Proofgold Address

Param nat_primrec^nat_primrec : ι → (ι → ι → ι) → ι → ι
Definition 1319b.. := λ x0 . λ x1 : ι → ι . λ x2 . nat_primrec x2 (λ x3 . x1) x0
Known nat_primrec_0^{nat_primrec_0} : ∀ x0 . ∀ x1 : ι → ι → ι . nat_primrec x0 x1 0 = x0
Theorem 76108.. : ∀ x0 : ι → ι . ∀ x1 . 1319b.. 0 x0 x1 = x1 (proof)
Param omega^omega : ι
Param ordsucc^ordsucc : ι → ι
Param nat_p^nat_p : ι → ο
Known nat_primrec_S^{nat_primrec_S} : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . nat_p x2 ⟶ nat_primrec x0 x1 (ordsucc x2) = x1 x2 (nat_primrec x0 x1 x2)
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Theorem 99a2b.. : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 : ι → ι . ∀ x2 . 1319b.. (ordsucc x0) x1 x2 = x1 (1319b.. x0 x1 x2) (proof)
Known nat_ind^nat_ind : ∀ x0 : ι → ο . x0 0 ⟶ (∀ x1 . nat_p x1 ⟶ x0 x1 ⟶ x0 (ordsucc x1)) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Theorem acd4e.. : ∀ x0 x1 . x1 ∈ x0 ⟶ ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x0) ⟶ ∀ x3 . nat_p x3 ⟶ 1319b.. x3 x2 x1 ∈ x0 (proof)
Theorem c6fca.. : ∀ x0 x1 . x1 ∈ x0 ⟶ ∀ x2 x3 : ι → ι . (∀ x4 . x4 ∈ x0 ⟶ x2 x4 ∈ x0) ⟶ (∀ x4 . x4 ∈ x0 ⟶ x2 x4 = x3 x4) ⟶ ∀ x4 . nat_p x4 ⟶ 1319b.. x4 x2 x1 = 1319b.. x4 x3 x1 (proof)
Theorem 9bb87.. : ∀ x0 x1 . x1 ∈ x0 ⟶ ∀ x2 x3 : ι → ι . (∀ x4 . x4 ∈ x0 ⟶ x2 x4 ∈ x0) ⟶ ∀ x4 : ι → ι . (∀ x5 . x5 ∈ x0 ⟶ x4 (x2 x5) = x3 (x4 x5)) ⟶ ∀ x5 . nat_p x5 ⟶ x4 (1319b.. x5 x2 x1) = 1319b.. x5 x3 (x4 x1) (proof)
Param lam^Sigma : ι → (ι → ι) → ι
Param If_i^If_i : ο → ι → ι → ι
Param ap^ap : ι → ι → ι
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
Theorem 60ada.. : ∀ x0 x1 . nat_p x1 ⟶ 1319b.. x1 (λ x3 . lam 2 (λ x4 . If_i (x4 = 0) (ordsucc (ap x3 0)) (ap x3 1))) (lam 2 (λ x3 . If_i (x3 = 0) 0 x0)) = lam 2 (λ x3 . If_i (x3 = 0) x1 x0) (proof)
Param add_SNo^add_SNo : ι → ι → ι
Param minus_SNo^minus_SNo : ι → ι
Known minus_SNo_0^minus_SNo_0 : minus_SNo 0 = 0
Param ordinal^ordinal : ι → ο
Known ordinal_ordsucc_SNo_eq^{ordinal_ordsucc_SNo_eq} : ∀ x0 . ordinal x0 ⟶ ordsucc x0 = add_SNo 1 x0
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Param SNo^SNo : ι → ο
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Known SNo_1^SNo_1 : SNo 1
Known minus_add_SNo_distr^{minus_add_SNo_distr} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ minus_SNo (add_SNo x0 x1) = add_SNo (minus_SNo x0) (minus_SNo x1)
Known omega_SNo^omega_SNo : ∀ x0 . x0 ∈ omega ⟶ SNo x0
Theorem 7c549.. : ∀ x0 x1 . nat_p x1 ⟶ 1319b.. x1 (λ x3 . lam 2 (λ x4 . If_i (x4 = 0) (add_SNo (ap x3 0) (minus_SNo 1)) (ap x3 1))) (lam 2 (λ x3 . If_i (x3 = 0) 0 x0)) = lam 2 (λ x3 . If_i (x3 = 0) (minus_SNo x1) x0) (proof)
Param pack_u^pack_u : ι → (ι → ι) → ι
Definition setprod^setprod := λ x0 x1 . lam x0 (λ x2 . x1)
Definition 340cb.. := λ x0 . pack_u (setprod omega x0) (λ x1 . lam 2 (λ x2 . If_i (x2 = 0) (ordsucc (ap x1 0)) (ap x1 1)))
Definition 9132f.. := λ x0 x1 x2 . lam (setprod omega x0) (λ x3 . lam 2 (λ x4 . If_i (x4 = 0) (ap x3 0) (ap x2 (ap x3 1))))
Definition 139e8.. := λ x0 . lam x0 (λ x1 . lam 2 (λ x2 . If_i (x2 = 0) 0 x1))
Param unpack_u_i^unpack_u_i : ι → (ι → (ι → ι) → ι) → ι
Definition 82409.. := λ x0 . unpack_u_i x0 (λ x1 . λ x2 : ι → ι . lam (setprod omega x1) (λ x3 . 1319b.. (ap x3 0) x2 (ap x3 1)))
Param MetaAdjunction_strict^{MetaAdjunction_strict} : (ι → ο) → (ι → ι → ι → ο) → (ι → ι) → (ι → ι → ι → ι → ι → ι) → (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) → ο
Param True^True : ο
Param Pi^Pi : ι → (ι → ι) → ι
Definition setexp^setexp := λ x0 x1 . Pi x1 (λ x2 . x0)
Definition HomSet^SetHom := λ x0 x1 x2 . x2 ∈ setexp x1 x0
Definition lam_id^lam_id := λ x0 . lam x0 (λ x1 . x1)
Definition lam_comp^lam_comp := λ x0 x1 x2 . lam x0 (λ x3 . ap x1 (ap x2 x3))
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
Param UnaryFuncHom^Hom_struct_u : ι → ι → ι → ο
Definition struct_id^struct_id := λ x0 . lam_id (ap x0 0)
Definition struct_comp^struct_comp := λ x0 x1 x2 . lam_comp (ap x0 0)
Param MetaFunctor_strict^{MetaFunctor_strict} : (ι → ο) → (ι → ι → ι → ο) → (ι → ι) → (ι → ι → ι → ι → ι → ι) → (ι → ο) → (ι → ι → ι → ο) → (ι → ι) → (ι → ι → ι → ι → ι → ι) → (ι → ι) → (ι → ι → ι → ι) → ο
Param MetaFunctor^MetaFunctor : (ι → ο) → (ι → ι → ι → ο) → (ι → ι) → (ι → ι → ι → ι → ι → ι) → (ι → ο) → (ι → ι → ι → ο) → (ι → ι) → (ι → ι → ι → ι → ι → ι) → (ι → ι) → (ι → ι → ι → ι) → ο
Param MetaNatTrans^MetaNatTrans : (ι → ο) → (ι → ι → ι → ο) → (ι → ι) → (ι → ι → ι → ι → ι → ι) → (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) → ο
Param MetaAdjunction^{MetaAdjunction} : (ι → ο) → (ι → ι → ι → ο) → (ι → ι) → (ι → ι → ι → ι → ι → ι) → (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) → ο
Known d6aa5..^{MetaAdjunction_strict_I} : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ι → ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 : ι → ι → ι → ο . ∀ x6 : ι → ι . ∀ x7 : ι → ι → ι → ι → ι → ι . ∀ x8 : ι → ι . ∀ x9 : ι → ι → ι → ι . ∀ x10 : ι → ι . ∀ x11 : ι → ι → ι → ι . ∀ x12 x13 : ι → ι . MetaFunctor_strict x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 ⟶ MetaFunctor x4 x5 x6 x7 x0 x1 x2 x3 x10 x11 ⟶ MetaNatTrans x0 x1 x2 x3 x0 x1 x2 x3 (λ x14 . x14) (λ x14 x15 x16 . x16) (λ x14 . x10 (x8 x14)) (λ x14 x15 x16 . x11 (x8 x14) (x8 x15) (x9 x14 x15 x16)) x12 ⟶ MetaNatTrans x4 x5 x6 x7 x4 x5 x6 x7 (λ x14 . x8 (x10 x14)) (λ x14 x15 x16 . x9 (x10 x14) (x10 x15) (x11 x14 x15 x16)) (λ x14 . x14) (λ x14 x15 x16 . x16) x13 ⟶ MetaAdjunction x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 ⟶ MetaAdjunction_strict x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13
Param MetaCat^MetaCat : (ι → ο) → (ι → ι → ι → ο) → (ι → ι) → (ι → ι → ι → ι → ι → ι) → ο
Known 5cbb4..^{MetaFunctor_strict_I} : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ι → ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 : ι → ι → ι → ο . ∀ x6 : ι → ι . ∀ x7 : ι → ι → ι → ι → ι → ι . ∀ x8 : ι → ι . ∀ x9 : ι → ι → ι → ι . MetaCat x0 x1 x2 x3 ⟶ MetaCat x4 x5 x6 x7 ⟶ MetaFunctor x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 ⟶ MetaFunctor_strict x0 x1 x2 x3 x4 x5 x6 x7 x8 x9
Known e4125..^MetaCatSet : MetaCat (λ x0 . True) HomSet lam_id (λ x0 x1 x2 . lam_comp x0)
Known 73eab..^{MetaCat_struct_u} : MetaCat struct_u UnaryFuncHom struct_id struct_comp
Known 2cb62..^MetaFunctorI : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ι → ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 : ι → ι → ι → ο . ∀ x6 : ι → ι . ∀ x7 : ι → ι → ι → ι → ι → ι . ∀ x8 : ι → ι . ∀ x9 : ι → ι → ι → ι . (∀ x10 . x0 x10 ⟶ x4 (x8 x10)) ⟶ (∀ x10 x11 x12 . x0 x10 ⟶ x0 x11 ⟶ x1 x10 x11 x12 ⟶ x5 (x8 x10) (x8 x11) (x9 x10 x11 x12)) ⟶ (∀ x10 . x0 x10 ⟶ x9 x10 x10 (x2 x10) = x6 (x8 x10)) ⟶ (∀ x10 x11 x12 x13 x14 . x0 x10 ⟶ x0 x11 ⟶ x0 x12 ⟶ x1 x10 x11 x13 ⟶ x1 x11 x12 x14 ⟶ x9 x10 x12 (x3 x10 x11 x12 x14 x13) = x7 (x8 x10) (x8 x11) (x8 x12) (x9 x11 x12 x14) (x9 x10 x11 x13)) ⟶ MetaFunctor x0 x1 x2 x3 x4 x5 x6 x7 x8 x9
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)
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
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 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 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
Known ap0_Sigma^ap0_Sigma : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ ap x2 0 ∈ x0
Known ap_Pi^ap_Pi : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ Pi x0 x1 ⟶ x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3
Known ap1_Sigma^ap1_Sigma : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ ap x2 1 ∈ x1 (ap x2 0)
Known beta^beta : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ ap (lam x0 x1) x2 = x1 x2
Known pack_u_0_eq2^pack_u_0_eq2 : ∀ x0 . ∀ x1 : ι → ι . x0 = ap (pack_u x0 x1) 0
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 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 0032d..^{MetaCat_struct_u_Forgetful} : MetaFunctor struct_u UnaryFuncHom struct_id struct_comp (λ x0 . True) HomSet lam_id (λ x0 x1 x2 . lam_comp x0) (λ x0 . ap x0 0) (λ x0 x1 x2 . x2)
Known c1d68..^{MetaNatTransI} : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ι → ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 : ι → ι → ι → ο . ∀ x6 : ι → ι . ∀ x7 : ι → ι → ι → ι → ι → ι . ∀ x8 : ι → ι . ∀ x9 : ι → ι → ι → ι . ∀ x10 : ι → ι . ∀ x11 : ι → ι → ι → ι . ∀ x12 : ι → ι . (∀ x13 . x0 x13 ⟶ x5 (x8 x13) (x10 x13) (x12 x13)) ⟶ (∀ x13 x14 x15 . x0 x13 ⟶ x0 x14 ⟶ x1 x13 x14 x15 ⟶ x7 (x8 x13) (x10 x13) (x10 x14) (x11 x13 x14 x15) (x12 x13) = x7 (x8 x13) (x8 x14) (x10 x14) (x12 x14) (x9 x13 x14 x15)) ⟶ MetaNatTrans x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12
Known nat_0^nat_0 : nat_p 0
Known omega_ordsucc^{omega_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ ordsucc x0 ∈ omega
Known fd494..^{MetaAdjunctionI} : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι → ο . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ι → ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 : ι → ι → ι → ο . ∀ x6 : ι → ι . ∀ x7 : ι → ι → ι → ι → ι → ι . ∀ x8 : ι → ι . ∀ x9 : ι → ι → ι → ι . ∀ x10 : ι → ι . ∀ x11 : ι → ι → ι → ι . ∀ x12 x13 : ι → ι . (∀ x14 . x0 x14 ⟶ x7 (x8 x14) (x8 (x10 (x8 x14))) (x8 x14) (x13 (x8 x14)) (x9 x14 (x10 (x8 x14)) (x12 x14)) = x6 (x8 x14)) ⟶ (∀ x14 . x4 x14 ⟶ x3 (x10 x14) (x10 (x8 (x10 x14))) (x10 x14) (x11 (x8 (x10 x14)) x14 (x13 x14)) (x12 (x10 x14)) = x2 (x10 x14)) ⟶ MetaAdjunction x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13
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
Theorem 6722a.. : MetaAdjunction_strict (λ x0 . True) HomSet lam_id (λ x0 x1 x2 . lam_comp x0) struct_u UnaryFuncHom struct_id struct_comp 340cb.. 9132f.. (λ x0 . ap x0 0) (λ x0 x1 x2 . x2) 139e8.. 82409.. (proof)
Theorem ed3d2..^{MetaCat_struct_u_left_adjoint_forgetful} : ∀ x0 : ο . (∀ x1 : ι → ι . (∀ x2 : ο . (∀ x3 : ι → ι → ι → ι . (∀ x4 : ο . (∀ x5 : ι → ι . (∀ x6 : ο . (∀ x7 : ι → ι . MetaAdjunction_strict (λ x8 . True) HomSet lam_id (λ x8 x9 x10 . lam_comp x8) struct_u UnaryFuncHom struct_id struct_comp x1 x3 (λ x8 . ap x8 0) (λ x8 x9 x10 . x10) x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)
Param unpack_u_o^unpack_u_o : ι → (ι → (ι → ι) → ο) → ο
Definition inj^inj := λ x0 x1 . λ x2 : ι → ι . and (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1) (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4)
Definition SelfInjection^struct_u_inj := λ x0 . and (struct_u x0) (unpack_u_o x0 (λ x1 . inj x1 x1))
Known 6bcb5..^{MetaCat_struct_u_inj} : MetaCat SelfInjection UnaryFuncHom struct_id struct_comp
Known unpack_u_o_eq^{unpack_u_o_eq} : ∀ x0 : ι → (ι → ι) → ο . ∀ x1 . ∀ x2 : ι → ι . (∀ x3 : ι → ι . (∀ x4 . x4 ∈ x1 ⟶ x2 x4 = x3 x4) ⟶ x0 x1 x3 = x0 x1 x2) ⟶ unpack_u_o (pack_u x1 x2) x0 = x0 x1 x2
Known tuple_2_inj^tuple_2_inj : ∀ x0 x1 x2 x3 . lam 2 (λ x5 . If_i (x5 = 0) x0 x1) = lam 2 (λ x5 . If_i (x5 = 0) x2 x3) ⟶ and (x0 = x2) (x1 = x3)
Known ordsucc_inj^ordsucc_inj : ∀ x0 x1 . ordsucc x0 = ordsucc x1 ⟶ x0 = x1
Known 1687d..^{MetaCat_struct_u_inj_Forgetful} : MetaFunctor SelfInjection UnaryFuncHom struct_id struct_comp (λ x0 . True) HomSet lam_id (λ x0 x1 x2 . lam_comp x0) (λ x0 . ap x0 0) (λ x0 x1 x2 . x2)
Known prop_ext_2^prop_ext_2 : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ x0 = x1
Theorem 68f23.. : MetaAdjunction_strict (λ x0 . True) HomSet lam_id (λ x0 x1 x2 . lam_comp x0) SelfInjection UnaryFuncHom struct_id struct_comp 340cb.. 9132f.. (λ x0 . ap x0 0) (λ x0 x1 x2 . x2) 139e8.. 82409.. (proof)
Theorem 8822b..^{MetaCat_struct_u_inj_left_adjoint_forgetful} : ∀ x0 : ο . (∀ x1 : ι → ι . (∀ x2 : ο . (∀ x3 : ι → ι → ι → ι . (∀ x4 : ο . (∀ x5 : ι → ι . (∀ x6 : ο . (∀ x7 : ι → ι . MetaAdjunction_strict (λ x8 . True) HomSet lam_id (λ x8 x9 x10 . lam_comp x8) SelfInjection UnaryFuncHom struct_id struct_comp x1 x3 (λ x8 . ap x8 0) (λ x8 x9 x10 . x10) x5 x7 ⟶ x6) ⟶ x6) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2) ⟶ x0) ⟶ x0 (proof)