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Proofgold Asset

asset id
ce6c01c126cefcf92c3c13d5f027ee8477622d986bb02bba9e875938bfa0cef4
asset hash
32f4aabf0c22e4d00ca880536c62ff0d96f1bf34d41a0b59c4ef1fe839c45336
bday / block
4914
tx
a8341..
preasset
doc published by Pr6Pc..
Param lamSigma : ι(ιι) → ι
Param ordsuccordsucc : ιι
Param If_iIf_i : οιιι
Param SepSep : ι(ιο) → 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 apap : ιιι
Known tuple_3_0_eqtuple_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 x3x1 = 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_pdecode_p : ιιο
Known tuple_3_1_eqtuple_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_pdecode_encode_p : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2x0decode_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 . x4x1x2 x4 = decode_p (ap x0 1) x4 (proof)
Theorem pack_p_p_1_eq2 : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 . x3x0x1 x3 = decode_p (ap (pack_p_p x0 x1 x2) 1) x3 (proof)
Known tuple_3_2_eqtuple_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 . x4x1x3 x4 = decode_p (ap x0 2) x4 (proof)
Theorem pack_p_p_2_eq2 : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 . x3x0x2 x3 = decode_p (ap (pack_p_p x0 x1 x2) 2) x3 (proof)
Definition andand := λ x0 x1 : ο . ∀ x2 : ο . (x0x1x2)x2
Known and3Iand3I : ∀ x0 x1 x2 : ο . x0x1x2and (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 x5and (and (x0 = x1) (∀ x6 . x6x0x2 x6 = x3 x6)) (∀ x6 . x6x0x4 x6 = x5 x6) (proof)
Param iffiff : οοο
Known encode_p_extencode_p_ext : ∀ x0 . ∀ x1 x2 : ι → ο . (∀ x3 . x3x0iff (x1 x3) (x2 x3))Sep x0 x1 = Sep x0 x2
Theorem pack_p_p_ext : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ο . (∀ x5 . x5x0iff (x1 x5) (x2 x5))(∀ x5 . x5x0iff (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_refliff_refl : ∀ x0 : ο . iff x0 x0
Theorem struct_p_p_eta : ∀ x0 . struct_p_p x0x0 = 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 . x5x1iff (x2 x5) (x4 x5))∀ x5 : ι → ο . (∀ x6 . x6x1iff (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 . x5x1iff (x2 x5) (x4 x5))∀ x5 : ι → ο . (∀ x6 . x6x1iff (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 x3x1 = 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 . x4x1x2 x4 = decode_p (ap x0 1) x4 (proof)
Theorem pack_p_e_1_eq2 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 x3 . x3x0x1 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 x3x3 = 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 x5and (and (x0 = x1) (∀ x6 . x6x0x2 x6 = x3 x6)) (x4 = x5) (proof)
Theorem pack_p_e_ext : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 . (∀ x4 . x4x0iff (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 . x4x2x1 (pack_p_e x2 x3 x4))x1 x0
Theorem pack_struct_p_e_I : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2x0struct_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)x2x0 (proof)
Theorem struct_p_e_eta : ∀ x0 . struct_p_e x0x0 = 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 . x5x1iff (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 . x5x1iff (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 x3x1 = 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 x3x2 = 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 x3x3 = 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 x5and (and (x0 = x1) (x2 = x3)) (x4 = x5) (proof)
Definition struct_e_e := λ x0 . ∀ x1 : ι → ο . (∀ x2 x3 . x3x2∀ x4 . x4x2x1 (pack_e_e x2 x3 x4))x1 x0
Theorem pack_struct_e_e_I : ∀ x0 x1 . x1x0∀ x2 . x2x0struct_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)x1x0 (proof)
Theorem pack_struct_e_e_E2 : ∀ x0 x1 x2 . struct_e_e (pack_e_e x0 x1 x2)x2x0 (proof)
Theorem struct_e_e_eta : ∀ x0 . struct_e_e x0x0 = 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)