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
Param Sep^Sep : ι → (ι → ο) → ι
Definition pack_r_p_e := λ x0 . λ x1 : ι → ι → ο . λ x2 : ι → ο . λ x3 . lam 4 (λ x4 . If_i (x4 = 0) x0 (If_i (x4 = 1) (encode_r x0 x1) (If_i (x4 = 2) (Sep x0 x2) x3)))
Param ap^ap : ι → ι → ι
Known tuple_4_0_eq^tuple_4_0_eq : ∀ x0 x1 x2 x3 . ap (lam 4 (λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) x1 (If_i (x5 = 2) x2 x3)))) 0 = x0
Theorem pack_r_p_e_0_eq : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 . x0 = pack_r_p_e x1 x2 x3 x4 ⟶ x1 = ap x0 0 (proof)
Theorem pack_r_p_e_0_eq2 : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 . x0 = ap (pack_r_p_e x0 x1 x2 x3) 0 (proof)
Param decode_r^decode_r : ι → ι → ι → ο
Known tuple_4_1_eq^tuple_4_1_eq : ∀ x0 x1 x2 x3 . ap (lam 4 (λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) x1 (If_i (x5 = 2) x2 x3)))) 1 = x1
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_p_e_1_eq : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 . x0 = pack_r_p_e x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x1 ⟶ ∀ x6 . x6 ∈ x1 ⟶ x2 x5 x6 = decode_r (ap x0 1) x5 x6 (proof)
Theorem pack_r_p_e_1_eq2 : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x1 x4 x5 = decode_r (ap (pack_r_p_e x0 x1 x2 x3) 1) x4 x5 (proof)
Param decode_p^decode_p : ι → ι → ο
Known tuple_4_2_eq^tuple_4_2_eq : ∀ x0 x1 x2 x3 . ap (lam 4 (λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) x1 (If_i (x5 = 2) x2 x3)))) 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_r_p_e_2_eq : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 . x0 = pack_r_p_e x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x1 ⟶ x3 x5 = decode_p (ap x0 2) x5 (proof)
Theorem pack_r_p_e_2_eq2 : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 x4 . x4 ∈ x0 ⟶ x2 x4 = decode_p (ap (pack_r_p_e x0 x1 x2 x3) 2) x4 (proof)
Known tuple_4_3_eq^tuple_4_3_eq : ∀ x0 x1 x2 x3 . ap (lam 4 (λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) x1 (If_i (x5 = 2) x2 x3)))) 3 = x3
Theorem pack_r_p_e_3_eq : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 . x0 = pack_r_p_e x1 x2 x3 x4 ⟶ x4 = ap x0 3 (proof)
Theorem pack_r_p_e_3_eq2 : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 . x3 = ap (pack_r_p_e x0 x1 x2 x3) 3 (proof)
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known and4I^and4I : ∀ x0 x1 x2 x3 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ and (and (and x0 x1) x2) x3
Theorem pack_r_p_e_inj : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 x5 : ι → ο . ∀ x6 x7 . pack_r_p_e x0 x2 x4 x6 = pack_r_p_e x1 x3 x5 x7 ⟶ and (and (and (x0 = x1) (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x2 x8 x9 = x3 x8 x9)) (∀ x8 . x8 ∈ x0 ⟶ x4 x8 = x5 x8)) (x6 = x7) (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_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_p_e_ext : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 x4 : ι → ο . ∀ x5 . (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ iff (x1 x6 x7) (x2 x6 x7)) ⟶ (∀ x6 . x6 ∈ x0 ⟶ iff (x3 x6) (x4 x6)) ⟶ pack_r_p_e x0 x1 x3 x5 = pack_r_p_e x0 x2 x4 x5 (proof)
Definition struct_r_p_e := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . x5 ∈ x2 ⟶ x1 (pack_r_p_e x2 x3 x4 x5)) ⟶ x1 x0
Theorem pack_struct_r_p_e_I : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 . x3 ∈ x0 ⟶ struct_r_p_e (pack_r_p_e x0 x1 x2 x3) (proof)
Theorem pack_struct_r_p_e_E3 : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 . struct_r_p_e (pack_r_p_e x0 x1 x2 x3) ⟶ x3 ∈ x0 (proof)
Known iff_refl^iff_refl : ∀ x0 : ο . iff x0 x0
Theorem struct_r_p_e_eta : ∀ x0 . struct_r_p_e x0 ⟶ x0 = pack_r_p_e (ap x0 0) (decode_r (ap x0 1)) (decode_p (ap x0 2)) (ap x0 3) (proof)
Definition unpack_r_p_e_i := λ x0 . λ x1 : ι → (ι → ι → ο) → (ι → ο) → ι → ι . x1 (ap x0 0) (decode_r (ap x0 1)) (decode_p (ap x0 2)) (ap x0 3)
Theorem unpack_r_p_e_i_eq : ∀ x0 : ι → (ι → ι → ο) → (ι → ο) → ι → ι . ∀ x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 . (∀ x5 : ι → ι → ο . (∀ x6 . x6 ∈ x1 ⟶ ∀ x7 . x7 ∈ x1 ⟶ iff (x2 x6 x7) (x5 x6 x7)) ⟶ ∀ x6 : ι → ο . (∀ x7 . x7 ∈ x1 ⟶ iff (x3 x7) (x6 x7)) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4) ⟶ unpack_r_p_e_i (pack_r_p_e x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition unpack_r_p_e_o := λ x0 . λ x1 : ι → (ι → ι → ο) → (ι → ο) → ι → ο . x1 (ap x0 0) (decode_r (ap x0 1)) (decode_p (ap x0 2)) (ap x0 3)
Theorem unpack_r_p_e_o_eq : ∀ x0 : ι → (ι → ι → ο) → (ι → ο) → ι → ο . ∀ x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 . (∀ x5 : ι → ι → ο . (∀ x6 . x6 ∈ x1 ⟶ ∀ x7 . x7 ∈ x1 ⟶ iff (x2 x6 x7) (x5 x6 x7)) ⟶ ∀ x6 : ι → ο . (∀ x7 . x7 ∈ x1 ⟶ iff (x3 x7) (x6 x7)) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4) ⟶ unpack_r_p_e_o (pack_r_p_e x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition pack_p_p_e := λ x0 . λ x1 x2 : ι → ο . λ x3 . lam 4 (λ x4 . If_i (x4 = 0) x0 (If_i (x4 = 1) (Sep x0 x1) (If_i (x4 = 2) (Sep x0 x2) x3)))
Theorem pack_p_p_e_0_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ο . ∀ x4 . x0 = pack_p_p_e x1 x2 x3 x4 ⟶ x1 = ap x0 0 (proof)
Theorem pack_p_p_e_0_eq2 : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 . x0 = ap (pack_p_p_e x0 x1 x2 x3) 0 (proof)
Theorem pack_p_p_e_1_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ο . ∀ x4 . x0 = pack_p_p_e x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x1 ⟶ x2 x5 = decode_p (ap x0 1) x5 (proof)
Theorem pack_p_p_e_1_eq2 : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 x4 . x4 ∈ x0 ⟶ x1 x4 = decode_p (ap (pack_p_p_e x0 x1 x2 x3) 1) x4 (proof)
Theorem pack_p_p_e_2_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ο . ∀ x4 . x0 = pack_p_p_e x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x1 ⟶ x3 x5 = decode_p (ap x0 2) x5 (proof)
Theorem pack_p_p_e_2_eq2 : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 x4 . x4 ∈ x0 ⟶ x2 x4 = decode_p (ap (pack_p_p_e x0 x1 x2 x3) 2) x4 (proof)
Theorem pack_p_p_e_3_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ο . ∀ x4 . x0 = pack_p_p_e x1 x2 x3 x4 ⟶ x4 = ap x0 3 (proof)
Theorem pack_p_p_e_3_eq2 : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 . x3 = ap (pack_p_p_e x0 x1 x2 x3) 3 (proof)
Theorem pack_p_p_e_inj : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ο . ∀ x6 x7 . pack_p_p_e x0 x2 x4 x6 = pack_p_p_e x1 x3 x5 x7 ⟶ and (and (and (x0 = x1) (∀ x8 . x8 ∈ x0 ⟶ x2 x8 = x3 x8)) (∀ x8 . x8 ∈ x0 ⟶ x4 x8 = x5 x8)) (x6 = x7) (proof)
Theorem pack_p_p_e_ext : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ο . ∀ x5 . (∀ x6 . x6 ∈ x0 ⟶ iff (x1 x6) (x2 x6)) ⟶ (∀ x6 . x6 ∈ x0 ⟶ iff (x3 x6) (x4 x6)) ⟶ pack_p_p_e x0 x1 x3 x5 = pack_p_p_e x0 x2 x4 x5 (proof)
Definition struct_p_p_e := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 x4 : ι → ο . ∀ x5 . x5 ∈ x2 ⟶ x1 (pack_p_p_e x2 x3 x4 x5)) ⟶ x1 x0
Theorem pack_struct_p_p_e_I : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 . x3 ∈ x0 ⟶ struct_p_p_e (pack_p_p_e x0 x1 x2 x3) (proof)
Theorem pack_struct_p_p_e_E3 : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 . struct_p_p_e (pack_p_p_e x0 x1 x2 x3) ⟶ x3 ∈ x0 (proof)
Theorem struct_p_p_e_eta : ∀ x0 . struct_p_p_e x0 ⟶ x0 = pack_p_p_e (ap x0 0) (decode_p (ap x0 1)) (decode_p (ap x0 2)) (ap x0 3) (proof)
Definition unpack_p_p_e_i := λ x0 . λ x1 : ι → (ι → ο) → (ι → ο) → ι → ι . x1 (ap x0 0) (decode_p (ap x0 1)) (decode_p (ap x0 2)) (ap x0 3)
Theorem unpack_p_p_e_i_eq : ∀ x0 : ι → (ι → ο) → (ι → ο) → ι → ι . ∀ x1 . ∀ x2 x3 : ι → ο . ∀ x4 . (∀ x5 : ι → ο . (∀ x6 . x6 ∈ x1 ⟶ iff (x2 x6) (x5 x6)) ⟶ ∀ x6 : ι → ο . (∀ x7 . x7 ∈ x1 ⟶ iff (x3 x7) (x6 x7)) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4) ⟶ unpack_p_p_e_i (pack_p_p_e x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition unpack_p_p_e_o := λ x0 . λ x1 : ι → (ι → ο) → (ι → ο) → ι → ο . x1 (ap x0 0) (decode_p (ap x0 1)) (decode_p (ap x0 2)) (ap x0 3)
Theorem unpack_p_p_e_o_eq : ∀ x0 : ι → (ι → ο) → (ι → ο) → ι → ο . ∀ x1 . ∀ x2 x3 : ι → ο . ∀ x4 . (∀ x5 : ι → ο . (∀ x6 . x6 ∈ x1 ⟶ iff (x2 x6) (x5 x6)) ⟶ ∀ x6 : ι → ο . (∀ x7 . x7 ∈ x1 ⟶ iff (x3 x7) (x6 x7)) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4) ⟶ unpack_p_p_e_o (pack_p_p_e x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)