Proofgold Address

Param lam^Sigma : ι → (ι → ι) → ι
Param ordsucc^ordsucc : ι → ι
Param If_i^If_i : ο → ι → ι → ι
Param encode_r^encode_r : ι → (ι → ι → ο) → ι
Definition pack_u_r := λ x0 . λ x1 : ι → ι . λ x2 : ι → ι → ο . lam 3 (λ x3 . If_i (x3 = 0) x0 (If_i (x3 = 1) (lam x0 x1) (encode_r x0 x2)))
Param ap^ap : ι → ι → ι
Known tuple_3_0_eq^tuple_3_0_eq : ∀ x0 x1 x2 . ap (lam 3 (λ x4 . If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2))) 0 = x0
Theorem pack_u_r_0_eq : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . x0 = pack_u_r x1 x2 x3 ⟶ x1 = ap x0 0
Theorem pack_u_r_0_eq2 : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 : ι → ι → ο . x3 x0 (ap (pack_u_r x0 x1 x2) 0) ⟶ x3 (ap (pack_u_r x0 x1 x2) 0) x0
Known tuple_3_1_eq^tuple_3_1_eq : ∀ x0 x1 x2 . ap (lam 3 (λ x4 . If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2))) 1 = x1
Known beta^beta : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ ap (lam x0 x1) x2 = x1 x2
Theorem pack_u_r_1_eq : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . x0 = pack_u_r x1 x2 x3 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x4 = ap (ap x0 1) x4
Theorem pack_u_r_1_eq2 : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 . x3 ∈ x0 ⟶ x1 x3 = ap (ap (pack_u_r x0 x1 x2) 1) x3
Param decode_r^decode_r : ι → ι → ι → ο
Known tuple_3_2_eq^tuple_3_2_eq : ∀ x0 x1 x2 . ap (lam 3 (λ x4 . If_i (x4 = 0) x0 (If_i (x4 = 1) x1 x2))) 2 = x2
Known decode_encode_r^{decode_encode_r} : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ decode_r (encode_r x0 x1) x2 x3 = x1 x2 x3
Theorem pack_u_r_2_eq : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . x0 = pack_u_r x1 x2 x3 ⟶ ∀ x4 . x4 ∈ x1 ⟶ ∀ x5 . x5 ∈ x1 ⟶ x3 x4 x5 = decode_r (ap x0 2) x4 x5
Theorem pack_u_r_2_eq2 : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 x4 = decode_r (ap (pack_u_r x0 x1 x2) 2) x3 x4
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Theorem pack_u_r_inj : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . pack_u_r x0 x2 x4 = pack_u_r x1 x3 x5 ⟶ and (and (x0 = x1) (∀ x6 . x6 ∈ x0 ⟶ x2 x6 = x3 x6)) (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x6 x7 = x5 x6 x7)
Param iff^iff : ο → ο → ο
Known encode_r_ext^encode_r_ext : ∀ x0 . ∀ x1 x2 : ι → ι → ο . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ iff (x1 x3 x4) (x2 x3 x4)) ⟶ encode_r x0 x1 = encode_r x0 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 pack_u_r_ext : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . (∀ x5 . x5 ∈ x0 ⟶ x1 x5 = x2 x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ iff (x3 x5 x6) (x4 x5 x6)) ⟶ pack_u_r x0 x1 x3 = pack_u_r x0 x2 x4
Definition struct_u_r := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι . (∀ x4 . x4 ∈ x2 ⟶ x3 x4 ∈ x2) ⟶ ∀ x4 : ι → ι → ο . x1 (pack_u_r x2 x3 x4)) ⟶ x1 x0
Theorem pack_struct_u_r_I : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ x0) ⟶ ∀ x2 : ι → ι → ο . struct_u_r (pack_u_r x0 x1 x2)
Theorem pack_struct_u_r_E1 : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ο . struct_u_r (pack_u_r x0 x1 x2) ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x3 ∈ x0
Known iff_refl^iff_refl : ∀ x0 : ο . iff x0 x0
Theorem struct_u_r_eta : ∀ x0 . struct_u_r x0 ⟶ x0 = pack_u_r (ap x0 0) (ap (ap x0 1)) (decode_r (ap x0 2))
Definition unpack_u_r_i := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι → ο) → ι . x1 (ap x0 0) (ap (ap x0 1)) (decode_r (ap x0 2))
Theorem unpack_u_r_i_eq : ∀ x0 : ι → (ι → ι) → (ι → ι → ο) → ι . ∀ x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . (∀ x4 : ι → ι . (∀ x5 . x5 ∈ x1 ⟶ x2 x5 = x4 x5) ⟶ ∀ x5 : ι → ι → ο . (∀ x6 . x6 ∈ x1 ⟶ ∀ x7 . x7 ∈ x1 ⟶ iff (x3 x6 x7) (x5 x6 x7)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ unpack_u_r_i (pack_u_r x1 x2 x3) x0 = x0 x1 x2 x3
Definition unpack_u_r_o := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι → ο) → ο . x1 (ap x0 0) (ap (ap x0 1)) (decode_r (ap x0 2))
Theorem unpack_u_r_o_eq : ∀ x0 : ι → (ι → ι) → (ι → ι → ο) → ο . ∀ x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . (∀ x4 : ι → ι . (∀ x5 . x5 ∈ x1 ⟶ x2 x5 = x4 x5) ⟶ ∀ x5 : ι → ι → ο . (∀ x6 . x6 ∈ x1 ⟶ ∀ x7 . x7 ∈ x1 ⟶ iff (x3 x6 x7) (x5 x6 x7)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ unpack_u_r_o (pack_u_r x1 x2 x3) x0 = x0 x1 x2 x3
Param Sep^Sep : ι → (ι → ο) → ι
Definition pack_u_p := λ x0 . λ x1 : ι → ι . λ x2 : ι → ο . lam 3 (λ x3 . If_i (x3 = 0) x0 (If_i (x3 = 1) (lam x0 x1) (Sep x0 x2)))
Theorem pack_u_p_0_eq : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ο . x0 = pack_u_p x1 x2 x3 ⟶ x1 = ap x0 0
Theorem pack_u_p_0_eq2 : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ο . x0 = ap (pack_u_p x0 x1 x2) 0
Theorem pack_u_p_1_eq : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ο . x0 = pack_u_p x1 x2 x3 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x4 = ap (ap x0 1) x4
Theorem pack_u_p_1_eq2 : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ο . ∀ x3 . x3 ∈ x0 ⟶ x1 x3 = ap (ap (pack_u_p x0 x1 x2) 1) x3
Param decode_p^decode_p : ι → ι → ο
Known decode_encode_p^{decode_encode_p} : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ decode_p (Sep x0 x1) x2 = x1 x2
Theorem pack_u_p_2_eq : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ο . x0 = pack_u_p x1 x2 x3 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x3 x4 = decode_p (ap x0 2) x4
Theorem pack_u_p_2_eq2 : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ο . ∀ x3 . x3 ∈ x0 ⟶ x2 x3 = decode_p (ap (pack_u_p x0 x1 x2) 2) x3
Theorem pack_u_p_inj : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ο . pack_u_p x0 x2 x4 = pack_u_p x1 x3 x5 ⟶ and (and (x0 = x1) (∀ x6 . x6 ∈ x0 ⟶ x2 x6 = x3 x6)) (∀ x6 . x6 ∈ x0 ⟶ x4 x6 = x5 x6)
Known encode_p_ext^encode_p_ext : ∀ x0 . ∀ x1 x2 : ι → ο . (∀ x3 . x3 ∈ x0 ⟶ iff (x1 x3) (x2 x3)) ⟶ Sep x0 x1 = Sep x0 x2
Theorem pack_u_p_ext : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ο . (∀ x5 . x5 ∈ x0 ⟶ x1 x5 = x2 x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ iff (x3 x5) (x4 x5)) ⟶ pack_u_p x0 x1 x3 = pack_u_p x0 x2 x4
Definition struct_u_p := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι . (∀ x4 . x4 ∈ x2 ⟶ x3 x4 ∈ x2) ⟶ ∀ x4 : ι → ο . x1 (pack_u_p x2 x3 x4)) ⟶ x1 x0
Theorem pack_struct_u_p_I : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ x0) ⟶ ∀ x2 : ι → ο . struct_u_p (pack_u_p x0 x1 x2)
Theorem pack_struct_u_p_E1 : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ο . struct_u_p (pack_u_p x0 x1 x2) ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x3 ∈ x0
Theorem struct_u_p_eta : ∀ x0 . struct_u_p x0 ⟶ x0 = pack_u_p (ap x0 0) (ap (ap x0 1)) (decode_p (ap x0 2))
Definition unpack_u_p_i := λ x0 . λ x1 : ι → (ι → ι) → (ι → ο) → ι . x1 (ap x0 0) (ap (ap x0 1)) (decode_p (ap x0 2))
Theorem unpack_u_p_i_eq : ∀ x0 : ι → (ι → ι) → (ι → ο) → ι . ∀ x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ο . (∀ x4 : ι → ι . (∀ x5 . x5 ∈ x1 ⟶ x2 x5 = x4 x5) ⟶ ∀ x5 : ι → ο . (∀ x6 . x6 ∈ x1 ⟶ iff (x3 x6) (x5 x6)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ unpack_u_p_i (pack_u_p x1 x2 x3) x0 = x0 x1 x2 x3
Definition unpack_u_p_o := λ x0 . λ x1 : ι → (ι → ι) → (ι → ο) → ο . x1 (ap x0 0) (ap (ap x0 1)) (decode_p (ap x0 2))
Theorem unpack_u_p_o_eq : ∀ x0 : ι → (ι → ι) → (ι → ο) → ο . ∀ x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ο . (∀ x4 : ι → ι . (∀ x5 . x5 ∈ x1 ⟶ x2 x5 = x4 x5) ⟶ ∀ x5 : ι → ο . (∀ x6 . x6 ∈ x1 ⟶ iff (x3 x6) (x5 x6)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ unpack_u_p_o (pack_u_p x1 x2 x3) x0 = x0 x1 x2 x3
Definition pack_u_e := λ x0 . λ x1 : ι → ι . λ x2 . lam 3 (λ x3 . If_i (x3 = 0) x0 (If_i (x3 = 1) (lam x0 x1) x2))
Theorem pack_u_e_0_eq : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 . x0 = pack_u_e x1 x2 x3 ⟶ x1 = ap x0 0
Theorem pack_u_e_0_eq2 : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x0 = ap (pack_u_e x0 x1 x2) 0
Theorem pack_u_e_1_eq : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 . x0 = pack_u_e x1 x2 x3 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x4 = ap (ap x0 1) x4
Theorem pack_u_e_1_eq2 : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x3 ∈ x0 ⟶ x1 x3 = ap (ap (pack_u_e x0 x1 x2) 1) x3
Theorem pack_u_e_2_eq : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 . x0 = pack_u_e x1 x2 x3 ⟶ x3 = ap x0 2
Theorem pack_u_e_2_eq2 : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 = ap (pack_u_e x0 x1 x2) 2
Theorem pack_u_e_inj : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 . pack_u_e x0 x2 x4 = pack_u_e x1 x3 x5 ⟶ and (and (x0 = x1) (∀ x6 . x6 ∈ x0 ⟶ x2 x6 = x3 x6)) (x4 = x5)
Theorem pack_u_e_ext : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 . (∀ x4 . x4 ∈ x0 ⟶ x1 x4 = x2 x4) ⟶ pack_u_e x0 x1 x3 = pack_u_e x0 x2 x3
Definition struct_u_e := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι . (∀ x4 . x4 ∈ x2 ⟶ x3 x4 ∈ x2) ⟶ ∀ x4 . x4 ∈ x2 ⟶ x1 (pack_u_e x2 x3 x4)) ⟶ x1 x0
Theorem pack_struct_u_e_I : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ x0) ⟶ ∀ x2 . x2 ∈ x0 ⟶ struct_u_e (pack_u_e x0 x1 x2)
Theorem pack_struct_u_e_E1 : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . struct_u_e (pack_u_e x0 x1 x2) ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x3 ∈ x0
Theorem pack_struct_u_e_E2 : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . struct_u_e (pack_u_e x0 x1 x2) ⟶ x2 ∈ x0
Theorem struct_u_e_eta : ∀ x0 . struct_u_e x0 ⟶ x0 = pack_u_e (ap x0 0) (ap (ap x0 1)) (ap x0 2)
Definition unpack_u_e_i := λ x0 . λ x1 : ι → (ι → ι) → ι → ι . x1 (ap x0 0) (ap (ap x0 1)) (ap x0 2)
Theorem unpack_u_e_i_eq : ∀ x0 : ι → (ι → ι) → ι → ι . ∀ x1 . ∀ x2 : ι → ι . ∀ x3 . (∀ x4 : ι → ι . (∀ x5 . x5 ∈ x1 ⟶ x2 x5 = x4 x5) ⟶ x0 x1 x4 x3 = x0 x1 x2 x3) ⟶ unpack_u_e_i (pack_u_e x1 x2 x3) x0 = x0 x1 x2 x3
Definition unpack_u_e_o := λ x0 . λ x1 : ι → (ι → ι) → ι → ο . x1 (ap x0 0) (ap (ap x0 1)) (ap x0 2)
Theorem unpack_u_e_o_eq : ∀ x0 : ι → (ι → ι) → ι → ο . ∀ x1 . ∀ x2 : ι → ι . ∀ x3 . (∀ x4 : ι → ι . (∀ x5 . x5 ∈ x1 ⟶ x2 x5 = x4 x5) ⟶ x0 x1 x4 x3 = x0 x1 x2 x3) ⟶ unpack_u_e_o (pack_u_e x1 x2 x3) x0 = x0 x1 x2 x3
Definition pack_r_r := λ x0 . λ x1 x2 : ι → ι → ο . lam 3 (λ x3 . If_i (x3 = 0) x0 (If_i (x3 = 1) (encode_r x0 x1) (encode_r x0 x2)))
Theorem pack_r_r_0_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . x0 = pack_r_r x1 x2 x3 ⟶ x1 = ap x0 0
Theorem pack_r_r_0_eq2 : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ο . x3 x0 (ap (pack_r_r x0 x1 x2) 0) ⟶ x3 (ap (pack_r_r x0 x1 x2) 0) x0
Theorem pack_r_r_1_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . x0 = pack_r_r x1 x2 x3 ⟶ ∀ x4 . x4 ∈ x1 ⟶ ∀ x5 . x5 ∈ x1 ⟶ x2 x4 x5 = decode_r (ap x0 1) x4 x5
Theorem pack_r_r_1_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = decode_r (ap (pack_r_r x0 x1 x2) 1) x3 x4
Theorem pack_r_r_2_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . x0 = pack_r_r x1 x2 x3 ⟶ ∀ x4 . x4 ∈ x1 ⟶ ∀ x5 . x5 ∈ x1 ⟶ x3 x4 x5 = decode_r (ap x0 2) x4 x5
Theorem pack_r_r_2_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 x4 = decode_r (ap (pack_r_r x0 x1 x2) 2) x3 x4
Theorem pack_r_r_inj : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι → ο . pack_r_r x0 x2 x4 = pack_r_r x1 x3 x5 ⟶ and (and (x0 = x1) (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x2 x6 x7 = x3 x6 x7)) (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x6 x7 = x5 x6 x7)
Theorem pack_r_r_ext : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι → ο . (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ iff (x1 x5 x6) (x2 x5 x6)) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ iff (x3 x5 x6) (x4 x5 x6)) ⟶ pack_r_r x0 x1 x3 = pack_r_r x0 x2 x4
Definition struct_r_r := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 x4 : ι → ι → ο . x1 (pack_r_r x2 x3 x4)) ⟶ x1 x0
Theorem pack_struct_r_r_I : ∀ x0 . ∀ x1 x2 : ι → ι → ο . struct_r_r (pack_r_r x0 x1 x2)
Theorem struct_r_r_eta : ∀ x0 . struct_r_r x0 ⟶ x0 = pack_r_r (ap x0 0) (decode_r (ap x0 1)) (decode_r (ap x0 2))
Definition unpack_r_r_i := λ x0 . λ x1 : ι → (ι → ι → ο) → (ι → ι → ο) → ι . x1 (ap x0 0) (decode_r (ap x0 1)) (decode_r (ap x0 2))
Theorem unpack_r_r_i_eq : ∀ x0 : ι → (ι → ι → ο) → (ι → ι → ο) → ι . ∀ x1 . ∀ x2 x3 : ι → ι → ο . (∀ x4 : ι → ι → ο . (∀ x5 . x5 ∈ x1 ⟶ ∀ x6 . x6 ∈ x1 ⟶ iff (x2 x5 x6) (x4 x5 x6)) ⟶ ∀ x5 : ι → ι → ο . (∀ x6 . x6 ∈ x1 ⟶ ∀ x7 . x7 ∈ x1 ⟶ iff (x3 x6 x7) (x5 x6 x7)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ unpack_r_r_i (pack_r_r x1 x2 x3) x0 = x0 x1 x2 x3
Definition unpack_r_r_o := λ x0 . λ x1 : ι → (ι → ι → ο) → (ι → ι → ο) → ο . x1 (ap x0 0) (decode_r (ap x0 1)) (decode_r (ap x0 2))
Theorem unpack_r_r_o_eq : ∀ x0 : ι → (ι → ι → ο) → (ι → ι → ο) → ο . ∀ x1 . ∀ x2 x3 : ι → ι → ο . (∀ x4 : ι → ι → ο . (∀ x5 . x5 ∈ x1 ⟶ ∀ x6 . x6 ∈ x1 ⟶ iff (x2 x5 x6) (x4 x5 x6)) ⟶ ∀ x5 : ι → ι → ο . (∀ x6 . x6 ∈ x1 ⟶ ∀ x7 . x7 ∈ x1 ⟶ iff (x3 x6 x7) (x5 x6 x7)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ unpack_r_r_o (pack_r_r x1 x2 x3) x0 = x0 x1 x2 x3
Definition pack_r_p := λ x0 . λ x1 : ι → ι → ο . λ x2 : ι → ο . lam 3 (λ x3 . If_i (x3 = 0) x0 (If_i (x3 = 1) (encode_r x0 x1) (Sep x0 x2)))
Theorem pack_r_p_0_eq : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . x0 = pack_r_p x1 x2 x3 ⟶ x1 = ap x0 0
Theorem pack_r_p_0_eq2 : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . x0 = ap (pack_r_p x0 x1 x2) 0
Theorem pack_r_p_1_eq : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . x0 = pack_r_p x1 x2 x3 ⟶ ∀ x4 . x4 ∈ x1 ⟶ ∀ x5 . x5 ∈ x1 ⟶ x2 x4 x5 = decode_r (ap x0 1) x4 x5
Theorem pack_r_p_1_eq2 : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = decode_r (ap (pack_r_p x0 x1 x2) 1) x3 x4
Theorem pack_r_p_2_eq : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . x0 = pack_r_p x1 x2 x3 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x3 x4 = decode_p (ap x0 2) x4
Theorem pack_r_p_2_eq2 : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 . x3 ∈ x0 ⟶ x2 x3 = decode_p (ap (pack_r_p x0 x1 x2) 2) x3
Theorem pack_r_p_inj : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 x5 : ι → ο . pack_r_p x0 x2 x4 = pack_r_p x1 x3 x5 ⟶ and (and (x0 = x1) (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x2 x6 x7 = x3 x6 x7)) (∀ x6 . x6 ∈ x0 ⟶ x4 x6 = x5 x6)
Theorem pack_r_p_ext : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 x4 : ι → ο . (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ iff (x1 x5 x6) (x2 x5 x6)) ⟶ (∀ x5 . x5 ∈ x0 ⟶ iff (x3 x5) (x4 x5)) ⟶ pack_r_p x0 x1 x3 = pack_r_p x0 x2 x4
Definition struct_r_p := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . x1 (pack_r_p x2 x3 x4)) ⟶ x1 x0
Theorem pack_struct_r_p_I : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . struct_r_p (pack_r_p x0 x1 x2)
Theorem struct_r_p_eta : ∀ x0 . struct_r_p x0 ⟶ x0 = pack_r_p (ap x0 0) (decode_r (ap x0 1)) (decode_p (ap x0 2))
Definition unpack_r_p_i := λ x0 . λ x1 : ι → (ι → ι → ο) → (ι → ο) → ι . x1 (ap x0 0) (decode_r (ap x0 1)) (decode_p (ap x0 2))
Theorem unpack_r_p_i_eq : ∀ x0 : ι → (ι → ι → ο) → (ι → ο) → ι . ∀ x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . (∀ x4 : ι → ι → ο . (∀ x5 . x5 ∈ x1 ⟶ ∀ x6 . x6 ∈ x1 ⟶ iff (x2 x5 x6) (x4 x5 x6)) ⟶ ∀ x5 : ι → ο . (∀ x6 . x6 ∈ x1 ⟶ iff (x3 x6) (x5 x6)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ unpack_r_p_i (pack_r_p x1 x2 x3) x0 = x0 x1 x2 x3
Definition unpack_r_p_o := λ x0 . λ x1 : ι → (ι → ι → ο) → (ι → ο) → ο . x1 (ap x0 0) (decode_r (ap x0 1)) (decode_p (ap x0 2))
Theorem unpack_r_p_o_eq : ∀ x0 : ι → (ι → ι → ο) → (ι → ο) → ο . ∀ x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . (∀ x4 : ι → ι → ο . (∀ x5 . x5 ∈ x1 ⟶ ∀ x6 . x6 ∈ x1 ⟶ iff (x2 x5 x6) (x4 x5 x6)) ⟶ ∀ x5 : ι → ο . (∀ x6 . x6 ∈ x1 ⟶ iff (x3 x6) (x5 x6)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ unpack_r_p_o (pack_r_p x1 x2 x3) x0 = x0 x1 x2 x3
Definition pack_r_e := λ x0 . λ x1 : ι → ι → ο . λ x2 . lam 3 (λ x3 . If_i (x3 = 0) x0 (If_i (x3 = 1) (encode_r x0 x1) x2))
Theorem pack_r_e_0_eq : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 . x0 = pack_r_e x1 x2 x3 ⟶ x1 = ap x0 0
Theorem pack_r_e_0_eq2 : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 . x0 = ap (pack_r_e x0 x1 x2) 0
Theorem pack_r_e_1_eq : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 . x0 = pack_r_e x1 x2 x3 ⟶ ∀ x4 . x4 ∈ x1 ⟶ ∀ x5 . x5 ∈ x1 ⟶ x2 x4 x5 = decode_r (ap x0 1) x4 x5
Theorem pack_r_e_1_eq2 : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = decode_r (ap (pack_r_e x0 x1 x2) 1) x3 x4
Theorem pack_r_e_2_eq : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 . x0 = pack_r_e x1 x2 x3 ⟶ x3 = ap x0 2
Theorem pack_r_e_2_eq2 : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 . x2 = ap (pack_r_e x0 x1 x2) 2
Theorem pack_r_e_inj : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 x5 . pack_r_e x0 x2 x4 = pack_r_e x1 x3 x5 ⟶ and (and (x0 = x1) (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x2 x6 x7 = x3 x6 x7)) (x4 = x5)
Theorem pack_r_e_ext : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 . (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ iff (x1 x4 x5) (x2 x4 x5)) ⟶ pack_r_e x0 x1 x3 = pack_r_e x0 x2 x3
Definition struct_r_e := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ο . ∀ x4 . x4 ∈ x2 ⟶ x1 (pack_r_e x2 x3 x4)) ⟶ x1 x0
Theorem pack_struct_r_e_I : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 . x2 ∈ x0 ⟶ struct_r_e (pack_r_e x0 x1 x2)
Theorem pack_struct_r_e_E2 : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 . struct_r_e (pack_r_e x0 x1 x2) ⟶ x2 ∈ x0
Theorem struct_r_e_eta : ∀ x0 . struct_r_e x0 ⟶ x0 = pack_r_e (ap x0 0) (decode_r (ap x0 1)) (ap x0 2)
Definition unpack_r_e_i := λ x0 . λ x1 : ι → (ι → ι → ο) → ι → ι . x1 (ap x0 0) (decode_r (ap x0 1)) (ap x0 2)
Theorem unpack_r_e_i_eq : ∀ x0 : ι → (ι → ι → ο) → ι → ι . ∀ x1 . ∀ x2 : ι → ι → ο . ∀ x3 . (∀ x4 : ι → ι → ο . (∀ x5 . x5 ∈ x1 ⟶ ∀ x6 . x6 ∈ x1 ⟶ iff (x2 x5 x6) (x4 x5 x6)) ⟶ x0 x1 x4 x3 = x0 x1 x2 x3) ⟶ unpack_r_e_i (pack_r_e x1 x2 x3) x0 = x0 x1 x2 x3
Definition unpack_r_e_o := λ x0 . λ x1 : ι → (ι → ι → ο) → ι → ο . x1 (ap x0 0) (decode_r (ap x0 1)) (ap x0 2)
Theorem unpack_r_e_o_eq : ∀ x0 : ι → (ι → ι → ο) → ι → ο . ∀ x1 . ∀ x2 : ι → ι → ο . ∀ x3 . (∀ x4 : ι → ι → ο . (∀ x5 . x5 ∈ x1 ⟶ ∀ x6 . x6 ∈ x1 ⟶ iff (x2 x5 x6) (x4 x5 x6)) ⟶ x0 x1 x4 x3 = x0 x1 x2 x3) ⟶ unpack_r_e_o (pack_r_e x1 x2 x3) x0 = x0 x1 x2 x3