Proofgold Asset

Param lam^Sigma : ι → (ι → ι) → ι
Param ordsucc^ordsucc : ι → ι
Param If_i^If_i : ο → ι → ι → ι
Param Sep^Sep : ι → (ι → ο) → ι
Definition pack_p_p := λ x0 . λ x1 x2 : ι → ο . lam 3 (λ x3 . If_i (x3 = 0) x0 (If_i (x3 = 1) (Sep x0 x1) (Sep 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_p_p_0_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ο . x0 = pack_p_p x1 x2 x3 ⟶ x1 = ap x0 0 (proof)
Theorem pack_p_p_0_eq2 : ∀ x0 . ∀ x1 x2 : ι → ο . x0 = ap (pack_p_p x0 x1 x2) 0 (proof)
Param decode_p^decode_p : ι → ι → ο
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 decode_encode_p^{decode_encode_p} : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ decode_p (Sep x0 x1) x2 = x1 x2
Theorem pack_p_p_1_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ο . x0 = pack_p_p x1 x2 x3 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x4 = decode_p (ap x0 1) x4 (proof)
Theorem pack_p_p_1_eq2 : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 . x3 ∈ x0 ⟶ x1 x3 = decode_p (ap (pack_p_p x0 x1 x2) 1) x3 (proof)
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
Theorem pack_p_p_2_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ο . x0 = pack_p_p x1 x2 x3 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x3 x4 = decode_p (ap x0 2) x4 (proof)
Theorem pack_p_p_2_eq2 : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 . x3 ∈ x0 ⟶ x2 x3 = decode_p (ap (pack_p_p x0 x1 x2) 2) x3 (proof)
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_p_p_inj : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ο . pack_p_p x0 x2 x4 = pack_p_p x1 x3 x5 ⟶ and (and (x0 = x1) (∀ x6 . x6 ∈ x0 ⟶ x2 x6 = x3 x6)) (∀ x6 . x6 ∈ x0 ⟶ x4 x6 = x5 x6) (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_p_ext : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ο . (∀ x5 . x5 ∈ x0 ⟶ iff (x1 x5) (x2 x5)) ⟶ (∀ x5 . x5 ∈ x0 ⟶ iff (x3 x5) (x4 x5)) ⟶ pack_p_p x0 x1 x3 = pack_p_p x0 x2 x4 (proof)
Definition struct_p_p := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 x4 : ι → ο . x1 (pack_p_p x2 x3 x4)) ⟶ x1 x0
Theorem pack_struct_p_p_I : ∀ x0 . ∀ x1 x2 : ι → ο . struct_p_p (pack_p_p x0 x1 x2) (proof)
Known iff_refl^iff_refl : ∀ x0 : ο . iff x0 x0
Theorem struct_p_p_eta : ∀ x0 . struct_p_p x0 ⟶ x0 = pack_p_p (ap x0 0) (decode_p (ap x0 1)) (decode_p (ap x0 2)) (proof)
Definition unpack_p_p_i := λ x0 . λ x1 : ι → (ι → ο) → (ι → ο) → ι . x1 (ap x0 0) (decode_p (ap x0 1)) (decode_p (ap x0 2))
Theorem unpack_p_p_i_eq : ∀ x0 : ι → (ι → ο) → (ι → ο) → ι . ∀ x1 . ∀ x2 x3 : ι → ο . (∀ x4 : ι → ο . (∀ x5 . x5 ∈ x1 ⟶ iff (x2 x5) (x4 x5)) ⟶ ∀ x5 : ι → ο . (∀ x6 . x6 ∈ x1 ⟶ iff (x3 x6) (x5 x6)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ unpack_p_p_i (pack_p_p x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition unpack_p_p_o := λ x0 . λ x1 : ι → (ι → ο) → (ι → ο) → ο . x1 (ap x0 0) (decode_p (ap x0 1)) (decode_p (ap x0 2))
Theorem unpack_p_p_o_eq : ∀ x0 : ι → (ι → ο) → (ι → ο) → ο . ∀ x1 . ∀ x2 x3 : ι → ο . (∀ x4 : ι → ο . (∀ x5 . x5 ∈ x1 ⟶ iff (x2 x5) (x4 x5)) ⟶ ∀ x5 : ι → ο . (∀ x6 . x6 ∈ x1 ⟶ iff (x3 x6) (x5 x6)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ unpack_p_p_o (pack_p_p x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition pack_p_e := λ x0 . λ x1 : ι → ο . λ x2 . lam 3 (λ x3 . If_i (x3 = 0) x0 (If_i (x3 = 1) (Sep x0 x1) x2))
Theorem pack_p_e_0_eq : ∀ x0 x1 . ∀ x2 : ι → ο . ∀ x3 . x0 = pack_p_e x1 x2 x3 ⟶ x1 = ap x0 0 (proof)
Theorem pack_p_e_0_eq2 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x0 = ap (pack_p_e x0 x1 x2) 0 (proof)
Theorem pack_p_e_1_eq : ∀ x0 x1 . ∀ x2 : ι → ο . ∀ x3 . x0 = pack_p_e x1 x2 x3 ⟶ ∀ x4 . x4 ∈ x1 ⟶ x2 x4 = decode_p (ap x0 1) x4 (proof)
Theorem pack_p_e_1_eq2 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 x3 . x3 ∈ x0 ⟶ x1 x3 = decode_p (ap (pack_p_e x0 x1 x2) 1) x3 (proof)
Theorem pack_p_e_2_eq : ∀ x0 x1 . ∀ x2 : ι → ο . ∀ x3 . x0 = pack_p_e x1 x2 x3 ⟶ x3 = ap x0 2 (proof)
Theorem pack_p_e_2_eq2 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 = ap (pack_p_e x0 x1 x2) 2 (proof)
Theorem pack_p_e_inj : ∀ x0 x1 . ∀ x2 x3 : ι → ο . ∀ x4 x5 . pack_p_e x0 x2 x4 = pack_p_e x1 x3 x5 ⟶ and (and (x0 = x1) (∀ x6 . x6 ∈ x0 ⟶ x2 x6 = x3 x6)) (x4 = x5) (proof)
Theorem pack_p_e_ext : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 . (∀ x4 . x4 ∈ x0 ⟶ iff (x1 x4) (x2 x4)) ⟶ pack_p_e x0 x1 x3 = pack_p_e x0 x2 x3 (proof)
Definition struct_p_e := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ο . ∀ x4 . x4 ∈ x2 ⟶ x1 (pack_p_e x2 x3 x4)) ⟶ x1 x0
Theorem pack_struct_p_e_I : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ struct_p_e (pack_p_e x0 x1 x2) (proof)
Theorem pack_struct_p_e_E2 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . struct_p_e (pack_p_e x0 x1 x2) ⟶ x2 ∈ x0 (proof)
Theorem struct_p_e_eta : ∀ x0 . struct_p_e x0 ⟶ x0 = pack_p_e (ap x0 0) (decode_p (ap x0 1)) (ap x0 2) (proof)
Definition unpack_p_e_i := λ x0 . λ x1 : ι → (ι → ο) → ι → ι . x1 (ap x0 0) (decode_p (ap x0 1)) (ap x0 2)
Theorem unpack_p_e_i_eq : ∀ x0 : ι → (ι → ο) → ι → ι . ∀ x1 . ∀ x2 : ι → ο . ∀ x3 . (∀ x4 : ι → ο . (∀ x5 . x5 ∈ x1 ⟶ iff (x2 x5) (x4 x5)) ⟶ x0 x1 x4 x3 = x0 x1 x2 x3) ⟶ unpack_p_e_i (pack_p_e x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition unpack_p_e_o := λ x0 . λ x1 : ι → (ι → ο) → ι → ο . x1 (ap x0 0) (decode_p (ap x0 1)) (ap x0 2)
Theorem unpack_p_e_o_eq : ∀ x0 : ι → (ι → ο) → ι → ο . ∀ x1 . ∀ x2 : ι → ο . ∀ x3 . (∀ x4 : ι → ο . (∀ x5 . x5 ∈ x1 ⟶ iff (x2 x5) (x4 x5)) ⟶ x0 x1 x4 x3 = x0 x1 x2 x3) ⟶ unpack_p_e_o (pack_p_e x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition pack_e_e := λ x0 x1 x2 . lam 3 (λ x3 . If_i (x3 = 0) x0 (If_i (x3 = 1) x1 x2))
Theorem pack_e_e_0_eq : ∀ x0 x1 x2 x3 . x0 = pack_e_e x1 x2 x3 ⟶ x1 = ap x0 0 (proof)
Theorem pack_e_e_0_eq2 : ∀ x0 x1 x2 . x0 = ap (pack_e_e x0 x1 x2) 0 (proof)
Theorem pack_e_e_1_eq : ∀ x0 x1 x2 x3 . x0 = pack_e_e x1 x2 x3 ⟶ x2 = ap x0 1 (proof)
Theorem pack_e_e_1_eq2 : ∀ x0 x1 x2 . x1 = ap (pack_e_e x0 x1 x2) 1 (proof)
Theorem pack_e_e_2_eq : ∀ x0 x1 x2 x3 . x0 = pack_e_e x1 x2 x3 ⟶ x3 = ap x0 2 (proof)
Theorem pack_e_e_2_eq2 : ∀ x0 x1 x2 . x2 = ap (pack_e_e x0 x1 x2) 2 (proof)
Theorem pack_e_e_inj : ∀ x0 x1 x2 x3 x4 x5 . pack_e_e x0 x2 x4 = pack_e_e x1 x3 x5 ⟶ and (and (x0 = x1) (x2 = x3)) (x4 = x5) (proof)
Definition struct_e_e := λ x0 . ∀ x1 : ι → ο . (∀ x2 x3 . x3 ∈ x2 ⟶ ∀ x4 . x4 ∈ x2 ⟶ x1 (pack_e_e x2 x3 x4)) ⟶ x1 x0
Theorem pack_struct_e_e_I : ∀ x0 x1 . x1 ∈ x0 ⟶ ∀ x2 . x2 ∈ x0 ⟶ struct_e_e (pack_e_e x0 x1 x2) (proof)
Theorem pack_struct_e_e_E1 : ∀ x0 x1 x2 . struct_e_e (pack_e_e x0 x1 x2) ⟶ x1 ∈ x0 (proof)
Theorem pack_struct_e_e_E2 : ∀ x0 x1 x2 . struct_e_e (pack_e_e x0 x1 x2) ⟶ x2 ∈ x0 (proof)
Theorem struct_e_e_eta : ∀ x0 . struct_e_e x0 ⟶ x0 = pack_e_e (ap x0 0) (ap x0 1) (ap x0 2) (proof)
Definition unpack_e_e_i := λ x0 . λ x1 : ι → ι → ι → ι . x1 (ap x0 0) (ap x0 1) (ap x0 2)
Theorem unpack_e_e_i_eq : ∀ x0 : ι → ι → ι → ι . ∀ x1 x2 x3 . unpack_e_e_i (pack_e_e x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition unpack_e_e_o := λ x0 . λ x1 : ι → ι → ι → ο . x1 (ap x0 0) (ap x0 1) (ap x0 2)
Theorem unpack_e_e_o_eq : ∀ x0 : ι → ι → ι → ο . ∀ x1 x2 x3 . unpack_e_e_o (pack_e_e x1 x2 x3) x0 = x0 x1 x2 x3 (proof)