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Proofgold Asset

asset id
9124e30f51559002fdfb1e958f868feac4db76a4ff84e2df4b30d8b27918ae0b
asset hash
d742b1053d289b830e25a8fe6d6afb214f37a83bdb6832ef2791002db06f535f
bday / block
2896
tx
166b2..
preasset
doc published by PrGxv..
Param 0fc90.. : ι(ιι) → ι
Param 4ae4a.. : ιι
Param 4a7ef.. : ι
Param If_i : οιιι
Param e0e40.. : ι((ιο) → ο) → ι
Param eb53d.. : ιCT2 ι
Definition a0d6b.. := λ x0 . λ x1 : (ι → ο) → ο . λ x2 : ι → ι → ι . λ x3 : ι → ι . λ x4 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (eb53d.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (0fc90.. x0 x3) x4))))
Param f482f.. : ιιι
Known 7d2e2.. : ∀ x0 x1 x2 x3 x4 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x6 . If_i (x6 = 4a7ef..) x0 (If_i (x6 = 4ae4a.. 4a7ef..) x1 (If_i (x6 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 (If_i (x6 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4))))) 4a7ef.. = x0
Theorem 97762.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ x5 . x0 = a0d6b.. x1 x2 x3 x4 x5x1 = f482f.. x0 4a7ef.. (proof)
Theorem 13649.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 . x0 = f482f.. (a0d6b.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Param decode_c : ι(ιο) → ο
Known 504a8.. : ∀ x0 x1 x2 x3 x4 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x6 . If_i (x6 = 4a7ef..) x0 (If_i (x6 = 4ae4a.. 4a7ef..) x1 (If_i (x6 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 (If_i (x6 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4))))) (4ae4a.. 4a7ef..) = x1
Known 81500.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ο . (∀ x3 . x2 x3prim1 x3 x0)decode_c (e0e40.. x0 x1) x2 = x1 x2
Theorem 96ff7.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ x5 . x0 = a0d6b.. x1 x2 x3 x4 x5∀ x6 : ι → ο . (∀ x7 . x6 x7prim1 x7 x1)x2 x6 = decode_c (f482f.. x0 (4ae4a.. 4a7ef..)) x6 (proof)
Theorem 6c751.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 . ∀ x5 : ι → ο . (∀ x6 . x5 x6prim1 x6 x0)x1 x5 = decode_c (f482f.. (a0d6b.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 (proof)
Param e3162.. : ιιιι
Known fb20c.. : ∀ x0 x1 x2 x3 x4 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x6 . If_i (x6 = 4a7ef..) x0 (If_i (x6 = 4ae4a.. 4a7ef..) x1 (If_i (x6 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 (If_i (x6 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4))))) (4ae4a.. (4ae4a.. 4a7ef..)) = x2
Known 35054.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . prim1 x2 x0∀ x3 . prim1 x3 x0e3162.. (eb53d.. x0 x1) x2 x3 = x1 x2 x3
Theorem 6f8b9.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ x5 . x0 = a0d6b.. x1 x2 x3 x4 x5∀ x6 . prim1 x6 x1∀ x7 . prim1 x7 x1x3 x6 x7 = e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 x7 (proof)
Theorem 07b63.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 x5 . prim1 x5 x0∀ x6 . prim1 x6 x0x2 x5 x6 = e3162.. (f482f.. (a0d6b.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Known 431f3.. : ∀ x0 x1 x2 x3 x4 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x6 . If_i (x6 = 4a7ef..) x0 (If_i (x6 = 4ae4a.. 4a7ef..) x1 (If_i (x6 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 (If_i (x6 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4))))) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) = x3
Known f22ec.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 x0f482f.. (0fc90.. x0 x1) x2 = x1 x2
Theorem 6901c.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ x5 . x0 = a0d6b.. x1 x2 x3 x4 x5∀ x6 . prim1 x6 x1x4 x6 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 (proof)
Theorem 5777c.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 x5 . prim1 x5 x0x3 x5 = f482f.. (f482f.. (a0d6b.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 (proof)
Known ffdcd.. : ∀ x0 x1 x2 x3 x4 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x6 . If_i (x6 = 4a7ef..) x0 (If_i (x6 = 4ae4a.. 4a7ef..) x1 (If_i (x6 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 (If_i (x6 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4))))) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) = x4
Theorem f447c.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ x5 . x0 = a0d6b.. x1 x2 x3 x4 x5x5 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem 2849e.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 . x4 = f482f.. (a0d6b.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Definition and := λ x0 x1 : ο . ∀ x2 : ο . (x0x1x2)x2
Known and5I : ∀ x0 x1 x2 x3 x4 : ο . x0x1x2x3x4and (and (and (and x0 x1) x2) x3) x4
Theorem 0b40b.. : ∀ x0 x1 . ∀ x2 x3 : (ι → ο) → ο . ∀ x4 x5 : ι → ι → ι . ∀ x6 x7 : ι → ι . ∀ x8 x9 . a0d6b.. x0 x2 x4 x6 x8 = a0d6b.. x1 x3 x5 x7 x9and (and (and (and (x0 = x1) (∀ x10 : ι → ο . (∀ x11 . x10 x11prim1 x11 x0)x2 x10 = x3 x10)) (∀ x10 . prim1 x10 x0∀ x11 . prim1 x11 x0x4 x10 x11 = x5 x10 x11)) (∀ x10 . prim1 x10 x0x6 x10 = x7 x10)) (x8 = x9) (proof)
Param iff : οοο
Known 4402a.. : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . prim1 x3 x0x1 x3 = x2 x3)0fc90.. x0 x1 = 0fc90.. x0 x2
Known 8fdaf.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . (∀ x3 . prim1 x3 x0∀ x4 . prim1 x4 x0x1 x3 x4 = x2 x3 x4)eb53d.. x0 x1 = eb53d.. x0 x2
Known fe043.. : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . (∀ x3 : ι → ο . (∀ x4 . x3 x4prim1 x4 x0)iff (x1 x3) (x2 x3))e0e40.. x0 x1 = e0e40.. x0 x2
Theorem 4222f.. : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . ∀ x3 x4 : ι → ι → ι . ∀ x5 x6 : ι → ι . ∀ x7 . (∀ x8 : ι → ο . (∀ x9 . x8 x9prim1 x9 x0)iff (x1 x8) (x2 x8))(∀ x8 . prim1 x8 x0∀ x9 . prim1 x9 x0x3 x8 x9 = x4 x8 x9)(∀ x8 . prim1 x8 x0x5 x8 = x6 x8)a0d6b.. x0 x1 x3 x5 x7 = a0d6b.. x0 x2 x4 x6 x7 (proof)
Definition 1b748.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : (ι → ο) → ο . ∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x2∀ x6 . prim1 x6 x2prim1 (x4 x5 x6) x2)∀ x5 : ι → ι . (∀ x6 . prim1 x6 x2prim1 (x5 x6) x2)∀ x6 . prim1 x6 x2x1 (a0d6b.. x2 x3 x4 x5 x6))x1 x0
Theorem f9e4c.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . (∀ x3 . prim1 x3 x0∀ x4 . prim1 x4 x0prim1 (x2 x3 x4) x0)∀ x3 : ι → ι . (∀ x4 . prim1 x4 x0prim1 (x3 x4) x0)∀ x4 . prim1 x4 x01b748.. (a0d6b.. x0 x1 x2 x3 x4) (proof)
Theorem 2bf47.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 . 1b748.. (a0d6b.. x0 x1 x2 x3 x4)∀ x5 . prim1 x5 x0∀ x6 . prim1 x6 x0prim1 (x2 x5 x6) x0 (proof)
Theorem 71580.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 . 1b748.. (a0d6b.. x0 x1 x2 x3 x4)∀ x5 . prim1 x5 x0prim1 (x3 x5) x0 (proof)
Theorem d2a28.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 . 1b748.. (a0d6b.. x0 x1 x2 x3 x4)prim1 x4 x0 (proof)
Known iff_refl : ∀ x0 : ο . iff x0 x0
Theorem 5fe65.. : ∀ x0 . 1b748.. x0x0 = a0d6b.. (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition 373f2.. := λ x0 . λ x1 : ι → ((ι → ο) → ο)(ι → ι → ι)(ι → ι)ι → ι . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 9e504.. : ∀ x0 : ι → ((ι → ο) → ο)(ι → ι → ι)(ι → ι)ι → ι . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ x5 . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8prim1 x8 x1)iff (x2 x7) (x6 x7))∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1∀ x9 . prim1 x9 x1x3 x8 x9 = x7 x8 x9)∀ x8 : ι → ι . (∀ x9 . prim1 x9 x1x4 x9 = x8 x9)x0 x1 x6 x7 x8 x5 = x0 x1 x2 x3 x4 x5)373f2.. (a0d6b.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition c596a.. := λ x0 . λ x1 : ι → ((ι → ο) → ο)(ι → ι → ι)(ι → ι)ι → ο . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 8bc8a.. : ∀ x0 : ι → ((ι → ο) → ο)(ι → ι → ι)(ι → ι)ι → ο . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ x5 . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8prim1 x8 x1)iff (x2 x7) (x6 x7))∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1∀ x9 . prim1 x9 x1x3 x8 x9 = x7 x8 x9)∀ x8 : ι → ι . (∀ x9 . prim1 x9 x1x4 x9 = x8 x9)x0 x1 x6 x7 x8 x5 = x0 x1 x2 x3 x4 x5)c596a.. (a0d6b.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Param d2155.. : ι(ιιο) → ι
Definition 38265.. := λ x0 . λ x1 : (ι → ο) → ο . λ x2 : ι → ι → ι . λ x3 x4 : ι → ι → ο . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (eb53d.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (d2155.. x0 x3) (d2155.. x0 x4)))))
Theorem 91633.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = 38265.. x1 x2 x3 x4 x5x1 = f482f.. x0 4a7ef.. (proof)
Theorem 74f86.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 x4 x5 : ι → ι → ο . x5 x0 (f482f.. (38265.. x0 x1 x2 x3 x4) 4a7ef..)x5 (f482f.. (38265.. x0 x1 x2 x3 x4) 4a7ef..) x0 (proof)
Theorem 5ec19.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = 38265.. x1 x2 x3 x4 x5∀ x6 : ι → ο . (∀ x7 . x6 x7prim1 x7 x1)x2 x6 = decode_c (f482f.. x0 (4ae4a.. 4a7ef..)) x6 (proof)
Theorem 91610.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ x5 : ι → ο . (∀ x6 . x5 x6prim1 x6 x0)x1 x5 = decode_c (f482f.. (38265.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 (proof)
Theorem 1019c.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = 38265.. x1 x2 x3 x4 x5∀ x6 . prim1 x6 x1∀ x7 . prim1 x7 x1x3 x6 x7 = e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 x7 (proof)
Theorem 5acf4.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ x5 . prim1 x5 x0∀ x6 . prim1 x6 x0x2 x5 x6 = e3162.. (f482f.. (38265.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Param 2b2e3.. : ιιιο
Known 67416.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 . prim1 x2 x0∀ x3 . prim1 x3 x02b2e3.. (d2155.. x0 x1) x2 x3 = x1 x2 x3
Theorem fc16f.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = 38265.. x1 x2 x3 x4 x5∀ x6 . prim1 x6 x1∀ x7 . prim1 x7 x1x4 x6 x7 = 2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 x7 (proof)
Theorem 8f14c.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ x5 . prim1 x5 x0∀ x6 . prim1 x6 x0x3 x5 x6 = 2b2e3.. (f482f.. (38265.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 x6 (proof)
Theorem 98a70.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = 38265.. x1 x2 x3 x4 x5∀ x6 . prim1 x6 x1∀ x7 . prim1 x7 x1x5 x6 x7 = 2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x6 x7 (proof)
Theorem caff9.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ x5 . prim1 x5 x0∀ x6 . prim1 x6 x0x4 x5 x6 = 2b2e3.. (f482f.. (38265.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x5 x6 (proof)
Theorem 9ef04.. : ∀ x0 x1 . ∀ x2 x3 : (ι → ο) → ο . ∀ x4 x5 : ι → ι → ι . ∀ x6 x7 x8 x9 : ι → ι → ο . 38265.. x0 x2 x4 x6 x8 = 38265.. x1 x3 x5 x7 x9and (and (and (and (x0 = x1) (∀ x10 : ι → ο . (∀ x11 . x10 x11prim1 x11 x0)x2 x10 = x3 x10)) (∀ x10 . prim1 x10 x0∀ x11 . prim1 x11 x0x4 x10 x11 = x5 x10 x11)) (∀ x10 . prim1 x10 x0∀ x11 . prim1 x11 x0x6 x10 x11 = x7 x10 x11)) (∀ x10 . prim1 x10 x0∀ x11 . prim1 x11 x0x8 x10 x11 = x9 x10 x11) (proof)
Known 62ef7.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . (∀ x3 . prim1 x3 x0∀ x4 . prim1 x4 x0iff (x1 x3 x4) (x2 x3 x4))d2155.. x0 x1 = d2155.. x0 x2
Theorem 9b86c.. : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . ∀ x3 x4 : ι → ι → ι . ∀ x5 x6 x7 x8 : ι → ι → ο . (∀ x9 : ι → ο . (∀ x10 . x9 x10prim1 x10 x0)iff (x1 x9) (x2 x9))(∀ x9 . prim1 x9 x0∀ x10 . prim1 x10 x0x3 x9 x10 = x4 x9 x10)(∀ x9 . prim1 x9 x0∀ x10 . prim1 x10 x0iff (x5 x9 x10) (x6 x9 x10))(∀ x9 . prim1 x9 x0∀ x10 . prim1 x10 x0iff (x7 x9 x10) (x8 x9 x10))38265.. x0 x1 x3 x5 x7 = 38265.. x0 x2 x4 x6 x8 (proof)
Definition 3d094.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : (ι → ο) → ο . ∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x2∀ x6 . prim1 x6 x2prim1 (x4 x5 x6) x2)∀ x5 x6 : ι → ι → ο . x1 (38265.. x2 x3 x4 x5 x6))x1 x0
Theorem 73744.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . (∀ x3 . prim1 x3 x0∀ x4 . prim1 x4 x0prim1 (x2 x3 x4) x0)∀ x3 x4 : ι → ι → ο . 3d094.. (38265.. x0 x1 x2 x3 x4) (proof)
Theorem 289de.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 x4 : ι → ι → ο . 3d094.. (38265.. x0 x1 x2 x3 x4)∀ x5 . prim1 x5 x0∀ x6 . prim1 x6 x0prim1 (x2 x5 x6) x0 (proof)
Theorem 683f8.. : ∀ x0 . 3d094.. x0x0 = 38265.. (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))) (proof)
Definition c70b7.. := λ x0 . λ x1 : ι → ((ι → ο) → ο)(ι → ι → ι)(ι → ι → ο)(ι → ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem 3d8f2.. : ∀ x0 : ι → ((ι → ο) → ο)(ι → ι → ι)(ι → ι → ο)(ι → ι → ο) → ι . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 x5 : ι → ι → ο . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8prim1 x8 x1)iff (x2 x7) (x6 x7))∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1∀ x9 . prim1 x9 x1x3 x8 x9 = x7 x8 x9)∀ x8 : ι → ι → ο . (∀ x9 . prim1 x9 x1∀ x10 . prim1 x10 x1iff (x4 x9 x10) (x8 x9 x10))∀ x9 : ι → ι → ο . (∀ x10 . prim1 x10 x1∀ x11 . prim1 x11 x1iff (x5 x10 x11) (x9 x10 x11))x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5)c70b7.. (38265.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition d976e.. := λ x0 . λ x1 : ι → ((ι → ο) → ο)(ι → ι → ι)(ι → ι → ο)(ι → ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem 63542.. : ∀ x0 : ι → ((ι → ο) → ο)(ι → ι → ι)(ι → ι → ο)(ι → ι → ο) → ο . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 x5 : ι → ι → ο . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8prim1 x8 x1)iff (x2 x7) (x6 x7))∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1∀ x9 . prim1 x9 x1x3 x8 x9 = x7 x8 x9)∀ x8 : ι → ι → ο . (∀ x9 . prim1 x9 x1∀ x10 . prim1 x10 x1iff (x4 x9 x10) (x8 x9 x10))∀ x9 : ι → ι → ο . (∀ x10 . prim1 x10 x1∀ x11 . prim1 x11 x1iff (x5 x10 x11) (x9 x10 x11))x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5)d976e.. (38265.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Param 1216a.. : ι(ιο) → ι
Definition d4d11.. := λ x0 . λ x1 : (ι → ο) → ο . λ x2 : ι → ι → ι . λ x3 : ι → ι → ο . λ x4 : ι → ο . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (eb53d.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (d2155.. x0 x3) (1216a.. x0 x4)))))
Theorem c7fd8.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . x0 = d4d11.. x1 x2 x3 x4 x5x1 = f482f.. x0 4a7ef.. (proof)
Theorem 65809.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . x0 = f482f.. (d4d11.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Theorem a4102.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . x0 = d4d11.. x1 x2 x3 x4 x5∀ x6 : ι → ο . (∀ x7 . x6 x7prim1 x7 x1)x2 x6 = decode_c (f482f.. x0 (4ae4a.. 4a7ef..)) x6 (proof)
Theorem cab07.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 : ι → ο . (∀ x6 . x5 x6prim1 x6 x0)x1 x5 = decode_c (f482f.. (d4d11.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 (proof)
Theorem 773af.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . x0 = d4d11.. x1 x2 x3 x4 x5∀ x6 . prim1 x6 x1∀ x7 . prim1 x7 x1x3 x6 x7 = e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 x7 (proof)
Theorem b869f.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x0∀ x6 . prim1 x6 x0x2 x5 x6 = e3162.. (f482f.. (d4d11.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Theorem 45095.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . x0 = d4d11.. x1 x2 x3 x4 x5∀ x6 . prim1 x6 x1∀ x7 . prim1 x7 x1x4 x6 x7 = 2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 x7 (proof)
Theorem 58312.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x0∀ x6 . prim1 x6 x0x3 x5 x6 = 2b2e3.. (f482f.. (d4d11.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 x6 (proof)
Param decode_p : ιιο
Known 931fe.. : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . prim1 x2 x0decode_p (1216a.. x0 x1) x2 = x1 x2
Theorem 0b346.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . x0 = d4d11.. x1 x2 x3 x4 x5∀ x6 . prim1 x6 x1x5 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x6 (proof)
Theorem e86e6.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x0x4 x5 = decode_p (f482f.. (d4d11.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x5 (proof)
Theorem 49f62.. : ∀ x0 x1 . ∀ x2 x3 : (ι → ο) → ο . ∀ x4 x5 : ι → ι → ι . ∀ x6 x7 : ι → ι → ο . ∀ x8 x9 : ι → ο . d4d11.. x0 x2 x4 x6 x8 = d4d11.. x1 x3 x5 x7 x9and (and (and (and (x0 = x1) (∀ x10 : ι → ο . (∀ x11 . x10 x11prim1 x11 x0)x2 x10 = x3 x10)) (∀ x10 . prim1 x10 x0∀ x11 . prim1 x11 x0x4 x10 x11 = x5 x10 x11)) (∀ x10 . prim1 x10 x0∀ x11 . prim1 x11 x0x6 x10 x11 = x7 x10 x11)) (∀ x10 . prim1 x10 x0x8 x10 = x9 x10) (proof)
Known ee7ef.. : ∀ x0 . ∀ x1 x2 : ι → ο . (∀ x3 . prim1 x3 x0iff (x1 x3) (x2 x3))1216a.. x0 x1 = 1216a.. x0 x2
Theorem 6b57f.. : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . ∀ x3 x4 : ι → ι → ι . ∀ x5 x6 : ι → ι → ο . ∀ x7 x8 : ι → ο . (∀ x9 : ι → ο . (∀ x10 . x9 x10prim1 x10 x0)iff (x1 x9) (x2 x9))(∀ x9 . prim1 x9 x0∀ x10 . prim1 x10 x0x3 x9 x10 = x4 x9 x10)(∀ x9 . prim1 x9 x0∀ x10 . prim1 x10 x0iff (x5 x9 x10) (x6 x9 x10))(∀ x9 . prim1 x9 x0iff (x7 x9) (x8 x9))d4d11.. x0 x1 x3 x5 x7 = d4d11.. x0 x2 x4 x6 x8 (proof)
Definition 135b2.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : (ι → ο) → ο . ∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x2∀ x6 . prim1 x6 x2prim1 (x4 x5 x6) x2)∀ x5 : ι → ι → ο . ∀ x6 : ι → ο . x1 (d4d11.. x2 x3 x4 x5 x6))x1 x0
Theorem 7ef48.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . (∀ x3 . prim1 x3 x0∀ x4 . prim1 x4 x0prim1 (x2 x3 x4) x0)∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . 135b2.. (d4d11.. x0 x1 x2 x3 x4) (proof)
Theorem b2101.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . 135b2.. (d4d11.. x0 x1 x2 x3 x4)∀ x5 . prim1 x5 x0∀ x6 . prim1 x6 x0prim1 (x2 x5 x6) x0 (proof)
Theorem 06e33.. : ∀ x0 . 135b2.. x0x0 = d4d11.. (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))) (proof)
Definition 1c0d8.. := λ x0 . λ x1 : ι → ((ι → ο) → ο)(ι → ι → ι)(ι → ι → ο)(ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem fcc31.. : ∀ x0 : ι → ((ι → ο) → ο)(ι → ι → ι)(ι → ι → ο)(ι → ο) → ι . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8prim1 x8 x1)iff (x2 x7) (x6 x7))∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1∀ x9 . prim1 x9 x1x3 x8 x9 = x7 x8 x9)∀ x8 : ι → ι → ο . (∀ x9 . prim1 x9 x1∀ x10 . prim1 x10 x1iff (x4 x9 x10) (x8 x9 x10))∀ x9 : ι → ο . (∀ x10 . prim1 x10 x1iff (x5 x10) (x9 x10))x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5)1c0d8.. (d4d11.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 1d1e5.. := λ x0 . λ x1 : ι → ((ι → ο) → ο)(ι → ι → ι)(ι → ι → ο)(ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem 9df88.. : ∀ x0 : ι → ((ι → ο) → ο)(ι → ι → ι)(ι → ι → ο)(ι → ο) → ο . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 : ι → ο . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8prim1 x8 x1)iff (x2 x7) (x6 x7))∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1∀ x9 . prim1 x9 x1x3 x8 x9 = x7 x8 x9)∀ x8 : ι → ι → ο . (∀ x9 . prim1 x9 x1∀ x10 . prim1 x10 x1iff (x4 x9 x10) (x8 x9 x10))∀ x9 : ι → ο . (∀ x10 . prim1 x10 x1iff (x5 x10) (x9 x10))x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5)1d1e5.. (d4d11.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 04077.. := λ x0 . λ x1 : (ι → ο) → ο . λ x2 : ι → ι → ι . λ x3 : ι → ι → ο . λ x4 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (e0e40.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (eb53d.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (d2155.. x0 x3) x4))))
Theorem 18def.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . x0 = 04077.. x1 x2 x3 x4 x5x1 = f482f.. x0 4a7ef.. (proof)
Theorem 69b45.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . x0 = f482f.. (04077.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Theorem 0987d.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . x0 = 04077.. x1 x2 x3 x4 x5∀ x6 : ι → ο . (∀ x7 . x6 x7prim1 x7 x1)x2 x6 = decode_c (f482f.. x0 (4ae4a.. 4a7ef..)) x6 (proof)
Theorem e66bf.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . ∀ x5 : ι → ο . (∀ x6 . x5 x6prim1 x6 x0)x1 x5 = decode_c (f482f.. (04077.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 (proof)
Theorem 12b50.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . x0 = 04077.. x1 x2 x3 x4 x5∀ x6 . prim1 x6 x1∀ x7 . prim1 x7 x1x3 x6 x7 = e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 x7 (proof)
Theorem 70d81.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . prim1 x5 x0∀ x6 . prim1 x6 x0x2 x5 x6 = e3162.. (f482f.. (04077.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Theorem e9aad.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . x0 = 04077.. x1 x2 x3 x4 x5∀ x6 . prim1 x6 x1∀ x7 . prim1 x7 x1x4 x6 x7 = 2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 x7 (proof)
Theorem 8ab77.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 . prim1 x5 x0∀ x6 . prim1 x6 x0x3 x5 x6 = 2b2e3.. (f482f.. (04077.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 x6 (proof)
Theorem 80085.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . x0 = 04077.. x1 x2 x3 x4 x5x5 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem d3f5a.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . x4 = f482f.. (04077.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem 5e479.. : ∀ x0 x1 . ∀ x2 x3 : (ι → ο) → ο . ∀ x4 x5 : ι → ι → ι . ∀ x6 x7 : ι → ι → ο . ∀ x8 x9 . 04077.. x0 x2 x4 x6 x8 = 04077.. x1 x3 x5 x7 x9and (and (and (and (x0 = x1) (∀ x10 : ι → ο . (∀ x11 . x10 x11prim1 x11 x0)x2 x10 = x3 x10)) (∀ x10 . prim1 x10 x0∀ x11 . prim1 x11 x0x4 x10 x11 = x5 x10 x11)) (∀ x10 . prim1 x10 x0∀ x11 . prim1 x11 x0x6 x10 x11 = x7 x10 x11)) (x8 = x9) (proof)
Theorem 72750.. : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . ∀ x3 x4 : ι → ι → ι . ∀ x5 x6 : ι → ι → ο . ∀ x7 . (∀ x8 : ι → ο . (∀ x9 . x8 x9prim1 x9 x0)iff (x1 x8) (x2 x8))(∀ x8 . prim1 x8 x0∀ x9 . prim1 x9 x0x3 x8 x9 = x4 x8 x9)(∀ x8 . prim1 x8 x0∀ x9 . prim1 x9 x0iff (x5 x8 x9) (x6 x8 x9))04077.. x0 x1 x3 x5 x7 = 04077.. x0 x2 x4 x6 x7 (proof)
Definition 3bcdb.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : (ι → ο) → ο . ∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x2∀ x6 . prim1 x6 x2prim1 (x4 x5 x6) x2)∀ x5 : ι → ι → ο . ∀ x6 . prim1 x6 x2x1 (04077.. x2 x3 x4 x5 x6))x1 x0
Theorem e9809.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . (∀ x3 . prim1 x3 x0∀ x4 . prim1 x4 x0prim1 (x2 x3 x4) x0)∀ x3 : ι → ι → ο . ∀ x4 . prim1 x4 x03bcdb.. (04077.. x0 x1 x2 x3 x4) (proof)
Theorem 5d7d8.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . 3bcdb.. (04077.. x0 x1 x2 x3 x4)∀ x5 . prim1 x5 x0∀ x6 . prim1 x6 x0prim1 (x2 x5 x6) x0 (proof)
Theorem fc3fe.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . 3bcdb.. (04077.. x0 x1 x2 x3 x4)prim1 x4 x0 (proof)
Theorem b8260.. : ∀ x0 . 3bcdb.. x0x0 = 04077.. (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition c14f1.. := λ x0 . λ x1 : ι → ((ι → ο) → ο)(ι → ι → ι)(ι → ι → ο)ι → ι . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem cf0a4.. : ∀ x0 : ι → ((ι → ο) → ο)(ι → ι → ι)(ι → ι → ο)ι → ι . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8prim1 x8 x1)iff (x2 x7) (x6 x7))∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1∀ x9 . prim1 x9 x1x3 x8 x9 = x7 x8 x9)∀ x8 : ι → ι → ο . (∀ x9 . prim1 x9 x1∀ x10 . prim1 x10 x1iff (x4 x9 x10) (x8 x9 x10))x0 x1 x6 x7 x8 x5 = x0 x1 x2 x3 x4 x5)c14f1.. (04077.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 01947.. := λ x0 . λ x1 : ι → ((ι → ο) → ο)(ι → ι → ι)(ι → ι → ο)ι → ο . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 51334.. : ∀ x0 : ι → ((ι → ο) → ο)(ι → ι → ι)(ι → ι → ο)ι → ο . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8prim1 x8 x1)iff (x2 x7) (x6 x7))∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1∀ x9 . prim1 x9 x1x3 x8 x9 = x7 x8 x9)∀ x8 : ι → ι → ο . (∀ x9 . prim1 x9 x1∀ x10 . prim1 x10 x1iff (x4 x9 x10) (x8 x9 x10))x0 x1 x6 x7 x8 x5 = x0 x1 x2 x3 x4 x5)01947.. (04077.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)