Proofgold Address

Param lam^Sigma : ι → (ι → ι) → ι
Param ordsucc^ordsucc : ι → ι
Param If_i^If_i : ο → ι → ι → ι
Definition pack_e^pack_e := λ x0 x1 . lam 2 (λ x2 . If_i (x2 = 0) x0 x1)
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
Theorem pack_e_0_eq^pack_e_0_eq : ∀ x0 x1 x2 . x0 = pack_e x1 x2 ⟶ x1 = ap x0 0 (proof)
Theorem pack_e_0_eq2^pack_e_0_eq2 : ∀ x0 x1 . x0 = ap (pack_e x0 x1) 0 (proof)
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 pack_e_1_eq^pack_e_1_eq : ∀ x0 x1 x2 . x0 = pack_e x1 x2 ⟶ x2 = ap x0 1 (proof)
Theorem pack_e_1_eq2^pack_e_1_eq2 : ∀ x0 x1 . x1 = ap (pack_e x0 x1) 1 (proof)
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Theorem pack_e_inj^pack_e_inj : ∀ x0 x1 x2 x3 . pack_e x0 x2 = pack_e x1 x3 ⟶ and (x0 = x1) (x2 = x3) (proof)
Definition struct_e^struct_e := λ x0 . ∀ x1 : ι → ο . (∀ x2 x3 . x3 ∈ x2 ⟶ x1 (pack_e x2 x3)) ⟶ x1 x0
Theorem pack_struct_e_I^{pack_struct_e_I} : ∀ x0 x1 . x1 ∈ x0 ⟶ struct_e (pack_e x0 x1) (proof)
Theorem pack_struct_e_E1^{pack_struct_e_E1} : ∀ x0 x1 . struct_e (pack_e x0 x1) ⟶ x1 ∈ x0 (proof)
Theorem struct_e_eta^struct_e_eta : ∀ x0 . struct_e x0 ⟶ x0 = pack_e (ap x0 0) (ap x0 1) (proof)
Definition unpack_e_i^unpack_e_i := λ x0 . λ x1 : ι → ι → ι . x1 (ap x0 0) (ap x0 1)
Theorem unpack_e_i_eq^{unpack_e_i_eq} : ∀ x0 : ι → ι → ι . ∀ x1 x2 . unpack_e_i (pack_e x1 x2) x0 = x0 x1 x2 (proof)
Definition unpack_e_o^unpack_e_o := λ x0 . λ x1 : ι → ι → ο . x1 (ap x0 0) (ap x0 1)
Theorem unpack_e_o_eq^{unpack_e_o_eq} : ∀ x0 : ι → ι → ο . ∀ x1 x2 . unpack_e_o (pack_e x1 x2) x0 = x0 x1 x2 (proof)
Definition pack_u^pack_u := λ x0 . λ x1 : ι → ι . lam 2 (λ x2 . If_i (x2 = 0) x0 (lam x0 x1))
Theorem pack_u_0_eq^pack_u_0_eq : ∀ x0 x1 . ∀ x2 : ι → ι . x0 = pack_u x1 x2 ⟶ x1 = ap x0 0 (proof)
Theorem pack_u_0_eq2^pack_u_0_eq2 : ∀ x0 . ∀ x1 : ι → ι . x0 = ap (pack_u x0 x1) 0 (proof)
Known beta^beta : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ ap (lam x0 x1) x2 = x1 x2
Theorem pack_u_1_eq^pack_u_1_eq : ∀ x0 x1 . ∀ x2 : ι → ι . x0 = pack_u x1 x2 ⟶ ∀ x3 . x3 ∈ x1 ⟶ x2 x3 = ap (ap x0 1) x3 (proof)
Theorem pack_u_1_eq2^pack_u_1_eq2 : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 = ap (ap (pack_u x0 x1) 1) x2 (proof)
Theorem pack_u_inj^pack_u_inj : ∀ x0 x1 . ∀ x2 x3 : ι → ι . pack_u x0 x2 = pack_u x1 x3 ⟶ and (x0 = x1) (∀ x4 . x4 ∈ x0 ⟶ x2 x4 = x3 x4) (proof)
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_ext^pack_u_ext : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x1 x3 = x2 x3) ⟶ pack_u x0 x1 = pack_u x0 x2 (proof)
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
Theorem pack_struct_u_I^{pack_struct_u_I} : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ x0) ⟶ struct_u (pack_u x0 x1) (proof)
Theorem pack_struct_u_E1^{pack_struct_u_E1} : ∀ x0 . ∀ x1 : ι → ι . struct_u (pack_u x0 x1) ⟶ ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ x0 (proof)
Theorem struct_u_eta^struct_u_eta : ∀ x0 . struct_u x0 ⟶ x0 = pack_u (ap x0 0) (ap (ap x0 1)) (proof)
Definition unpack_u_i^unpack_u_i := λ x0 . λ x1 : ι → (ι → ι) → ι . x1 (ap x0 0) (ap (ap x0 1))
Theorem 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 (proof)
Definition unpack_u_o^unpack_u_o := λ x0 . λ x1 : ι → (ι → ι) → ο . x1 (ap x0 0) (ap (ap x0 1))
Theorem 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 (proof)
Param encode_b^encode_b : ι → CT2 ι
Definition pack_b^pack_b := λ x0 . λ x1 : ι → ι → ι . lam 2 (λ x2 . If_i (x2 = 0) x0 (encode_b x0 x1))
Theorem pack_b_0_eq^pack_b_0_eq : ∀ x0 x1 . ∀ x2 : ι → ι → ι . x0 = pack_b x1 x2 ⟶ x1 = ap x0 0 (proof)
Theorem pack_b_0_eq2^pack_b_0_eq2 : ∀ x0 . ∀ x1 : ι → ι → ι . x0 = ap (pack_b x0 x1) 0 (proof)
Param decode_b^decode_b : ι → ι → ι → ι
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_1_eq^pack_b_1_eq : ∀ x0 x1 . ∀ x2 : ι → ι → ι . x0 = pack_b x1 x2 ⟶ ∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x3 x4 = decode_b (ap x0 1) x3 x4 (proof)
Theorem pack_b_1_eq2^pack_b_1_eq2 : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 = decode_b (ap (pack_b x0 x1) 1) x2 x3 (proof)
Theorem pack_b_inj^pack_b_inj : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . pack_b x0 x2 = pack_b x1 x3 ⟶ and (x0 = x1) (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 x5 = x3 x4 x5) (proof)
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_ext^pack_b_ext : ∀ x0 . ∀ x1 x2 : ι → ι → ι . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x1 x3 x4 = x2 x3 x4) ⟶ pack_b x0 x1 = pack_b x0 x2 (proof)
Definition struct_b^struct_b := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ι . (∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x2 ⟶ x3 x4 x5 ∈ x2) ⟶ x1 (pack_b x2 x3)) ⟶ x1 x0
Theorem pack_struct_b_I^{pack_struct_b_I} : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ∈ x0) ⟶ struct_b (pack_b x0 x1) (proof)
Theorem pack_struct_b_E1^{pack_struct_b_E1} : ∀ x0 . ∀ x1 : ι → ι → ι . struct_b (pack_b x0 x1) ⟶ ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ∈ x0 (proof)
Theorem struct_b_eta^struct_b_eta : ∀ x0 . struct_b x0 ⟶ x0 = pack_b (ap x0 0) (decode_b (ap x0 1)) (proof)
Definition unpack_b_i^unpack_b_i := λ x0 . λ x1 : ι → (ι → ι → ι) → ι . x1 (ap x0 0) (decode_b (ap x0 1))
Theorem unpack_b_i_eq^{unpack_b_i_eq} : ∀ x0 : ι → (ι → ι → ι) → ι . ∀ x1 . ∀ x2 : ι → ι → ι . (∀ x3 : ι → ι → ι . (∀ x4 . x4 ∈ x1 ⟶ ∀ x5 . x5 ∈ x1 ⟶ x2 x4 x5 = x3 x4 x5) ⟶ x0 x1 x3 = x0 x1 x2) ⟶ unpack_b_i (pack_b x1 x2) x0 = x0 x1 x2 (proof)
Definition unpack_b_o^unpack_b_o := λ x0 . λ x1 : ι → (ι → ι → ι) → ο . x1 (ap x0 0) (decode_b (ap x0 1))
Theorem unpack_b_o_eq^{unpack_b_o_eq} : ∀ x0 : ι → (ι → ι → ι) → ο . ∀ x1 . ∀ x2 : ι → ι → ι . (∀ x3 : ι → ι → ι . (∀ x4 . x4 ∈ x1 ⟶ ∀ x5 . x5 ∈ x1 ⟶ x2 x4 x5 = x3 x4 x5) ⟶ x0 x1 x3 = x0 x1 x2) ⟶ unpack_b_o (pack_b x1 x2) x0 = x0 x1 x2 (proof)
Param Sep^Sep : ι → (ι → ο) → ι
Definition pack_p^pack_p := λ x0 . λ x1 : ι → ο . lam 2 (λ x2 . If_i (x2 = 0) x0 (Sep x0 x1))
Theorem pack_p_0_eq^pack_p_0_eq : ∀ x0 x1 . ∀ x2 : ι → ο . x0 = pack_p x1 x2 ⟶ x1 = ap x0 0 (proof)
Theorem pack_p_0_eq2^pack_p_0_eq2 : ∀ x0 . ∀ x1 : ι → ο . x0 = ap (pack_p x0 x1) 0 (proof)
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_p_1_eq^pack_p_1_eq : ∀ x0 x1 . ∀ x2 : ι → ο . x0 = pack_p x1 x2 ⟶ ∀ x3 . x3 ∈ x1 ⟶ x2 x3 = decode_p (ap x0 1) x3 (proof)
Theorem pack_p_1_eq2^pack_p_1_eq2 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 = decode_p (ap (pack_p x0 x1) 1) x2 (proof)
Theorem pack_p_inj^pack_p_inj : ∀ x0 x1 . ∀ x2 x3 : ι → ο . pack_p x0 x2 = pack_p x1 x3 ⟶ and (x0 = x1) (∀ x4 . x4 ∈ x0 ⟶ x2 x4 = x3 x4) (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
Theorem pack_p_ext^pack_p_ext : ∀ x0 . ∀ x1 x2 : ι → ο . (∀ x3 . x3 ∈ x0 ⟶ iff (x1 x3) (x2 x3)) ⟶ pack_p x0 x1 = pack_p x0 x2 (proof)
Definition struct_p^struct_p := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ο . x1 (pack_p x2 x3)) ⟶ x1 x0
Theorem pack_struct_p_I^{pack_struct_p_I} : ∀ x0 . ∀ x1 : ι → ο . struct_p (pack_p x0 x1) (proof)
Known iff_refl^iff_refl : ∀ x0 : ο . iff x0 x0
Theorem struct_p_eta^struct_p_eta : ∀ x0 . struct_p x0 ⟶ x0 = pack_p (ap x0 0) (decode_p (ap x0 1)) (proof)
Definition unpack_p_i^unpack_p_i := λ x0 . λ x1 : ι → (ι → ο) → ι . x1 (ap x0 0) (decode_p (ap x0 1))
Theorem unpack_p_i_eq^{unpack_p_i_eq} : ∀ x0 : ι → (ι → ο) → ι . ∀ x1 . ∀ x2 : ι → ο . (∀ x3 : ι → ο . (∀ x4 . x4 ∈ x1 ⟶ iff (x2 x4) (x3 x4)) ⟶ x0 x1 x3 = x0 x1 x2) ⟶ unpack_p_i (pack_p x1 x2) x0 = x0 x1 x2 (proof)
Definition unpack_p_o^unpack_p_o := λ x0 . λ x1 : ι → (ι → ο) → ο . x1 (ap x0 0) (decode_p (ap x0 1))
Theorem unpack_p_o_eq^{unpack_p_o_eq} : ∀ x0 : ι → (ι → ο) → ο . ∀ x1 . ∀ x2 : ι → ο . (∀ x3 : ι → ο . (∀ x4 . x4 ∈ x1 ⟶ iff (x2 x4) (x3 x4)) ⟶ x0 x1 x3 = x0 x1 x2) ⟶ unpack_p_o (pack_p x1 x2) x0 = x0 x1 x2 (proof)
Param encode_r^encode_r : ι → (ι → ι → ο) → ι
Definition pack_r^pack_r := λ x0 . λ x1 : ι → ι → ο . lam 2 (λ x2 . If_i (x2 = 0) x0 (encode_r x0 x1))
Theorem pack_r_0_eq^pack_r_0_eq : ∀ x0 x1 . ∀ x2 : ι → ι → ο . x0 = pack_r x1 x2 ⟶ x1 = ap x0 0 (proof)
Theorem pack_r_0_eq2^pack_r_0_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι → ο . x2 x0 (ap (pack_r x0 x1) 0) ⟶ x2 (ap (pack_r x0 x1) 0) x0 (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_r_1_eq^pack_r_1_eq : ∀ x0 x1 . ∀ x2 : ι → ι → ο . x0 = pack_r x1 x2 ⟶ ∀ x3 . x3 ∈ x1 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x3 x4 = decode_r (ap x0 1) x3 x4 (proof)
Theorem pack_r_1_eq2^pack_r_1_eq2 : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 = decode_r (ap (pack_r x0 x1) 1) x2 x3 (proof)
Theorem pack_r_inj^pack_r_inj : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . pack_r x0 x2 = pack_r x1 x3 ⟶ and (x0 = x1) (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 x5 = x3 x4 x5) (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
Theorem pack_r_ext^pack_r_ext : ∀ x0 . ∀ x1 x2 : ι → ι → ο . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ iff (x1 x3 x4) (x2 x3 x4)) ⟶ pack_r x0 x1 = pack_r x0 x2 (proof)
Definition struct_r^struct_r := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ο . x1 (pack_r x2 x3)) ⟶ x1 x0
Theorem pack_struct_r_I^{pack_struct_r_I} : ∀ x0 . ∀ x1 : ι → ι → ο . struct_r (pack_r x0 x1) (proof)
Theorem struct_r_eta^struct_r_eta : ∀ x0 . struct_r x0 ⟶ x0 = pack_r (ap x0 0) (decode_r (ap x0 1)) (proof)
Definition unpack_r_i^unpack_r_i := λ x0 . λ x1 : ι → (ι → ι → ο) → ι . x1 (ap x0 0) (decode_r (ap x0 1))
Theorem unpack_r_i_eq^{unpack_r_i_eq} : ∀ x0 : ι → (ι → ι → ο) → ι . ∀ x1 . ∀ x2 : ι → ι → ο . (∀ x3 : ι → ι → ο . (∀ x4 . x4 ∈ x1 ⟶ ∀ x5 . x5 ∈ x1 ⟶ iff (x2 x4 x5) (x3 x4 x5)) ⟶ x0 x1 x3 = x0 x1 x2) ⟶ unpack_r_i (pack_r x1 x2) x0 = x0 x1 x2 (proof)
Definition unpack_r_o^unpack_r_o := λ x0 . λ x1 : ι → (ι → ι → ο) → ο . x1 (ap x0 0) (decode_r (ap x0 1))
Theorem unpack_r_o_eq^{unpack_r_o_eq} : ∀ x0 : ι → (ι → ι → ο) → ο . ∀ x1 . ∀ x2 : ι → ι → ο . (∀ x3 : ι → ι → ο . (∀ x4 . x4 ∈ x1 ⟶ ∀ x5 . x5 ∈ x1 ⟶ iff (x2 x4 x5) (x3 x4 x5)) ⟶ x0 x1 x3 = x0 x1 x2) ⟶ unpack_r_o (pack_r x1 x2) x0 = x0 x1 x2 (proof)
Param encode_c^encode_c : ι → ((ι → ο) → ο) → ι
Definition pack_c^pack_c := λ x0 . λ x1 : (ι → ο) → ο . lam 2 (λ x2 . If_i (x2 = 0) x0 (encode_c x0 x1))
Theorem pack_c_0_eq^pack_c_0_eq : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . x0 = pack_c x1 x2 ⟶ x1 = ap x0 0 (proof)
Theorem pack_c_0_eq2^pack_c_0_eq2 : ∀ x0 . ∀ x1 : (ι → ο) → ο . x0 = ap (pack_c x0 x1) 0 (proof)
Param decode_c^decode_c : ι → (ι → ο) → ο
Known decode_encode_c^{decode_encode_c} : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ο . (∀ x3 . x2 x3 ⟶ x3 ∈ x0) ⟶ decode_c (encode_c x0 x1) x2 = x1 x2
Theorem pack_c_1_eq^pack_c_1_eq : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . x0 = pack_c x1 x2 ⟶ ∀ x3 : ι → ο . (∀ x4 . x3 x4 ⟶ x4 ∈ x1) ⟶ x2 x3 = decode_c (ap x0 1) x3 (proof)
Theorem pack_c_1_eq2^pack_c_1_eq2 : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ο . (∀ x3 . x2 x3 ⟶ x3 ∈ x0) ⟶ x1 x2 = decode_c (ap (pack_c x0 x1) 1) x2 (proof)
Theorem pack_c_inj^pack_c_inj : ∀ x0 x1 . ∀ x2 x3 : (ι → ο) → ο . pack_c x0 x2 = pack_c x1 x3 ⟶ and (x0 = x1) (∀ x4 : ι → ο . (∀ x5 . x4 x5 ⟶ x5 ∈ x0) ⟶ x2 x4 = x3 x4) (proof)
Known encode_c_ext^encode_c_ext : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . (∀ x3 : ι → ο . (∀ x4 . x3 x4 ⟶ x4 ∈ x0) ⟶ iff (x1 x3) (x2 x3)) ⟶ encode_c x0 x1 = encode_c x0 x2
Theorem pack_c_ext^pack_c_ext : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . (∀ x3 : ι → ο . (∀ x4 . x3 x4 ⟶ x4 ∈ x0) ⟶ iff (x1 x3) (x2 x3)) ⟶ pack_c x0 x1 = pack_c x0 x2 (proof)
Definition struct_c^struct_c := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : (ι → ο) → ο . x1 (pack_c x2 x3)) ⟶ x1 x0
Theorem pack_struct_c_I^{pack_struct_c_I} : ∀ x0 . ∀ x1 : (ι → ο) → ο . struct_c (pack_c x0 x1) (proof)
Theorem struct_c_eta^struct_c_eta : ∀ x0 . struct_c x0 ⟶ x0 = pack_c (ap x0 0) (decode_c (ap x0 1)) (proof)
Definition unpack_c_i^unpack_c_i := λ x0 . λ x1 : ι → ((ι → ο) → ο) → ι . x1 (ap x0 0) (decode_c (ap x0 1))
Theorem unpack_c_i_eq^{unpack_c_i_eq} : ∀ x0 : ι → ((ι → ο) → ο) → ι . ∀ x1 . ∀ x2 : (ι → ο) → ο . (∀ x3 : (ι → ο) → ο . (∀ x4 : ι → ο . (∀ x5 . x4 x5 ⟶ x5 ∈ x1) ⟶ iff (x2 x4) (x3 x4)) ⟶ x0 x1 x3 = x0 x1 x2) ⟶ unpack_c_i (pack_c x1 x2) x0 = x0 x1 x2 (proof)
Definition unpack_c_o^unpack_c_o := λ x0 . λ x1 : ι → ((ι → ο) → ο) → ο . x1 (ap x0 0) (decode_c (ap x0 1))
Theorem unpack_c_o_eq^{unpack_c_o_eq} : ∀ x0 : ι → ((ι → ο) → ο) → ο . ∀ x1 . ∀ x2 : (ι → ο) → ο . (∀ x3 : (ι → ο) → ο . (∀ x4 : ι → ο . (∀ x5 . x4 x5 ⟶ x5 ∈ x1) ⟶ iff (x2 x4) (x3 x4)) ⟶ x0 x1 x3 = x0 x1 x2) ⟶ unpack_c_o (pack_c x1 x2) x0 = x0 x1 x2 (proof)