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Proofgold Asset

asset id
7d134de85dd18d0a263492a4e254e65c89384a9a269a90cce446e5a01a7f8918
asset hash
1e6d750dc62abfe4e53626f0e046cc2e77b86d56341b1222e747b7609e04ceec
bday / block
4079
tx
74c65..
preasset
doc published by PrGxv..
Param ordinal : ιο
Definition or := λ x0 x1 : ο . ∀ x2 : ο . (x0x2)(x1x2)x2
Known ordinal_trichotomy_or : ∀ x0 x1 . ordinal x0ordinal x1or (or (prim1 x0 x1) (x0 = x1)) (prim1 x1 x0)
Theorem ordinal_trichotomy_or_impred : ∀ x0 x1 . ordinal x0ordinal x1∀ x2 : ο . (prim1 x0 x1x2)(x0 = x1x2)(prim1 x1 x0x2)x2 (proof)
Param 80242.. : ιο
Param 099f3.. : ιιο
Param 23e07.. : ιι
Param 5246e.. : ιι
Param e4431.. : ιι
Param d3786.. : ιιι
Param SNoEq_ : ιιιο
Param nIn : ιιο
Known 36ff9.. : ∀ x0 x1 . 80242.. x080242.. x1099f3.. x0 x1∀ x2 : ο . (∀ x3 . 80242.. x3prim1 (e4431.. x3) (d3786.. (e4431.. x0) (e4431.. x1))SNoEq_ (e4431.. x3) x3 x0SNoEq_ (e4431.. x3) x3 x1099f3.. x0 x3099f3.. x3 x1nIn (e4431.. x3) x0prim1 (e4431.. x3) x1x2)(prim1 (e4431.. x0) (e4431.. x1)SNoEq_ (e4431.. x0) x0 x1prim1 (e4431.. x0) x1x2)(prim1 (e4431.. x1) (e4431.. x0)SNoEq_ (e4431.. x1) x0 x1nIn (e4431.. x1) x0x2)x2
Definition and := λ x0 x1 : ο . ∀ x2 : ο . (x0x1x2)x2
Known 3eff2.. : ∀ x0 x1 x2 . prim1 x2 (d3786.. x0 x1)and (prim1 x2 x0) (prim1 x2 x1)
Known f5194.. : ∀ x0 . 80242.. x0∀ x1 . 80242.. x1prim1 (e4431.. x1) (e4431.. x0)099f3.. x1 x0prim1 x1 (23e07.. x0)
Known 18a76.. : ∀ x0 . 80242.. x0∀ x1 . 80242.. x1prim1 (e4431.. x1) (e4431.. x0)099f3.. x0 x1prim1 x1 (5246e.. x0)
Theorem 6f4d6.. : ∀ x0 x1 . 80242.. x080242.. x1099f3.. x0 x1∀ x2 : ο . (∀ x3 . prim1 x3 (23e07.. x1)prim1 x3 (5246e.. x0)x2)(prim1 x0 (23e07.. x1)x2)(prim1 x1 (5246e.. x0)x2)x2 (proof)
Known bbdbf.. : ∀ x0 x1 . 80242.. x080242.. x1∀ x2 : ο . (099f3.. x0 x1x2)(x0 = x1x2)(099f3.. x1 x0x2)x2
Theorem 7abb3.. : ∀ x0 x1 . 80242.. x080242.. x1∀ x2 : ο . (x0 = x1x2)(∀ x3 . prim1 x3 (23e07.. x1)prim1 x3 (5246e.. x0)x2)(prim1 x0 (23e07.. x1)x2)(prim1 x1 (5246e.. x0)x2)(∀ x3 . prim1 x3 (5246e.. x1)prim1 x3 (23e07.. x0)x2)(prim1 x0 (5246e.. x1)x2)(prim1 x1 (23e07.. x0)x2)x2 (proof)
Param bc82c.. : ιιι
Param f4dc0.. : ιι
Known 368c5.. : ∀ x0 x1 x2 . 80242.. x080242.. x180242.. x2bc82c.. x0 (bc82c.. x1 x2) = bc82c.. (bc82c.. x0 x1) x2
Known 706f7.. : ∀ x0 . 80242.. x080242.. (f4dc0.. x0)
Param 4a7ef.. : ι
Known b5313.. : ∀ x0 . 80242.. x0bc82c.. x0 (f4dc0.. x0) = 4a7ef..
Known 97325.. : ∀ x0 . 80242.. x0bc82c.. x0 4a7ef.. = x0
Theorem e7471.. : ∀ x0 x1 . 80242.. x080242.. x1bc82c.. (bc82c.. x0 x1) (f4dc0.. x1) = x0 (proof)
Known 8782d.. : ∀ x0 x1 x2 . 80242.. x080242.. x180242.. x2099f3.. x0 x2099f3.. (bc82c.. x0 x1) (bc82c.. x2 x1)
Known b71d0.. : ∀ x0 x1 . 80242.. x080242.. x180242.. (bc82c.. x0 x1)
Theorem 08d14.. : ∀ x0 x1 x2 . 80242.. x080242.. x180242.. x2099f3.. (bc82c.. x0 x1) (bc82c.. x2 x1)099f3.. x0 x2 (proof)
Theorem f24af.. : ∀ x0 x1 x2 x3 . 80242.. x080242.. x180242.. x280242.. x3bc82c.. x0 (bc82c.. x1 (bc82c.. x2 x3)) = bc82c.. (bc82c.. x0 (bc82c.. x1 x2)) x3 (proof)
Known f3bd7.. : ∀ x0 x1 . 80242.. x080242.. x1bc82c.. x0 x1 = bc82c.. x1 x0
Theorem eb5ba.. : ∀ x0 x1 x2 . 80242.. x080242.. x180242.. x2bc82c.. x0 (bc82c.. x1 x2) = bc82c.. x1 (bc82c.. x0 x2) (proof)
Theorem 1906a.. : ∀ x0 x1 x2 x3 . 80242.. x180242.. x280242.. x3bc82c.. x0 (bc82c.. x1 (bc82c.. x2 x3)) = bc82c.. x0 (bc82c.. x2 (bc82c.. x1 x3)) (proof)
Theorem cba25.. : ∀ x0 x1 x2 . 80242.. x080242.. x180242.. x2bc82c.. (bc82c.. x0 x1) x2 = bc82c.. (bc82c.. x0 x2) x1 (proof)
Theorem 40681.. : ∀ x0 x1 x2 x3 . 80242.. x080242.. x180242.. x280242.. x3bc82c.. (bc82c.. x0 x1) (bc82c.. x2 x3) = bc82c.. (bc82c.. x0 x2) (bc82c.. x1 x3) (proof)
Theorem 5144e.. : ∀ x0 x1 x2 . 80242.. x080242.. x180242.. x2bc82c.. x0 (bc82c.. x1 x2) = bc82c.. x2 (bc82c.. x0 x1) (proof)
Known 5c481.. : ∀ x0 . 80242.. x0bc82c.. (f4dc0.. x0) x0 = 4a7ef..
Known 85a07.. : ∀ x0 . 80242.. x0bc82c.. 4a7ef.. x0 = x0
Theorem 5d9d4.. : ∀ x0 x1 . 80242.. x080242.. x1bc82c.. (f4dc0.. x0) (bc82c.. x0 x1) = x1 (proof)
Theorem baee2.. : ∀ x0 x1 . 80242.. x080242.. x1bc82c.. x0 (bc82c.. (f4dc0.. x0) x1) = x1 (proof)
Theorem 65db8.. : ∀ x0 x1 x2 x3 . 80242.. x080242.. x180242.. x280242.. x3bc82c.. (bc82c.. x0 (bc82c.. x1 x2)) (bc82c.. (f4dc0.. x2) x3) = bc82c.. x0 (bc82c.. x1 x3) (proof)
Theorem 2a18c.. : ∀ x0 x1 x2 x3 . 80242.. x080242.. x180242.. x280242.. x3bc82c.. (bc82c.. x0 (bc82c.. x1 x2)) (bc82c.. x3 (f4dc0.. x2)) = bc82c.. x0 (bc82c.. x1 x3) (proof)
Known 8948a.. : ∀ x0 . 80242.. x0f4dc0.. (f4dc0.. x0) = x0
Theorem 8b23b.. : ∀ x0 x1 x2 x3 . 80242.. x080242.. x180242.. x280242.. x3bc82c.. (bc82c.. x0 (bc82c.. x1 (f4dc0.. x2))) (bc82c.. x2 x3) = bc82c.. x0 (bc82c.. x1 x3) (proof)
Theorem 1038f.. : ∀ x0 x1 x2 . 80242.. x080242.. x180242.. x2099f3.. (bc82c.. x0 (f4dc0.. x1)) x2099f3.. x0 (bc82c.. x2 x1) (proof)
Theorem 3350d.. : ∀ x0 x1 x2 . 80242.. x080242.. x180242.. x2099f3.. x2 (bc82c.. x0 (f4dc0.. x1))099f3.. (bc82c.. x2 x1) x0 (proof)
Known ba2a1.. : ∀ x0 x1 x2 x3 . 80242.. x080242.. x180242.. x280242.. x3099f3.. x0 x2099f3.. x1 x3099f3.. (bc82c.. x0 x1) (bc82c.. x2 x3)
Theorem 70f5e.. : ∀ x0 x1 x2 x3 x4 x5 . 80242.. x080242.. x180242.. x280242.. x380242.. x480242.. x5099f3.. (bc82c.. x0 x4) (bc82c.. x2 x5)099f3.. (bc82c.. x1 x5) (bc82c.. x3 x4)099f3.. (bc82c.. x0 x1) (bc82c.. x2 x3) (proof)
Known c7cc7.. : ∀ x0 x1 x2 . 80242.. x080242.. x180242.. x2099f3.. x0 x1099f3.. x1 x2099f3.. x0 x2
Known 831d4.. : ∀ x0 x1 x2 . 80242.. x080242.. x180242.. x2099f3.. x1 x2099f3.. (bc82c.. x0 x1) (bc82c.. x0 x2)
Theorem fa00b.. : ∀ x0 x1 x2 x3 x4 x5 . 80242.. x080242.. x180242.. x280242.. x380242.. x480242.. x5099f3.. (bc82c.. x0 x2) (bc82c.. x3 x5)099f3.. (bc82c.. x1 x5) x4099f3.. (bc82c.. x0 (bc82c.. x1 x2)) (bc82c.. x3 x4) (proof)
Known ebb60.. : 80242.. 4a7ef..
Theorem 073b4.. : ∀ x0 x1 x2 x3 x4 x5 . 80242.. x080242.. x180242.. x280242.. x380242.. x480242.. x5099f3.. (bc82c.. x0 x5) (bc82c.. x2 x4)099f3.. x1 (bc82c.. x5 x3)099f3.. (bc82c.. x0 x1) (bc82c.. x2 (bc82c.. x3 x4)) (proof)
Theorem ad682.. : ∀ x0 x1 x2 x3 x4 x5 x6 x7 . 80242.. x080242.. x180242.. x280242.. x380242.. x480242.. x580242.. x680242.. x7099f3.. (bc82c.. x0 x5) (bc82c.. x6 x7)099f3.. (bc82c.. x1 x7) x4099f3.. (bc82c.. x6 x2) (bc82c.. x3 x5)099f3.. (bc82c.. x0 (bc82c.. x1 x2)) (bc82c.. x3 x4) (proof)
Theorem 4a843.. : ∀ x0 x1 x2 x3 x4 x5 x6 x7 . 80242.. x080242.. x180242.. x280242.. x380242.. x480242.. x580242.. x680242.. x7099f3.. (bc82c.. x0 x5) (bc82c.. x6 x4)099f3.. x1 (bc82c.. x7 x3)099f3.. (bc82c.. x6 x7) (bc82c.. x2 x5)099f3.. (bc82c.. x0 x1) (bc82c.. x2 (bc82c.. x3 x4)) (proof)
Param e6316.. : ιιι
Param 02a50.. : ιιι
Param 0ac37.. : ιιι
Param 94f9e.. : ι(ιι) → ι
Param 0fc90.. : ι(ιι) → ι
Definition ac767.. := λ x0 x1 . 0fc90.. x0 (λ x2 . x1)
Param f482f.. : ιιι
Param 4ae4a.. : ιι
Known edd11.. : ∀ x0 x1 x2 . prim1 x2 (0ac37.. x0 x1)or (prim1 x2 x0) (prim1 x2 x1)
Known 8c6f6.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 (94f9e.. x0 x1)∀ x3 : ο . (∀ x4 . prim1 x4 x0x2 = x1 x4x3)x3
Known 33e74.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 (0fc90.. x0 x1)prim1 (f482f.. x2 4a7ef..) x0
Known 35b50.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 (0fc90.. x0 x1)prim1 (f482f.. x2 (4ae4a.. 4a7ef..)) (x1 (f482f.. x2 4a7ef..))
Known da962.. : ∀ x0 x1 x2 . prim1 x2 x0prim1 x2 (0ac37.. x0 x1)
Param If_i : οιιι
Known 6f2e8.. : ∀ x0 x1 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x3 . If_i (x3 = 4a7ef..) x0 x1)) 4a7ef.. = x0
Known 15d37.. : ∀ x0 x1 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x3 . If_i (x3 = 4a7ef..) x0 x1)) (4ae4a.. 4a7ef..) = x1
Known 696c0.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 x0prim1 (x1 x2) (94f9e.. x0 x1)
Known d5115.. : ∀ x0 x1 x2 . prim1 x2 x0∀ x3 . prim1 x3 x1prim1 (0fc90.. (4ae4a.. (4ae4a.. 4a7ef..)) (λ x4 . If_i (x4 = 4a7ef..) x2 x3)) (ac767.. x0 x1)
Known 287ca.. : ∀ x0 x1 x2 . prim1 x2 x1prim1 x2 (0ac37.. x0 x1)
Known 6381b.. : ∀ x0 . 80242.. x0∀ x1 . 80242.. x1e6316.. x0 x1 = 02a50.. (0ac37.. (94f9e.. (ac767.. (23e07.. x0) (23e07.. x1)) (λ x3 . bc82c.. (e6316.. (f482f.. x3 4a7ef..) x1) (bc82c.. (e6316.. x0 (f482f.. x3 (4ae4a.. 4a7ef..))) (f4dc0.. (e6316.. (f482f.. x3 4a7ef..) (f482f.. x3 (4ae4a.. 4a7ef..))))))) (94f9e.. (ac767.. (5246e.. x0) (5246e.. x1)) (λ x3 . bc82c.. (e6316.. (f482f.. x3 4a7ef..) x1) (bc82c.. (e6316.. x0 (f482f.. x3 (4ae4a.. 4a7ef..))) (f4dc0.. (e6316.. (f482f.. x3 4a7ef..) (f482f.. x3 (4ae4a.. 4a7ef..)))))))) (0ac37.. (94f9e.. (ac767.. (23e07.. x0) (5246e.. x1)) (λ x3 . bc82c.. (e6316.. (f482f.. x3 4a7ef..) x1) (bc82c.. (e6316.. x0 (f482f.. x3 (4ae4a.. 4a7ef..))) (f4dc0.. (e6316.. (f482f.. x3 4a7ef..) (f482f.. x3 (4ae4a.. 4a7ef..))))))) (94f9e.. (ac767.. (5246e.. x0) (23e07.. x1)) (λ x3 . bc82c.. (e6316.. (f482f.. x3 4a7ef..) x1) (bc82c.. (e6316.. x0 (f482f.. x3 (4ae4a.. 4a7ef..))) (f4dc0.. (e6316.. (f482f.. x3 4a7ef..) (f482f.. x3 (4ae4a.. 4a7ef..))))))))
Theorem ca858.. : ∀ x0 x1 . 80242.. x080242.. x1∀ x2 : ο . (∀ x3 x4 . (∀ x5 . prim1 x5 x3∀ x6 : ο . (∀ x7 . prim1 x7 (23e07.. x0)∀ x8 . prim1 x8 (23e07.. x1)x5 = bc82c.. (e6316.. x7 x1) (bc82c.. (e6316.. x0 x8) (f4dc0.. (e6316.. x7 x8)))x6)(∀ x7 . prim1 x7 (5246e.. x0)∀ x8 . prim1 x8 (5246e.. x1)x5 = bc82c.. (e6316.. x7 x1) (bc82c.. (e6316.. x0 x8) (f4dc0.. (e6316.. x7 x8)))x6)x6)(∀ x5 . prim1 x5 (23e07.. x0)∀ x6 . prim1 x6 (23e07.. x1)prim1 (bc82c.. (e6316.. x5 x1) (bc82c.. (e6316.. x0 x6) (f4dc0.. (e6316.. x5 x6)))) x3)(∀ x5 . prim1 x5 (5246e.. x0)∀ x6 . prim1 x6 (5246e.. x1)prim1 (bc82c.. (e6316.. x5 x1) (bc82c.. (e6316.. x0 x6) (f4dc0.. (e6316.. x5 x6)))) x3)(∀ x5 . prim1 x5 x4∀ x6 : ο . (∀ x7 . prim1 x7 (23e07.. x0)∀ x8 . prim1 x8 (5246e.. x1)x5 = bc82c.. (e6316.. x7 x1) (bc82c.. (e6316.. x0 x8) (f4dc0.. (e6316.. x7 x8)))x6)(∀ x7 . prim1 x7 (5246e.. x0)∀ x8 . prim1 x8 (23e07.. x1)x5 = bc82c.. (e6316.. x7 x1) (bc82c.. (e6316.. x0 x8) (f4dc0.. (e6316.. x7 x8)))x6)x6)(∀ x5 . prim1 x5 (23e07.. x0)∀ x6 . prim1 x6 (5246e.. x1)prim1 (bc82c.. (e6316.. x5 x1) (bc82c.. (e6316.. x0 x6) (f4dc0.. (e6316.. x5 x6)))) x4)(∀ x5 . prim1 x5 (5246e.. x0)∀ x6 . prim1 x6 (23e07.. x1)prim1 (bc82c.. (e6316.. x5 x1) (bc82c.. (e6316.. x0 x6) (f4dc0.. (e6316.. x5 x6)))) x4)e6316.. x0 x1 = 02a50.. x3 x4x2)x2 (proof)
Param 56ded.. : ιι
Known 5ccff.. : ∀ x0 : ι → ο . (∀ x1 . 80242.. x1(∀ x2 . prim1 x2 (56ded.. (e4431.. x1))x0 x2)x0 x1)∀ x1 . 80242.. x1x0 x1
Definition 02b90.. := λ x0 x1 . and (and (∀ x2 . prim1 x2 x080242.. x2) (∀ x2 . prim1 x2 x180242.. x2)) (∀ x2 . prim1 x2 x0∀ x3 . prim1 x3 x1099f3.. x2 x3)
Known cbec9.. : ∀ x0 . 80242.. x0∀ x1 . prim1 x1 (23e07.. x0)∀ x2 : ο . (80242.. x1prim1 (e4431.. x1) (e4431.. x0)099f3.. x1 x0x2)x2
Known 0888b.. : ∀ x0 x1 . 02b90.. x0 x1∀ x2 . prim1 x2 x0099f3.. x2 (02a50.. x0 x1)
Known e76d1.. : ∀ x0 . 80242.. x0∀ x1 . prim1 x1 (5246e.. x0)∀ x2 : ο . (80242.. x1prim1 (e4431.. x1) (e4431.. x0)099f3.. x0 x1x2)x2
Known 9c8cc.. : ∀ x0 x1 . 02b90.. x0 x1∀ x2 . prim1 x2 x1099f3.. (02a50.. x0 x1) x2
Known 5a5d4.. : ∀ x0 x1 . 02b90.. x0 x180242.. (02a50.. x0 x1)
Known and3I : ∀ x0 x1 x2 : ο . x0x1x2and (and x0 x1) x2
Known 63df9.. : ∀ x0 . 80242.. x0∀ x1 . prim1 x1 (23e07.. x0)prim1 x1 (56ded.. (e4431.. x0))
Known 54843.. : ∀ x0 . 80242.. x0∀ x1 . prim1 x1 (5246e.. x0)prim1 x1 (56ded.. (e4431.. x0))
Definition Subq := λ x0 x1 . ∀ x2 . prim1 x2 x0prim1 x2 x1
Definition TransSet := λ x0 . ∀ x1 . prim1 x1 x0Subq x1 x0
Known ordinal_TransSet : ∀ x0 . ordinal x0TransSet x0
Known 4c9ee.. : ∀ x0 . 80242.. x0ordinal (e4431.. x0)
Known b50ea.. : ∀ x0 x1 . 80242.. x080242.. x1prim1 (e4431.. x0) (e4431.. x1)prim1 x0 (56ded.. (e4431.. x1))
Param 1beb9.. : ιιο
Known bfaa9.. : ∀ x0 . ordinal x0∀ x1 . prim1 x1 (56ded.. x0)∀ x2 : ο . (prim1 (e4431.. x1) x0ordinal (e4431.. x1)80242.. x11beb9.. (e4431.. x1) x1x2)x2
Theorem ed5b9.. : ∀ x0 . 80242.. x0∀ x1 . 80242.. x1∀ x2 : ο . (80242.. (e6316.. x0 x1)(∀ x3 . prim1 x3 (23e07.. x0)∀ x4 . prim1 x4 (23e07.. x1)099f3.. (bc82c.. (e6316.. x3 x1) (e6316.. x0 x4)) (bc82c.. (e6316.. x0 x1) (e6316.. x3 x4)))(∀ x3 . prim1 x3 (5246e.. x0)∀ x4 . prim1 x4 (5246e.. x1)099f3.. (bc82c.. (e6316.. x3 x1) (e6316.. x0 x4)) (bc82c.. (e6316.. x0 x1) (e6316.. x3 x4)))(∀ x3 . prim1 x3 (23e07.. x0)∀ x4 . prim1 x4 (5246e.. x1)099f3.. (bc82c.. (e6316.. x0 x1) (e6316.. x3 x4)) (bc82c.. (e6316.. x3 x1) (e6316.. x0 x4)))(∀ x3 . prim1 x3 (5246e.. x0)∀ x4 . prim1 x4 (23e07.. x1)099f3.. (bc82c.. (e6316.. x0 x1) (e6316.. x3 x4)) (bc82c.. (e6316.. x3 x1) (e6316.. x0 x4)))x2)x2 (proof)