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_b^encode_b : ι → CT2 ι
Param Sep^Sep : ι → (ι → ο) → ι
Definition pack_b_p_e_e := λ x0 . λ x1 : ι → ι → ι . λ x2 : ι → ο . λ x3 x4 . lam 5 (λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) (encode_b x0 x1) (If_i (x5 = 2) (Sep x0 x2) (If_i (x5 = 3) x3 x4))))
Param ap^ap : ι → ι → ι
Known tuple_5_0_eq^tuple_5_0_eq : ∀ x0 x1 x2 x3 x4 . ap (lam 5 (λ x6 . If_i (x6 = 0) x0 (If_i (x6 = 1) x1 (If_i (x6 = 2) x2 (If_i (x6 = 3) x3 x4))))) 0 = x0
Theorem pack_b_p_e_e_0_eq : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . x0 = pack_b_p_e_e x1 x2 x3 x4 x5 ⟶ x1 = ap x0 0 (proof)
Theorem pack_b_p_e_e_0_eq2 : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 x4 . x0 = ap (pack_b_p_e_e x0 x1 x2 x3 x4) 0 (proof)
Param decode_b^decode_b : ι → ι → ι → ι
Known tuple_5_1_eq^tuple_5_1_eq : ∀ x0 x1 x2 x3 x4 . ap (lam 5 (λ x6 . If_i (x6 = 0) x0 (If_i (x6 = 1) x1 (If_i (x6 = 2) x2 (If_i (x6 = 3) x3 x4))))) 1 = x1
Known decode_encode_b^{decode_encode_b} : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ decode_b (encode_b x0 x1) x2 x3 = x1 x2 x3
Theorem pack_b_p_e_e_1_eq : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . x0 = pack_b_p_e_e x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x1 ⟶ ∀ x7 . x7 ∈ x1 ⟶ x2 x6 x7 = decode_b (ap x0 1) x6 x7 (proof)
Theorem pack_b_p_e_e_1_eq2 : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 x4 x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x1 x5 x6 = decode_b (ap (pack_b_p_e_e x0 x1 x2 x3 x4) 1) x5 x6 (proof)
Param decode_p^decode_p : ι → ι → ο
Known tuple_5_2_eq^tuple_5_2_eq : ∀ x0 x1 x2 x3 x4 . ap (lam 5 (λ x6 . If_i (x6 = 0) x0 (If_i (x6 = 1) x1 (If_i (x6 = 2) x2 (If_i (x6 = 3) x3 x4))))) 2 = x2
Known decode_encode_p^{decode_encode_p} : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ decode_p (Sep x0 x1) x2 = x1 x2
Theorem pack_b_p_e_e_2_eq : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . x0 = pack_b_p_e_e x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x1 ⟶ x3 x6 = decode_p (ap x0 2) x6 (proof)
Theorem pack_b_p_e_e_2_eq2 : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 x4 x5 . x5 ∈ x0 ⟶ x2 x5 = decode_p (ap (pack_b_p_e_e x0 x1 x2 x3 x4) 2) x5 (proof)
Known tuple_5_3_eq^tuple_5_3_eq : ∀ x0 x1 x2 x3 x4 . ap (lam 5 (λ x6 . If_i (x6 = 0) x0 (If_i (x6 = 1) x1 (If_i (x6 = 2) x2 (If_i (x6 = 3) x3 x4))))) 3 = x3
Theorem pack_b_p_e_e_3_eq : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . x0 = pack_b_p_e_e x1 x2 x3 x4 x5 ⟶ x4 = ap x0 3 (proof)
Theorem pack_b_p_e_e_3_eq2 : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 x4 . x3 = ap (pack_b_p_e_e x0 x1 x2 x3 x4) 3 (proof)
Known tuple_5_4_eq^tuple_5_4_eq : ∀ x0 x1 x2 x3 x4 . ap (lam 5 (λ x6 . If_i (x6 = 0) x0 (If_i (x6 = 1) x1 (If_i (x6 = 2) x2 (If_i (x6 = 3) x3 x4))))) 4 = x4
Theorem pack_b_p_e_e_4_eq : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . x0 = pack_b_p_e_e x1 x2 x3 x4 x5 ⟶ x5 = ap x0 4 (proof)
Theorem pack_b_p_e_e_4_eq2 : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 x4 . x4 = ap (pack_b_p_e_e x0 x1 x2 x3 x4) 4 (proof)
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known and5I^and5I : ∀ x0 x1 x2 x3 x4 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ and (and (and (and x0 x1) x2) x3) x4
Theorem pack_b_p_e_e_inj : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ο . ∀ x6 x7 x8 x9 . pack_b_p_e_e x0 x2 x4 x6 x8 = pack_b_p_e_e x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ x2 x10 x11 = x3 x10 x11)) (∀ x10 . x10 ∈ x0 ⟶ x4 x10 = x5 x10)) (x6 = x7)) (x8 = x9) (proof)
Param iff^iff : ο → ο → ο
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
Known encode_b_ext^encode_b_ext : ∀ x0 . ∀ x1 x2 : ι → ι → ι . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = x2 x3 x4) ⟶ encode_b x0 x1 = encode_b x0 x2
Theorem pack_b_p_e_e_ext : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ο . ∀ x5 x6 . (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x1 x7 x8 = x2 x7 x8) ⟶ (∀ x7 . x7 ∈ x0 ⟶ iff (x3 x7) (x4 x7)) ⟶ pack_b_p_e_e x0 x1 x3 x5 x6 = pack_b_p_e_e x0 x2 x4 x5 x6 (proof)
Definition struct_b_p_e_e := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ι . (∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x2 ⟶ x3 x4 x5 ∈ x2) ⟶ ∀ x4 : ι → ο . ∀ x5 . x5 ∈ x2 ⟶ ∀ x6 . x6 ∈ x2 ⟶ x1 (pack_b_p_e_e x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem pack_struct_b_p_e_e_I : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ∈ x0) ⟶ ∀ x2 : ι → ο . ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ struct_b_p_e_e (pack_b_p_e_e x0 x1 x2 x3 x4) (proof)
Theorem pack_struct_b_p_e_e_E1 : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 x4 . struct_b_p_e_e (pack_b_p_e_e x0 x1 x2 x3 x4) ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x1 x5 x6 ∈ x0 (proof)
Theorem pack_struct_b_p_e_e_E3 : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 x4 . struct_b_p_e_e (pack_b_p_e_e x0 x1 x2 x3 x4) ⟶ x3 ∈ x0 (proof)
Theorem pack_struct_b_p_e_e_E4 : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 x4 . struct_b_p_e_e (pack_b_p_e_e x0 x1 x2 x3 x4) ⟶ x4 ∈ x0 (proof)
Known iff_refl^iff_refl : ∀ x0 : ο . iff x0 x0
Theorem struct_b_p_e_e_eta : ∀ x0 . struct_b_p_e_e x0 ⟶ x0 = pack_b_p_e_e (ap x0 0) (decode_b (ap x0 1)) (decode_p (ap x0 2)) (ap x0 3) (ap x0 4) (proof)
Definition unpack_b_p_e_e_i := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ο) → ι → ι → ι . x1 (ap x0 0) (decode_b (ap x0 1)) (decode_p (ap x0 2)) (ap x0 3) (ap x0 4)
Theorem unpack_b_p_e_e_i_eq : ∀ x0 : ι → (ι → ι → ι) → (ι → ο) → ι → ι → ι . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . (∀ x6 : ι → ι → ι . (∀ x7 . x7 ∈ x1 ⟶ ∀ x8 . x8 ∈ x1 ⟶ x2 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ο . (∀ x8 . x8 ∈ x1 ⟶ iff (x3 x8) (x7 x8)) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5) ⟶ unpack_b_p_e_e_i (pack_b_p_e_e x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition unpack_b_p_e_e_o := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ο) → ι → ι → ο . x1 (ap x0 0) (decode_b (ap x0 1)) (decode_p (ap x0 2)) (ap x0 3) (ap x0 4)
Theorem unpack_b_p_e_e_o_eq : ∀ x0 : ι → (ι → ι → ι) → (ι → ο) → ι → ι → ο . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . (∀ x6 : ι → ι → ι . (∀ x7 . x7 ∈ x1 ⟶ ∀ x8 . x8 ∈ x1 ⟶ x2 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ο . (∀ x8 . x8 ∈ x1 ⟶ iff (x3 x8) (x7 x8)) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5) ⟶ unpack_b_p_e_e_o (pack_b_p_e_e x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Param encode_r^encode_r : ι → (ι → ι → ο) → ι
Definition pack_u_u_r_r := λ x0 . λ x1 x2 : ι → ι . λ x3 x4 : ι → ι → ο . lam 5 (λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) (lam x0 x1) (If_i (x5 = 2) (lam x0 x2) (If_i (x5 = 3) (encode_r x0 x3) (encode_r x0 x4)))))
Theorem pack_u_u_r_r_0_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = pack_u_u_r_r x1 x2 x3 x4 x5 ⟶ x1 = ap x0 0 (proof)
Theorem pack_u_u_r_r_0_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 x5 : ι → ι → ο . x5 x0 (ap (pack_u_u_r_r x0 x1 x2 x3 x4) 0) ⟶ x5 (ap (pack_u_u_r_r x0 x1 x2 x3 x4) 0) x0 (proof)
Known beta^beta : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ ap (lam x0 x1) x2 = x1 x2
Theorem pack_u_u_r_r_1_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = pack_u_u_r_r x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x1 ⟶ x2 x6 = ap (ap x0 1) x6 (proof)
Theorem pack_u_u_r_r_1_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ x5 . x5 ∈ x0 ⟶ x1 x5 = ap (ap (pack_u_u_r_r x0 x1 x2 x3 x4) 1) x5 (proof)
Theorem pack_u_u_r_r_2_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = pack_u_u_r_r x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x1 ⟶ x3 x6 = ap (ap x0 2) x6 (proof)
Theorem pack_u_u_r_r_2_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ x5 . x5 ∈ x0 ⟶ x2 x5 = ap (ap (pack_u_u_r_r x0 x1 x2 x3 x4) 2) x5 (proof)
Param decode_r^decode_r : ι → ι → ι → ο
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_u_r_r_3_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = pack_u_u_r_r x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x1 ⟶ ∀ x7 . x7 ∈ x1 ⟶ x4 x6 x7 = decode_r (ap x0 3) x6 x7 (proof)
Theorem pack_u_u_r_r_3_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 = decode_r (ap (pack_u_u_r_r x0 x1 x2 x3 x4) 3) x5 x6 (proof)
Theorem pack_u_u_r_r_4_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = pack_u_u_r_r x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x1 ⟶ ∀ x7 . x7 ∈ x1 ⟶ x5 x6 x7 = decode_r (ap x0 4) x6 x7 (proof)
Theorem pack_u_u_r_r_4_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 x6 = decode_r (ap (pack_u_u_r_r x0 x1 x2 x3 x4) 4) x5 x6 (proof)
Theorem pack_u_u_r_r_inj : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι . ∀ x6 x7 x8 x9 : ι → ι → ο . pack_u_u_r_r x0 x2 x4 x6 x8 = pack_u_u_r_r x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 . x10 ∈ x0 ⟶ x2 x10 = x3 x10)) (∀ x10 . x10 ∈ x0 ⟶ x4 x10 = x5 x10)) (∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ x6 x10 x11 = x7 x10 x11)) (∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ x8 x10 x11 = x9 x10 x11) (proof)
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_u_r_r_ext : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι . ∀ x5 x6 x7 x8 : ι → ι → ο . (∀ x9 . x9 ∈ x0 ⟶ x1 x9 = x2 x9) ⟶ (∀ x9 . x9 ∈ x0 ⟶ x3 x9 = x4 x9) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ iff (x5 x9 x10) (x6 x9 x10)) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ iff (x7 x9 x10) (x8 x9 x10)) ⟶ pack_u_u_r_r x0 x1 x3 x5 x7 = pack_u_u_r_r x0 x2 x4 x6 x8 (proof)
Definition struct_u_u_r_r := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι . (∀ x4 . x4 ∈ x2 ⟶ x3 x4 ∈ x2) ⟶ ∀ x4 : ι → ι . (∀ x5 . x5 ∈ x2 ⟶ x4 x5 ∈ x2) ⟶ ∀ x5 x6 : ι → ι → ο . x1 (pack_u_u_r_r x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem pack_struct_u_u_r_r_I : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ x0) ⟶ ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x0) ⟶ ∀ x3 x4 : ι → ι → ο . struct_u_u_r_r (pack_u_u_r_r x0 x1 x2 x3 x4) (proof)
Theorem pack_struct_u_u_r_r_E1 : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . struct_u_u_r_r (pack_u_u_r_r x0 x1 x2 x3 x4) ⟶ ∀ x5 . x5 ∈ x0 ⟶ x1 x5 ∈ x0 (proof)
Theorem pack_struct_u_u_r_r_E2 : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . struct_u_u_r_r (pack_u_u_r_r x0 x1 x2 x3 x4) ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x5 ∈ x0 (proof)
Theorem struct_u_u_r_r_eta : ∀ x0 . struct_u_u_r_r x0 ⟶ x0 = pack_u_u_r_r (ap x0 0) (ap (ap x0 1)) (ap (ap x0 2)) (decode_r (ap x0 3)) (decode_r (ap x0 4)) (proof)
Definition unpack_u_u_r_r_i := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → (ι → ι → ο) → ι . x1 (ap x0 0) (ap (ap x0 1)) (ap (ap x0 2)) (decode_r (ap x0 3)) (decode_r (ap x0 4))
Theorem unpack_u_u_r_r_i_eq : ∀ x0 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → (ι → ι → ο) → ι . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . (∀ x6 : ι → ι . (∀ x7 . x7 ∈ x1 ⟶ x2 x7 = x6 x7) ⟶ ∀ x7 : ι → ι . (∀ x8 . x8 ∈ x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ x8 : ι → ι → ο . (∀ x9 . x9 ∈ x1 ⟶ ∀ x10 . x10 ∈ x1 ⟶ iff (x4 x9 x10) (x8 x9 x10)) ⟶ ∀ x9 : ι → ι → ο . (∀ x10 . x10 ∈ x1 ⟶ ∀ x11 . x11 ∈ x1 ⟶ iff (x5 x10 x11) (x9 x10 x11)) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ unpack_u_u_r_r_i (pack_u_u_r_r x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition unpack_u_u_r_r_o := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → (ι → ι → ο) → ο . x1 (ap x0 0) (ap (ap x0 1)) (ap (ap x0 2)) (decode_r (ap x0 3)) (decode_r (ap x0 4))
Theorem unpack_u_u_r_r_o_eq : ∀ x0 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → (ι → ι → ο) → ο . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . (∀ x6 : ι → ι . (∀ x7 . x7 ∈ x1 ⟶ x2 x7 = x6 x7) ⟶ ∀ x7 : ι → ι . (∀ x8 . x8 ∈ x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ x8 : ι → ι → ο . (∀ x9 . x9 ∈ x1 ⟶ ∀ x10 . x10 ∈ x1 ⟶ iff (x4 x9 x10) (x8 x9 x10)) ⟶ ∀ x9 : ι → ι → ο . (∀ x10 . x10 ∈ x1 ⟶ ∀ x11 . x11 ∈ x1 ⟶ iff (x5 x10 x11) (x9 x10 x11)) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ unpack_u_u_r_r_o (pack_u_u_r_r x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition pack_u_u_r_p := λ x0 . λ x1 x2 : ι → ι . λ x3 : ι → ι → ο . λ x4 : ι → ο . lam 5 (λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) (lam x0 x1) (If_i (x5 = 2) (lam x0 x2) (If_i (x5 = 3) (encode_r x0 x3) (Sep x0 x4)))))
Theorem pack_u_u_r_p_0_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . x0 = pack_u_u_r_p x1 x2 x3 x4 x5 ⟶ x1 = ap x0 0 (proof)
Theorem pack_u_u_r_p_0_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . x0 = ap (pack_u_u_r_p x0 x1 x2 x3 x4) 0 (proof)
Theorem pack_u_u_r_p_1_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . x0 = pack_u_u_r_p x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x1 ⟶ x2 x6 = ap (ap x0 1) x6 (proof)
Theorem pack_u_u_r_p_1_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . x5 ∈ x0 ⟶ x1 x5 = ap (ap (pack_u_u_r_p x0 x1 x2 x3 x4) 1) x5 (proof)
Theorem pack_u_u_r_p_2_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . x0 = pack_u_u_r_p x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x1 ⟶ x3 x6 = ap (ap x0 2) x6 (proof)
Theorem pack_u_u_r_p_2_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . x5 ∈ x0 ⟶ x2 x5 = ap (ap (pack_u_u_r_p x0 x1 x2 x3 x4) 2) x5 (proof)
Theorem pack_u_u_r_p_3_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . x0 = pack_u_u_r_p x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x1 ⟶ ∀ x7 . x7 ∈ x1 ⟶ x4 x6 x7 = decode_r (ap x0 3) x6 x7 (proof)
Theorem pack_u_u_r_p_3_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 = decode_r (ap (pack_u_u_r_p x0 x1 x2 x3 x4) 3) x5 x6 (proof)
Theorem pack_u_u_r_p_4_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . x0 = pack_u_u_r_p x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x1 ⟶ x5 x6 = decode_p (ap x0 4) x6 (proof)
Theorem pack_u_u_r_p_4_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . x5 ∈ x0 ⟶ x4 x5 = decode_p (ap (pack_u_u_r_p x0 x1 x2 x3 x4) 4) x5 (proof)
Theorem pack_u_u_r_p_inj : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι . ∀ x6 x7 : ι → ι → ο . ∀ x8 x9 : ι → ο . pack_u_u_r_p x0 x2 x4 x6 x8 = pack_u_u_r_p x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 . x10 ∈ x0 ⟶ x2 x10 = x3 x10)) (∀ x10 . x10 ∈ x0 ⟶ x4 x10 = x5 x10)) (∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ x6 x10 x11 = x7 x10 x11)) (∀ x10 . x10 ∈ x0 ⟶ x8 x10 = x9 x10) (proof)
Theorem pack_u_u_r_p_ext : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι . ∀ x5 x6 : ι → ι → ο . ∀ x7 x8 : ι → ο . (∀ x9 . x9 ∈ x0 ⟶ x1 x9 = x2 x9) ⟶ (∀ x9 . x9 ∈ x0 ⟶ x3 x9 = x4 x9) ⟶ (∀ x9 . x9 ∈ x0 ⟶ ∀ x10 . x10 ∈ x0 ⟶ iff (x5 x9 x10) (x6 x9 x10)) ⟶ (∀ x9 . x9 ∈ x0 ⟶ iff (x7 x9) (x8 x9)) ⟶ pack_u_u_r_p x0 x1 x3 x5 x7 = pack_u_u_r_p x0 x2 x4 x6 x8 (proof)
Definition struct_u_u_r_p := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι . (∀ x4 . x4 ∈ x2 ⟶ x3 x4 ∈ x2) ⟶ ∀ x4 : ι → ι . (∀ x5 . x5 ∈ x2 ⟶ x4 x5 ∈ x2) ⟶ ∀ x5 : ι → ι → ο . ∀ x6 : ι → ο . x1 (pack_u_u_r_p x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem pack_struct_u_u_r_p_I : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ x0) ⟶ ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x0) ⟶ ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . struct_u_u_r_p (pack_u_u_r_p x0 x1 x2 x3 x4) (proof)
Theorem pack_struct_u_u_r_p_E1 : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . struct_u_u_r_p (pack_u_u_r_p x0 x1 x2 x3 x4) ⟶ ∀ x5 . x5 ∈ x0 ⟶ x1 x5 ∈ x0 (proof)
Theorem pack_struct_u_u_r_p_E2 : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . struct_u_u_r_p (pack_u_u_r_p x0 x1 x2 x3 x4) ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x5 ∈ x0 (proof)
Theorem struct_u_u_r_p_eta : ∀ x0 . struct_u_u_r_p x0 ⟶ x0 = pack_u_u_r_p (ap x0 0) (ap (ap x0 1)) (ap (ap x0 2)) (decode_r (ap x0 3)) (decode_p (ap x0 4)) (proof)
Definition unpack_u_u_r_p_i := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → (ι → ο) → ι . x1 (ap x0 0) (ap (ap x0 1)) (ap (ap x0 2)) (decode_r (ap x0 3)) (decode_p (ap x0 4))
Theorem unpack_u_u_r_p_i_eq : ∀ x0 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → (ι → ο) → ι . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . (∀ x6 : ι → ι . (∀ x7 . x7 ∈ x1 ⟶ x2 x7 = x6 x7) ⟶ ∀ x7 : ι → ι . (∀ x8 . x8 ∈ x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ x8 : ι → ι → ο . (∀ x9 . x9 ∈ x1 ⟶ ∀ x10 . x10 ∈ x1 ⟶ iff (x4 x9 x10) (x8 x9 x10)) ⟶ ∀ x9 : ι → ο . (∀ x10 . x10 ∈ x1 ⟶ iff (x5 x10) (x9 x10)) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ unpack_u_u_r_p_i (pack_u_u_r_p x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition unpack_u_u_r_p_o := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → (ι → ο) → ο . x1 (ap x0 0) (ap (ap x0 1)) (ap (ap x0 2)) (decode_r (ap x0 3)) (decode_p (ap x0 4))
Theorem unpack_u_u_r_p_o_eq : ∀ x0 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → (ι → ο) → ο . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . (∀ x6 : ι → ι . (∀ x7 . x7 ∈ x1 ⟶ x2 x7 = x6 x7) ⟶ ∀ x7 : ι → ι . (∀ x8 . x8 ∈ x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ x8 : ι → ι → ο . (∀ x9 . x9 ∈ x1 ⟶ ∀ x10 . x10 ∈ x1 ⟶ iff (x4 x9 x10) (x8 x9 x10)) ⟶ ∀ x9 : ι → ο . (∀ x10 . x10 ∈ x1 ⟶ iff (x5 x10) (x9 x10)) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ unpack_u_u_r_p_o (pack_u_u_r_p x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition pack_u_u_r_e := λ x0 . λ x1 x2 : ι → ι . λ x3 : ι → ι → ο . λ x4 . lam 5 (λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) (lam x0 x1) (If_i (x5 = 2) (lam x0 x2) (If_i (x5 = 3) (encode_r x0 x3) x4))))
Theorem pack_u_u_r_e_0_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . x0 = pack_u_u_r_e x1 x2 x3 x4 x5 ⟶ x1 = ap x0 0 (proof)
Theorem pack_u_u_r_e_0_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . x0 = ap (pack_u_u_r_e x0 x1 x2 x3 x4) 0 (proof)
Theorem pack_u_u_r_e_1_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . x0 = pack_u_u_r_e x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x1 ⟶ x2 x6 = ap (ap x0 1) x6 (proof)
Theorem pack_u_u_r_e_1_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . x5 ∈ x0 ⟶ x1 x5 = ap (ap (pack_u_u_r_e x0 x1 x2 x3 x4) 1) x5 (proof)
Theorem pack_u_u_r_e_2_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . x0 = pack_u_u_r_e x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x1 ⟶ x3 x6 = ap (ap x0 2) x6 (proof)
Theorem pack_u_u_r_e_2_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . x5 ∈ x0 ⟶ x2 x5 = ap (ap (pack_u_u_r_e x0 x1 x2 x3 x4) 2) x5 (proof)
Theorem pack_u_u_r_e_3_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . x0 = pack_u_u_r_e x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x1 ⟶ ∀ x7 . x7 ∈ x1 ⟶ x4 x6 x7 = decode_r (ap x0 3) x6 x7 (proof)
Theorem pack_u_u_r_e_3_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 = decode_r (ap (pack_u_u_r_e x0 x1 x2 x3 x4) 3) x5 x6 (proof)
Theorem pack_u_u_r_e_4_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . x0 = pack_u_u_r_e x1 x2 x3 x4 x5 ⟶ x5 = ap x0 4 (proof)
Theorem pack_u_u_r_e_4_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . x4 = ap (pack_u_u_r_e x0 x1 x2 x3 x4) 4 (proof)
Theorem pack_u_u_r_e_inj : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι . ∀ x6 x7 : ι → ι → ο . ∀ x8 x9 . pack_u_u_r_e x0 x2 x4 x6 x8 = pack_u_u_r_e x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 . x10 ∈ x0 ⟶ x2 x10 = x3 x10)) (∀ x10 . x10 ∈ x0 ⟶ x4 x10 = x5 x10)) (∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ x6 x10 x11 = x7 x10 x11)) (x8 = x9) (proof)
Theorem pack_u_u_r_e_ext : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι . ∀ x5 x6 : ι → ι → ο . ∀ x7 . (∀ x8 . x8 ∈ x0 ⟶ x1 x8 = x2 x8) ⟶ (∀ x8 . x8 ∈ x0 ⟶ x3 x8 = x4 x8) ⟶ (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ iff (x5 x8 x9) (x6 x8 x9)) ⟶ pack_u_u_r_e x0 x1 x3 x5 x7 = pack_u_u_r_e x0 x2 x4 x6 x7 (proof)
Definition struct_u_u_r_e := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι . (∀ x4 . x4 ∈ x2 ⟶ x3 x4 ∈ x2) ⟶ ∀ x4 : ι → ι . (∀ x5 . x5 ∈ x2 ⟶ x4 x5 ∈ x2) ⟶ ∀ x5 : ι → ι → ο . ∀ x6 . x6 ∈ x2 ⟶ x1 (pack_u_u_r_e x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem pack_struct_u_u_r_e_I : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ x0) ⟶ ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x0) ⟶ ∀ x3 : ι → ι → ο . ∀ x4 . x4 ∈ x0 ⟶ struct_u_u_r_e (pack_u_u_r_e x0 x1 x2 x3 x4) (proof)
Theorem pack_struct_u_u_r_e_E1 : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . struct_u_u_r_e (pack_u_u_r_e x0 x1 x2 x3 x4) ⟶ ∀ x5 . x5 ∈ x0 ⟶ x1 x5 ∈ x0 (proof)
Theorem pack_struct_u_u_r_e_E2 : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . struct_u_u_r_e (pack_u_u_r_e x0 x1 x2 x3 x4) ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x5 ∈ x0 (proof)
Theorem pack_struct_u_u_r_e_E4 : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . struct_u_u_r_e (pack_u_u_r_e x0 x1 x2 x3 x4) ⟶ x4 ∈ x0 (proof)
Theorem struct_u_u_r_e_eta : ∀ x0 . struct_u_u_r_e x0 ⟶ x0 = pack_u_u_r_e (ap x0 0) (ap (ap x0 1)) (ap (ap x0 2)) (decode_r (ap x0 3)) (ap x0 4) (proof)
Definition unpack_u_u_r_e_i := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → ι → ι . x1 (ap x0 0) (ap (ap x0 1)) (ap (ap x0 2)) (decode_r (ap x0 3)) (ap x0 4)
Theorem unpack_u_u_r_e_i_eq : ∀ x0 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → ι → ι . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . (∀ x6 : ι → ι . (∀ x7 . x7 ∈ x1 ⟶ x2 x7 = x6 x7) ⟶ ∀ x7 : ι → ι . (∀ x8 . x8 ∈ x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ x8 : ι → ι → ο . (∀ x9 . x9 ∈ x1 ⟶ ∀ x10 . x10 ∈ x1 ⟶ iff (x4 x9 x10) (x8 x9 x10)) ⟶ x0 x1 x6 x7 x8 x5 = x0 x1 x2 x3 x4 x5) ⟶ unpack_u_u_r_e_i (pack_u_u_r_e x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition unpack_u_u_r_e_o := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → ι → ο . x1 (ap x0 0) (ap (ap x0 1)) (ap (ap x0 2)) (decode_r (ap x0 3)) (ap x0 4)
Theorem unpack_u_u_r_e_o_eq : ∀ x0 : ι → (ι → ι) → (ι → ι) → (ι → ι → ο) → ι → ο . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . (∀ x6 : ι → ι . (∀ x7 . x7 ∈ x1 ⟶ x2 x7 = x6 x7) ⟶ ∀ x7 : ι → ι . (∀ x8 . x8 ∈ x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ x8 : ι → ι → ο . (∀ x9 . x9 ∈ x1 ⟶ ∀ x10 . x10 ∈ x1 ⟶ iff (x4 x9 x10) (x8 x9 x10)) ⟶ x0 x1 x6 x7 x8 x5 = x0 x1 x2 x3 x4 x5) ⟶ unpack_u_u_r_e_o (pack_u_u_r_e x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)