theorem cf46e.. : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ι → ι → ι → ι . MetaCat x0 x1 x2 x3 ⟶ ∀ x4 x5 : ι → ι → ι → ι → ι . ∀ x6 : ι → ι → ι → ι → ι → ι → ι . MetaCat_equalizer_buggy_struct_p x0 x1 x2 x3 x4 x5 x6 ⟶ ∀ x7 x8 x9 : ι → ι → ι . ∀ x10 : ι → ι → ι → ι → ι → ι . MetaCat_product_constr_p x0 x1 x2 x3 x7 x8 x9 x10 ⟶ MetaCat_pullback_buggy_struct_p x0 x1 x2 x3 (λ x11 x12 x13 x14 x15 . x4 (x7 x11 x12) x13 (x3 (x7 x11 x12) x11 x13 x14 (x8 x11 x12)) (x3 (x7 x11 x12) x12 x13 x15 (x9 x11 x12))) (λ x11 x12 x13 x14 x15 . x3 (x4 (x7 x11 x12) x13 (x3 (x7 x11 x12) x11 x13 x14 (x8 x11 x12)) (x3 (x7 x11 x12) x12 x13 x15 (x9 x11 x12))) (x7 x11 x12) x11 (x8 x11 x12) (x5 (x7 x11 x12) x13 (x3 (x7 x11 x12) x11 x13 x14 (x8 x11 x12)) (x3 (x7 x11 x12) x12 x13 x15 (x9 x11 x12)))) (λ x11 x12 x13 x14 x15 . x3 (x4 (x7 x11 x12) x13 (x3 (x7 x11 x12) x11 x13 x14 (x8 x11 x12)) (x3 (x7 x11 x12) x12 x13 x15 (x9 x11 x12))) (x7 x11 x12) x12 (x9 x11 x12) (x5 (x7 x11 x12) x13 (x3 (x7 x11 x12) x11 x13 x14 (x8 x11 x12)) (x3 (x7 x11 x12) x12 x13 x15 (x9 x11 x12)))) (λ x11 x12 x13 x14 x15 x16 x17 x18 . x6 (x7 x11 x12) x13 (x3 (x7 x11 x12) x11 x13 x14 (x8 x11 x12)) (x3 (x7 x11 x12) x12 x13 x15 (x9 x11 x12)) x16 (x10 x11 x12 x16 x17 x18)) (proof)
Theorem 6f81c.. : ∀ 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_buggy_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_buggy_struct_p x0 x1 x2 x3 x5 x7 x9 x11 ⟶ x10) ⟶ x10) ⟶ x8) ⟶ x8) ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4 (proof)
Theorem f0f28.. : ∀ 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_coequalizer_buggy_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_coproduct_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_pushout_buggy_constr_p x0 x1 x2 x3 x5 x7 x9 x11 ⟶ x10) ⟶ x10) ⟶ x8) ⟶ x8) ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4 (proof)
Param Sep^Sep : ι → (ι → ο) → ι
Definition famunion^famunion := λ x0 . λ x1 : ι → ι . prim3 (prim5 x0 x1)
Param binunion^binunion : ι → ι → ι
Param Inj1^Inj1 : ι → ι
Definition Inj0^Inj0 := λ x0 . prim5 x0 Inj1
Definition setsum^setsum := λ x0 x1 . binunion (prim5 x0 Inj0) (prim5 x1 Inj1)
Definition lam^Sigma := λ x0 . λ x1 : ι → ι . famunion x0 (λ x2 . prim5 (x1 x2) (setsum x2))
Param ap^ap : ι → ι → ι
Definition Pi^Pi := λ x0 . λ x1 : ι → ι . {x2 ∈ prim4 (lam x0 (λ x2 . prim3 (x1 x2)))|∀ x3 . x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3}
Definition setexp^setexp := λ x0 x1 . Pi x1 (λ x2 . x0)
Definition HomSet^SetHom := λ x0 x1 x2 . x2 ∈ setexp x1 x0
Definition True^True := ∀ x0 : ο . x0 ⟶ x0
Conjecture 31571.. : MetaCat (λ x0 . True) HomSet (λ x0 . lam x0 (λ x1 . x1)) (λ x0 x1 x2 x3 x4 . lam x0 (λ x5 . ap x3 (ap x4 x5)))
Conjecture 637da.. : MetaCat (λ x0 . x0 ∈ prim6 0) HomSet (λ x0 . lam x0 (λ x1 . x1)) (λ x0 x1 x2 x3 x4 . lam x0 (λ x5 . ap x3 (ap x4 x5)))
Conjecture 95fe9.. : MetaCat (λ x0 . x0 ∈ prim6 (prim6 0)) HomSet (λ x0 . lam x0 (λ x1 . x1)) (λ x0 x1 x2 x3 x4 . lam x0 (λ x5 . ap x3 (ap x4 x5)))
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 fa807..^{MetaCatSet_initial_gen} : ∀ x0 : ι → ο . x0 0 ⟶ ∀ x1 : ο . (∀ x2 . (∀ x3 : ο . (∀ x4 : ι → ι . MetaCat_initial_p x0 HomSet (λ x5 . lam x5 (λ x6 . x6)) (λ x5 x6 x7 x8 x9 . lam x5 (λ x10 . ap x8 (ap x9 x10))) x2 x4 ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1 (proof)
Known TrueI^TrueI : True
Theorem 80cab..^{MetaCatSet_initial} : ∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . MetaCat_initial_p (λ x4 . True) HomSet (λ x4 . lam x4 (λ x5 . x5)) (λ x4 x5 x6 x7 x8 . lam x4 (λ x9 . ap x7 (ap x8 x9))) x1 x3 ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Param ordsucc^ordsucc : ι → ι
Known In_0_1^In_0_1 : 0 ∈ 1
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 4cd1b..^{MetaCatSet_terminal_gen} : ∀ x0 : ι → ο . x0 1 ⟶ ∀ x1 : ο . (∀ x2 . (∀ x3 : ο . (∀ x4 : ι → ι . MetaCat_terminal_p x0 HomSet (λ x5 . lam x5 (λ x6 . x6)) (λ x5 x6 x7 x8 x9 . lam x5 (λ x10 . ap x8 (ap x9 x10))) x2 x4 ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1 (proof)
Theorem 370ea..^{MetaCatSet_terminal} : ∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . MetaCat_terminal_p (λ x4 . True) HomSet (λ x4 . lam x4 (λ x5 . x5)) (λ x4 x5 x6 x7 x8 . lam x4 (λ x9 . ap x7 (ap x8 x9))) x1 x3 ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Param combine_funcs^{combine_funcs} : ι → ι → (ι → ι) → (ι → 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 Inj0_setsum^Inj0_setsum : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ Inj0 x2 ∈ setsum x0 x1
Known Inj1_setsum^Inj1_setsum : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ Inj1 x2 ∈ setsum x0 x1
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known setsum_Inj_inv^{setsum_Inj_inv} : ∀ x0 x1 x2 . x2 ∈ setsum x0 x1 ⟶ or (∀ x3 : ο . (∀ x4 . and (x4 ∈ x0) (x2 = Inj0 x4) ⟶ x3) ⟶ x3) (∀ x3 : ο . (∀ x4 . and (x4 ∈ x1) (x2 = Inj1 x4) ⟶ x3) ⟶ x3)
Known combine_funcs_eq1^{combine_funcs_eq1} : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 . combine_funcs x0 x1 x2 x3 (Inj0 x4) = x2 x4
Known combine_funcs_eq2^{combine_funcs_eq2} : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 . combine_funcs x0 x1 x2 x3 (Inj1 x4) = x3 x4
Known beta^beta : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ ap (lam x0 x1) x2 = x1 x2
Theorem 3c078..^{MetaCatSet_coproduct_gen} : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ ∀ x2 . x0 x2 ⟶ x0 (setsum x1 x2)) ⟶ ∀ x1 : ο . (∀ x2 : ι → ι → ι . (∀ x3 : ο . (∀ x4 : ι → ι → ι . (∀ x5 : ο . (∀ x6 : ι → ι → ι . (∀ x7 : ο . (∀ x8 : ι → ι → ι → ι → ι → ι . MetaCat_coproduct_constr_p x0 HomSet (λ x9 . lam x9 (λ x10 . x10)) (λ x9 x10 x11 x12 x13 . lam x9 (λ x14 . ap x12 (ap x13 x14))) x2 x4 x6 x8 ⟶ x7) ⟶ x7) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1 (proof)
Theorem c28ea..^{MetaCatSet_coproduct} : ∀ 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_coproduct_constr_p (λ x8 . True) HomSet (λ x8 . lam x8 (λ x9 . x9)) (λ x8 x9 x10 x11 x12 . lam x8 (λ x13 . ap x11 (ap x12 x13))) x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Definition setprod^setprod := λ x0 x1 . lam x0 (λ x2 . x1)
Param If_i^If_i : ο → ι → ι → ι
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)
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 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_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 21e78..^{MetaCatSet_product_gen} : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ ∀ x2 . x0 x2 ⟶ x0 (setprod x1 x2)) ⟶ ∀ x1 : ο . (∀ x2 : ι → ι → ι . (∀ x3 : ο . (∀ x4 : ι → ι → ι . (∀ x5 : ο . (∀ x6 : ι → ι → ι . (∀ x7 : ο . (∀ x8 : ι → ι → ι → ι → ι → ι . MetaCat_product_constr_p x0 HomSet (λ x9 . lam x9 (λ x10 . x10)) (λ x9 x10 x11 x12 x13 . lam x9 (λ x14 . ap x12 (ap x13 x14))) x2 x4 x6 x8 ⟶ x7) ⟶ x7) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1 (proof)
Theorem 14d11..^{MetaCatSet_product} : ∀ 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 (λ x8 . True) HomSet (λ x8 . lam x8 (λ x9 . x9)) (λ x8 x9 x10 x11 x12 . lam x8 (λ x13 . ap x11 (ap x12 x13))) x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param ZF_closed^ZF_closed : ι → ο
Definition TransSet^TransSet := λ x0 . ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Known UnivOf_TransSet^{UnivOf_TransSet} : ∀ x0 . TransSet (prim6 x0)
Known ZF_Power_closed^{ZF_Power_closed} : ∀ x0 . ZF_closed x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ prim4 x1 ∈ x0
Known PowerI^PowerI : ∀ x0 x1 . x1 ⊆ x0 ⟶ x1 ∈ prim4 x0
Known UnivOf_ZF_closed^{UnivOf_ZF_closed} : ∀ x0 . ZF_closed (prim6 x0)
Theorem 6aa34..^{UnivOf_Subq_closed} : ∀ x0 x1 . x1 ∈ prim6 x0 ⟶ ∀ x2 . x2 ⊆ x1 ⟶ x2 ∈ prim6 x0 (proof)
Definition famunion_closed^{famunion_closed} := λ x0 . ∀ x1 . x1 ∈ x0 ⟶ ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x1 ⟶ x2 x3 ∈ x0) ⟶ famunion x1 x2 ∈ x0
Definition Union_closed^Union_closed := λ x0 . ∀ x1 . x1 ∈ x0 ⟶ prim3 x1 ∈ x0
Definition Repl_closed^Repl_closed := λ x0 . ∀ x1 . x1 ∈ x0 ⟶ ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x1 ⟶ x2 x3 ∈ x0) ⟶ prim5 x1 x2 ∈ x0
Theorem Union_Repl_famunion_closed^{Union_Repl_famunion_closed} : ∀ x0 . Union_closed x0 ⟶ Repl_closed x0 ⟶ famunion_closed x0 (proof)
Definition Power_closed^Power_closed := λ x0 . ∀ x1 . x1 ∈ x0 ⟶ prim4 x1 ∈ x0
Known ZF_closed_E^ZF_closed_E : ∀ x0 . ZF_closed x0 ⟶ ∀ x1 : ο . (Union_closed x0 ⟶ Power_closed x0 ⟶ Repl_closed x0 ⟶ x1) ⟶ x1
Known Empty_In_Power^{Empty_In_Power} : ∀ x0 . 0 ∈ prim4 x0
Theorem 1037d..^ZF_closed_0 : ∀ x0 x1 . TransSet x0 ⟶ ZF_closed x0 ⟶ x1 ∈ x0 ⟶ 0 ∈ x0 (proof)
Known In_ind^In_ind : ∀ x0 : ι → ο . (∀ x1 . (∀ x2 . x2 ∈ x1 ⟶ x0 x2) ⟶ x0 x1) ⟶ ∀ x1 . x0 x1
Known Inj1_eq^Inj1_eq : ∀ x0 . Inj1 x0 = binunion (Sing 0) (prim5 x0 Inj1)
Known ZF_binunion_closed^{ZF_binunion_closed} : ∀ x0 . ZF_closed x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ∀ x2 . x2 ∈ x0 ⟶ binunion x1 x2 ∈ x0
Known ZF_Sing_closed^{ZF_Sing_closed} : ∀ x0 . ZF_closed x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ Sing x1 ∈ x0
Known ZF_Repl_closed^{ZF_Repl_closed} : ∀ x0 . ZF_closed x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x1 ⟶ x2 x3 ∈ x0) ⟶ prim5 x1 x2 ∈ x0
Theorem 7eedb..^{ZF_Inj1_closed} : ∀ x0 . TransSet x0 ⟶ ZF_closed x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ Inj1 x1 ∈ x0 (proof)
Theorem 1045a..^{ZF_Inj0_closed} : ∀ x0 . TransSet x0 ⟶ ZF_closed x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ Inj0 x1 ∈ x0 (proof)
Theorem 32cb8..^{ZF_setsum_closed} : ∀ x0 . TransSet x0 ⟶ ZF_closed x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ∀ x2 . x2 ∈ x0 ⟶ setsum x1 x2 ∈ x0 (proof)
Theorem c59b3..^{ZF_Sigma_closed} : ∀ x0 . TransSet x0 ⟶ ZF_closed x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x1 ⟶ x2 x3 ∈ x0) ⟶ lam x1 x2 ∈ x0 (proof)
Theorem ecfb5..^{ZF_setprod_closed} : ∀ x0 . TransSet x0 ⟶ ZF_closed x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ∀ x2 . x2 ∈ x0 ⟶ setprod x1 x2 ∈ x0 (proof)
Known Sep_In_Power^Sep_In_Power : ∀ x0 . ∀ x1 : ι → ο . Sep x0 x1 ∈ prim4 x0
Theorem 43ba5..^ZF_Pi_closed : ∀ x0 . TransSet x0 ⟶ ZF_closed x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x1 ⟶ x2 x3 ∈ x0) ⟶ Pi x1 x2 ∈ x0 (proof)
Theorem 33118..^{ZF_setexp_closed} : ∀ x0 . TransSet x0 ⟶ ZF_closed x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ∀ x2 . x2 ∈ x0 ⟶ setexp x2 x1 ∈ x0 (proof)
Known UnivOf_In^UnivOf_In : ∀ x0 . x0 ∈ prim6 x0
Theorem 6694e..^{MetaCatHFSet_initial} : ∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . MetaCat_initial_p (λ x4 . x4 ∈ prim6 0) HomSet (λ x4 . lam x4 (λ x5 . x5)) (λ x4 x5 x6 x7 x8 . lam x4 (λ x9 . ap x7 (ap x8 x9))) x1 x3 ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Theorem 4e15c..^{MetaCatSmallSet_initial} : ∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . MetaCat_initial_p (λ x4 . x4 ∈ prim6 (prim6 0)) HomSet (λ x4 . lam x4 (λ x5 . x5)) (λ x4 x5 x6 x7 x8 . lam x4 (λ x9 . ap x7 (ap x8 x9))) x1 x3 ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Known ZF_ordsucc_closed^{ZF_ordsucc_closed} : ∀ x0 . ZF_closed x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ∈ x0
Theorem d3646..^{MetaCatHFSet_terminal} : ∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . MetaCat_terminal_p (λ x4 . x4 ∈ prim6 0) HomSet (λ x4 . lam x4 (λ x5 . x5)) (λ x4 x5 x6 x7 x8 . lam x4 (λ x9 . ap x7 (ap x8 x9))) x1 x3 ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Theorem b5da2..^{MetaCatSmallSet_terminal} : ∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . MetaCat_terminal_p (λ x4 . x4 ∈ prim6 (prim6 0)) HomSet (λ x4 . lam x4 (λ x5 . x5)) (λ x4 x5 x6 x7 x8 . lam x4 (λ x9 . ap x7 (ap x8 x9))) x1 x3 ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Theorem 5604f..^{MetaCatHFSet_coproduct} : ∀ 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_coproduct_constr_p (λ x8 . x8 ∈ prim6 0) HomSet (λ x8 . lam x8 (λ x9 . x9)) (λ x8 x9 x10 x11 x12 . lam x8 (λ x13 . ap x11 (ap x12 x13))) x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Theorem 002ee..^{MetaCatSmallSet_coproduct} : ∀ 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_coproduct_constr_p (λ x8 . x8 ∈ prim6 (prim6 0)) HomSet (λ x8 . lam x8 (λ x9 . x9)) (λ x8 x9 x10 x11 x12 . lam x8 (λ x13 . ap x11 (ap x12 x13))) x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Theorem 8ae2a..^{MetaCatHFSet_product} : ∀ 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 (λ x8 . x8 ∈ prim6 0) HomSet (λ x8 . lam x8 (λ x9 . x9)) (λ x8 x9 x10 x11 x12 . lam x8 (λ x13 . ap x11 (ap x12 x13))) x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Theorem 4281b..^{MetaCatSmallSet_product} : ∀ 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 (λ x8 . x8 ∈ prim6 (prim6 0)) HomSet (λ x8 . lam x8 (λ x9 . x9)) (λ x8 x9 x10 x11 x12 . lam x8 (λ x13 . ap x11 (ap x12 x13))) x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Conjecture fec92.. : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ ∀ x2 . x2 ⊆ x1 ⟶ x0 x2) ⟶ ∀ 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_coequalizer_buggy_struct_p x0 HomSet (λ x7 . lam x7 (λ x8 . x8)) (λ x7 x8 x9 x10 x11 . lam x7 (λ x12 . ap x10 (ap x11 x12))) x2 x4 x6 ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1
Conjecture 03db4.. : ∀ 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_coequalizer_buggy_struct_p (λ x6 . True) HomSet (λ x6 . lam x6 (λ x7 . x7)) (λ x6 x7 x8 x9 x10 . lam x6 (λ x11 . ap x9 (ap x10 x11))) x1 x3 x5 ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0
Conjecture 7d4ee.. : ∀ 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_coequalizer_buggy_struct_p (λ x6 . x6 ∈ prim6 0) HomSet (λ x6 . lam x6 (λ x7 . x7)) (λ x6 x7 x8 x9 x10 . lam x6 (λ x11 . ap x9 (ap x10 x11))) x1 x3 x5 ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0
Conjecture e0823.. : ∀ 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_coequalizer_buggy_struct_p (λ x6 . x6 ∈ prim6 (prim6 0)) HomSet (λ x6 . lam x6 (λ x7 . x7)) (λ x6 x7 x8 x9 x10 . lam x6 (λ x11 . ap x9 (ap x10 x11))) x1 x3 x5 ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0
Conjecture 32409.. : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ ∀ x2 . x2 ⊆ x1 ⟶ x0 x2) ⟶ ∀ 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_buggy_struct_p x0 HomSet (λ x7 . lam x7 (λ x8 . x8)) (λ x7 x8 x9 x10 x11 . lam x7 (λ x12 . ap x10 (ap x11 x12))) x2 x4 x6 ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1
Conjecture 2c602.. : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ ∀ x2 . x2 ⊆ x1 ⟶ x0 x2) ⟶ ∀ 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_buggy_struct_p (λ x7 . x7 ∈ prim6 0) HomSet (λ x7 . lam x7 (λ x8 . x8)) (λ x7 x8 x9 x10 x11 . lam x7 (λ x12 . ap x10 (ap x11 x12))) x2 x4 x6 ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1
Conjecture 6fa47.. : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ ∀ x2 . x2 ⊆ x1 ⟶ x0 x2) ⟶ ∀ 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_buggy_struct_p (λ x7 . x7 ∈ prim6 (prim6 0)) HomSet (λ x7 . lam x7 (λ x8 . x8)) (λ x7 x8 x9 x10 x11 . lam x7 (λ x12 . ap x10 (ap x11 x12))) x2 x4 x6 ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1
Conjecture 9048a.. : ∀ x0 : ι → ο . MetaCat x0 HomSet (λ x1 . lam x1 (λ x2 . x2)) (λ x1 x2 x3 x4 x5 . lam x1 (λ x6 . ap x4 (ap x5 x6))) ⟶ (∀ x1 . x0 x1 ⟶ ∀ x2 . x2 ⊆ x1 ⟶ x0 x2) ⟶ (∀ x1 . x0 x1 ⟶ ∀ x2 . x0 x2 ⟶ x0 (setprod x1 x2)) ⟶ ∀ x1 : ο . (∀ x2 : ι → ι → ι → ι → ι → ι . (∀ x3 : ο . (∀ x4 : ι → ι → ι → ι → ι → ι . (∀ x5 : ο . (∀ x6 : ι → ι → ι → ι → ι → ι . (∀ x7 : ο . (∀ x8 : ι → ι → ι → ι → ι → ι → ι → ι → ι . MetaCat_pullback_buggy_struct_p x0 HomSet (λ x9 . lam x9 (λ x10 . x10)) (λ x9 x10 x11 x12 x13 . lam x9 (λ x14 . ap x12 (ap x13 x14))) x2 x4 x6 x8 ⟶ x7) ⟶ x7) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1
Conjecture e4971.. : ∀ 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_buggy_struct_p (λ x8 . True) HomSet (λ x8 . lam x8 (λ x9 . x9)) (λ x8 x9 x10 x11 x12 . lam x8 (λ x13 . ap x11 (ap x12 x13))) x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0
Conjecture f7865.. : ∀ 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_buggy_struct_p (λ x8 . x8 ∈ prim6 0) HomSet (λ x8 . lam x8 (λ x9 . x9)) (λ x8 x9 x10 x11 x12 . lam x8 (λ x13 . ap x11 (ap x12 x13))) x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0
Conjecture 27032.. : ∀ 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_buggy_struct_p (λ x8 . x8 ∈ prim6 (prim6 0)) HomSet (λ x8 . lam x8 (λ x9 . x9)) (λ x8 x9 x10 x11 x12 . lam x8 (λ x13 . ap x11 (ap x12 x13))) x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0
Conjecture 72fd2.. : ∀ x0 : ι → ο . MetaCat x0 HomSet (λ x1 . lam x1 (λ x2 . x2)) (λ x1 x2 x3 x4 x5 . lam x1 (λ x6 . ap x4 (ap x5 x6))) ⟶ (∀ x1 . x0 x1 ⟶ ∀ x2 . x2 ⊆ x1 ⟶ x0 x2) ⟶ (∀ x1 . x0 x1 ⟶ ∀ x2 . x0 x2 ⟶ x0 (setsum x1 x2)) ⟶ ∀ x1 : ο . (∀ x2 : ι → ι → ι → ι → ι → ι . (∀ x3 : ο . (∀ x4 : ι → ι → ι → ι → ι → ι . (∀ x5 : ο . (∀ x6 : ι → ι → ι → ι → ι → ι . (∀ x7 : ο . (∀ x8 : ι → ι → ι → ι → ι → ι → ι → ι → ι . MetaCat_pushout_buggy_constr_p x0 HomSet (λ x9 . lam x9 (λ x10 . x10)) (λ x9 x10 x11 x12 x13 . lam x9 (λ x14 . ap x12 (ap x13 x14))) x2 x4 x6 x8 ⟶ x7) ⟶ x7) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1
Conjecture 6cf2e.. : ∀ 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_pushout_buggy_constr_p (λ x8 . True) HomSet (λ x8 . lam x8 (λ x9 . x9)) (λ x8 x9 x10 x11 x12 . lam x8 (λ x13 . ap x11 (ap x12 x13))) x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0
Conjecture f5042.. : ∀ 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_pushout_buggy_constr_p (λ x8 . x8 ∈ prim6 0) HomSet (λ x8 . lam x8 (λ x9 . x9)) (λ x8 x9 x10 x11 x12 . lam x8 (λ x13 . ap x11 (ap x12 x13))) x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0
Conjecture 3cfce.. : ∀ 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_pushout_buggy_constr_p (λ x8 . x8 ∈ prim6 (prim6 0)) HomSet (λ x8 . lam x8 (λ x9 . x9)) (λ x8 x9 x10 x11 x12 . lam x8 (λ x13 . ap x11 (ap x12 x13))) x1 x3 x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0
Conjecture 4c8b8.. : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ ∀ x2 . x0 x2 ⟶ x0 (setprod x1 x2)) ⟶ (∀ x1 . x0 x1 ⟶ ∀ x2 . x0 x2 ⟶ x0 (setexp x2 x1)) ⟶ MetaCat_exp_constr_p x0 HomSet (λ x1 . lam x1 (λ x2 . x2)) (λ x1 x2 x3 x4 x5 . lam x1 (λ x6 . ap x4 (ap x5 x6))) setprod (λ x1 x2 . lam (setprod x1 x2) (λ x3 . ap x3 0)) (λ x1 x2 . lam (setprod x1 x2) (λ x3 . ap x3 1)) (λ x1 x2 x3 x4 x5 . lam x3 (λ x6 . lam 2 (λ x7 . If_i (x7 = 0) (ap x4 x6) (ap x5 x6)))) (λ x1 x2 . setexp x2 x1) (λ x1 x2 . lam (setprod (setexp x2 x1) x1) (λ x3 . ap (ap x3 0) (ap x3 1))) (λ x1 x2 x3 x4 . lam x3 (λ x5 . lam x1 (λ x6 . ap x4 (lam 2 (λ x7 . If_i (x7 = 0) x5 x6)))))
Conjecture e6b46.. : ∀ x0 : ι → ο . (∀ x1 . x0 x1 ⟶ ∀ x2 . x0 x2 ⟶ x0 (setprod x1 x2)) ⟶ (∀ x1 . x0 x1 ⟶ ∀ x2 . x0 x2 ⟶ x0 (setexp x2 x1)) ⟶ ∀ x1 : ο . (∀ x2 : ι → ι → ι . (∀ x3 : ο . (∀ x4 : ι → ι → ι . (∀ x5 : ο . (∀ x6 : ι → ι → ι . (∀ x7 : ο . (∀ x8 : ι → ι → ι → ι → ι → ι . (∀ x9 : ο . (∀ x10 : ι → ι → ι . (∀ x11 : ο . (∀ x12 : ι → ι → ι . (∀ x13 : ο . (∀ x14 : ι → ι → ι → ι → ι . MetaCat_exp_constr_p x0 HomSet (λ x15 . lam x15 (λ x16 . x16)) (λ x15 x16 x17 x18 x19 . lam x15 (λ x20 . ap x18 (ap x19 x20))) x2 x4 x6 x8 x10 x12 x14 ⟶ x13) ⟶ x13) ⟶ x11) ⟶ x11) ⟶ x9) ⟶ x9) ⟶ x7) ⟶ x7) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1
Conjecture 7a848.. : ∀ 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 (λ x14 . True) HomSet (λ x14 . lam x14 (λ x15 . x15)) (λ x14 x15 x16 x17 x18 . lam x14 (λ x19 . ap x17 (ap x18 x19))) x1 x3 x5 x7 x9 x11 x13 ⟶ x12) ⟶ x12) ⟶ x10) ⟶ x10) ⟶ x8) ⟶ x8) ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0
Conjecture b673c.. : ∀ 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 (λ x14 . x14 ∈ prim6 0) HomSet (λ x14 . lam x14 (λ x15 . x15)) (λ x14 x15 x16 x17 x18 . lam x14 (λ x19 . ap x17 (ap x18 x19))) x1 x3 x5 x7 x9 x11 x13 ⟶ x12) ⟶ x12) ⟶ x10) ⟶ x10) ⟶ x8) ⟶ x8) ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0
Conjecture 9c381.. : ∀ x0 : ι → ο . ∀ x1 x2 x3 . MetaCat_monic_p x0 HomSet (λ x4 . lam x4 (λ x5 . x5)) (λ x4 x5 x6 x7 x8 . lam x4 (λ x9 . ap x7 (ap x8 x9))) x1 x2 x3 ⟶ ∀ x4 . x4 ∈ x1 ⟶ ∀ x5 . x5 ∈ x1 ⟶ ap x3 x4 = ap x3 x5 ⟶ x4 = x5
Param inv^inv : ι → (ι → ι) → ι → ι
Conjecture f0ae4.. : ∀ x0 : ι → ο . x0 1 ⟶ x0 2 ⟶ MetaCat_subobject_classifier_buggy_p x0 HomSet (λ x1 . lam x1 (λ x2 . x2)) (λ x1 x2 x3 x4 x5 . lam x1 (λ x6 . ap x4 (ap x5 x6))) 1 (λ x1 . lam x1 (λ x2 . 0)) 2 (lam 1 (λ x1 . 1)) (λ x1 x2 x3 . lam x2 (λ x4 . If_i (∀ x5 : ο . (∀ x6 . and (x6 ∈ x1) (ap x3 x6 = x4) ⟶ x5) ⟶ x5) 1 0)) (λ x1 x2 x3 x4 x5 x6 . lam x4 (λ x7 . inv x1 (ap x3) (ap x6 x7)))
Conjecture d9711.. : ∀ x0 : ι → ο . x0 1 ⟶ x0 2 ⟶ ∀ x1 : ο . (∀ x2 . (∀ x3 : ο . (∀ x4 : ι → ι . (∀ x5 : ο . (∀ x6 . (∀ x7 : ο . (∀ x8 . (∀ x9 : ο . (∀ x10 : ι → ι → ι → ι . (∀ x11 : ο . (∀ x12 : ι → ι → ι → ι → ι → ι → ι . MetaCat_subobject_classifier_buggy_p x0 HomSet (λ x13 . lam x13 (λ x14 . x14)) (λ x13 x14 x15 x16 x17 . lam x13 (λ x18 . ap x16 (ap x17 x18))) x2 x4 x6 x8 x10 x12 ⟶ x11) ⟶ x11) ⟶ x9) ⟶ x9) ⟶ x7) ⟶ x7) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1
Conjecture 0f6a0.. : ∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . (∀ x4 : ο . (∀ x5 . (∀ x6 : ο . (∀ x7 . (∀ x8 : ο . (∀ x9 : ι → ι → ι → ι . (∀ x10 : ο . (∀ x11 : ι → ι → ι → ι → ι → ι → ι . MetaCat_subobject_classifier_buggy_p (λ x12 . True) HomSet (λ x12 . lam x12 (λ x13 . x13)) (λ x12 x13 x14 x15 x16 . lam x12 (λ x17 . ap x15 (ap x16 x17))) x1 x3 x5 x7 x9 x11 ⟶ x10) ⟶ x10) ⟶ x8) ⟶ x8) ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0
Conjecture 67463.. : ∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . (∀ x4 : ο . (∀ x5 . (∀ x6 : ο . (∀ x7 . (∀ x8 : ο . (∀ x9 : ι → ι → ι → ι . (∀ x10 : ο . (∀ x11 : ι → ι → ι → ι → ι → ι → ι . MetaCat_subobject_classifier_buggy_p (λ x12 . x12 ∈ prim6 0) HomSet (λ x12 . lam x12 (λ x13 . x13)) (λ x12 x13 x14 x15 x16 . lam x12 (λ x17 . ap x15 (ap x16 x17))) x1 x3 x5 x7 x9 x11 ⟶ x10) ⟶ x10) ⟶ x8) ⟶ x8) ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0
Conjecture 39f19.. : ∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . (∀ x4 : ο . (∀ x5 . (∀ x6 : ο . (∀ x7 . (∀ x8 : ο . (∀ x9 : ι → ι → ι → ι . (∀ x10 : ο . (∀ x11 : ι → ι → ι → ι → ι → ι → ι . MetaCat_subobject_classifier_buggy_p (λ x12 . x12 ∈ prim6 (prim6 0)) HomSet (λ x12 . lam x12 (λ x13 . x13)) (λ x12 x13 x14 x15 x16 . lam x12 (λ x17 . ap x15 (ap x16 x17))) x1 x3 x5 x7 x9 x11 ⟶ x10) ⟶ x10) ⟶ x8) ⟶ x8) ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0
Param omega^omega : ι
Param nat_primrec^nat_primrec : ι → (ι → ι → ι) → ι → ι
Conjecture e35b8.. : ∀ x0 : ι → ο . x0 1 ⟶ x0 omega ⟶ MetaCat_nno_p x0 HomSet (λ x1 . lam x1 (λ x2 . x2)) (λ x1 x2 x3 x4 x5 . lam x1 (λ x6 . ap x4 (ap x5 x6))) 1 (λ x1 . lam x1 (λ x2 . 0)) omega (lam 1 (λ x1 . 0)) (lam omega ordsucc) (λ x1 x2 x3 . lam omega (nat_primrec (ap x2 0) (λ x4 . ap x3)))
Conjecture 492d5.. : ∀ x0 : ι → ο . x0 1 ⟶ x0 omega ⟶ ∀ x1 : ο . (∀ x2 . (∀ x3 : ο . (∀ x4 : ι → ι . (∀ x5 : ο . (∀ x6 . (∀ x7 : ο . (∀ x8 . (∀ x9 : ο . (∀ x10 . (∀ x11 : ο . (∀ x12 : ι → ι → ι → ι . MetaCat_nno_p x0 HomSet (λ x13 . lam x13 (λ x14 . x14)) (λ x13 x14 x15 x16 x17 . lam x13 (λ x18 . ap x16 (ap x17 x18))) x2 x4 x6 x8 x10 x12 ⟶ x11) ⟶ x11) ⟶ x9) ⟶ x9) ⟶ x7) ⟶ x7) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1
Conjecture ed04c.. : ∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . (∀ x4 : ο . (∀ x5 . (∀ x6 : ο . (∀ x7 . (∀ x8 : ο . (∀ x9 . (∀ x10 : ο . (∀ x11 : ι → ι → ι → ι . MetaCat_nno_p (λ x12 . True) HomSet (λ x12 . lam x12 (λ x13 . x13)) (λ x12 x13 x14 x15 x16 . lam x12 (λ x17 . ap x15 (ap x16 x17))) x1 x3 x5 x7 x9 x11 ⟶ x10) ⟶ x10) ⟶ x8) ⟶ x8) ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0
Conjecture 1dac4.. : ∀ x0 : ο . (∀ x1 . (∀ x2 : ο . (∀ x3 : ι → ι . (∀ x4 : ο . (∀ x5 . (∀ x6 : ο . (∀ x7 . (∀ x8 : ο . (∀ x9 . (∀ x10 : ο . (∀ x11 : ι → ι → ι → ι . MetaCat_nno_p (λ x12 . x12 ∈ prim6 (prim6 0)) HomSet (λ x12 . lam x12 (λ x13 . x13)) (λ x12 x13 x14 x15 x16 . lam x12 (λ x17 . ap x15 (ap x16 x17))) x1 x3 x5 x7 x9 x11 ⟶ x10) ⟶ x10) ⟶ x8) ⟶ x8) ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0