Proofgold Asset

Param 0fc90.. : ι → (ι → ι) → ι
Param 4ae4a.. : ι → ι
Param 4a7ef.. : ι
Param If_i : ο → ι → ι → ι
Param d2155.. : ι → (ι → ι → ο) → ι
Param 1216a.. : ι → (ι → ο) → ι
Definition e4ab3.. := λ x0 . λ x1 : ι → ι → ο . λ x2 : ι → ο . λ x3 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x4 . If_i (x4 = 4a7ef..) x0 (If_i (x4 = 4ae4a.. 4a7ef..) (d2155.. x0 x1) (If_i (x4 = 4ae4a.. (4ae4a.. 4a7ef..)) (1216a.. x0 x2) x3)))
Param f482f.. : ι → ι → ι
Known 9f6be.. : ∀ x0 x1 x2 x3 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) x1 (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 x3)))) 4a7ef.. = x0
Theorem 805c9.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 . x0 = e4ab3.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem deb33.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 . x0 = f482f.. (e4ab3.. x0 x1 x2 x3) 4a7ef.. (proof)
Param 2b2e3.. : ι → ι → ι → ο
Known 8a328.. : ∀ x0 x1 x2 x3 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) x1 (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 x3)))) (4ae4a.. 4a7ef..) = x1
Known 67416.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ 2b2e3.. (d2155.. x0 x1) x2 x3 = x1 x2 x3
Theorem d443f.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 . x0 = e4ab3.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = 2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..)) x5 x6 (proof)
Theorem 4e2a6.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x1 x4 x5 = 2b2e3.. (f482f.. (e4ab3.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 x5 (proof)
Param decode_p : ι → ι → ο
Known 142e6.. : ∀ x0 x1 x2 x3 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) x1 (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 x3)))) (4ae4a.. (4ae4a.. 4a7ef..)) = x2
Known 931fe.. : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . prim1 x2 x0 ⟶ decode_p (1216a.. x0 x1) x2 = x1 x2
Theorem 7f210.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 . x0 = e4ab3.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x3 x5 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Theorem af949.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 x4 . prim1 x4 x0 ⟶ x2 x4 = decode_p (f482f.. (e4ab3.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) x4 (proof)
Known 62a6b.. : ∀ x0 x1 x2 x3 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) x1 (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 x3)))) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) = x3
Theorem f6dbf.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 . x0 = e4ab3.. x1 x2 x3 x4 ⟶ x4 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem eead3.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 . x3 = f482f.. (e4ab3.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Definition and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known and4I : ∀ x0 x1 x2 x3 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ and (and (and x0 x1) x2) x3
Theorem 97d9e.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 x5 : ι → ο . ∀ x6 x7 . e4ab3.. x0 x2 x4 x6 = e4ab3.. x1 x3 x5 x7 ⟶ and (and (and (x0 = x1) (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x2 x8 x9 = x3 x8 x9)) (∀ x8 . prim1 x8 x0 ⟶ x4 x8 = x5 x8)) (x6 = x7) (proof)
Param iff : ο → ο → ο
Known ee7ef.. : ∀ x0 . ∀ x1 x2 : ι → ο . (∀ x3 . prim1 x3 x0 ⟶ iff (x1 x3) (x2 x3)) ⟶ 1216a.. x0 x1 = 1216a.. x0 x2
Known 62ef7.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ iff (x1 x3 x4) (x2 x3 x4)) ⟶ d2155.. x0 x1 = d2155.. x0 x2
Theorem 6261d.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 x4 : ι → ο . ∀ x5 . (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ iff (x1 x6 x7) (x2 x6 x7)) ⟶ (∀ x6 . prim1 x6 x0 ⟶ iff (x3 x6) (x4 x6)) ⟶ e4ab3.. x0 x1 x3 x5 = e4ab3.. x0 x2 x4 x5 (proof)
Definition 60280.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x2 ⟶ x1 (e4ab3.. x2 x3 x4 x5)) ⟶ x1 x0
Theorem 77b7d.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 . prim1 x3 x0 ⟶ 60280.. (e4ab3.. x0 x1 x2 x3) (proof)
Theorem 694b1.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 . 60280.. (e4ab3.. x0 x1 x2 x3) ⟶ prim1 x3 x0 (proof)
Known iff_refl : ∀ x0 : ο . iff x0 x0
Theorem b3850.. : ∀ x0 . 60280.. x0 ⟶ x0 = e4ab3.. (f482f.. x0 4a7ef..) (2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Definition 07f33.. := λ x0 . λ x1 : ι → (ι → ι → ο) → (ι → ο) → ι → ι . x1 (f482f.. x0 4a7ef..) (2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem f5bac.. : ∀ x0 : ι → (ι → ι → ο) → (ι → ο) → ι → ι . ∀ x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 . (∀ x5 : ι → ι → ο . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ iff (x2 x6 x7) (x5 x6 x7)) ⟶ ∀ x6 : ι → ο . (∀ x7 . prim1 x7 x1 ⟶ iff (x3 x7) (x6 x7)) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4) ⟶ 07f33.. (e4ab3.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 3a2d6.. := λ x0 . λ x1 : ι → (ι → ι → ο) → (ι → ο) → ι → ο . x1 (f482f.. x0 4a7ef..) (2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem caace.. : ∀ x0 : ι → (ι → ι → ο) → (ι → ο) → ι → ο . ∀ x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 . (∀ x5 : ι → ι → ο . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ iff (x2 x6 x7) (x5 x6 x7)) ⟶ ∀ x6 : ι → ο . (∀ x7 . prim1 x7 x1 ⟶ iff (x3 x7) (x6 x7)) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4) ⟶ 3a2d6.. (e4ab3.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 9a89f.. := λ x0 . λ x1 x2 : ι → ο . λ x3 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x4 . If_i (x4 = 4a7ef..) x0 (If_i (x4 = 4ae4a.. 4a7ef..) (1216a.. x0 x1) (If_i (x4 = 4ae4a.. (4ae4a.. 4a7ef..)) (1216a.. x0 x2) x3)))
Theorem 6cd62.. : ∀ x0 x1 . ∀ x2 x3 : ι → ο . ∀ x4 . x0 = 9a89f.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem de5ce.. : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 . x0 = f482f.. (9a89f.. x0 x1 x2 x3) 4a7ef.. (proof)
Theorem 9c447.. : ∀ x0 x1 . ∀ x2 x3 : ι → ο . ∀ x4 . x0 = 9a89f.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x2 x5 = decode_p (f482f.. x0 (4ae4a.. 4a7ef..)) x5 (proof)
Theorem 6b100.. : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 x4 . prim1 x4 x0 ⟶ x1 x4 = decode_p (f482f.. (9a89f.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 (proof)
Theorem de13d.. : ∀ x0 x1 . ∀ x2 x3 : ι → ο . ∀ x4 . x0 = 9a89f.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x3 x5 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Theorem 29b74.. : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 x4 . prim1 x4 x0 ⟶ x2 x4 = decode_p (f482f.. (9a89f.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) x4 (proof)
Theorem ba999.. : ∀ x0 x1 . ∀ x2 x3 : ι → ο . ∀ x4 . x0 = 9a89f.. x1 x2 x3 x4 ⟶ x4 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem bb846.. : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 . x3 = f482f.. (9a89f.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem 1bc46.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ο . ∀ x6 x7 . 9a89f.. x0 x2 x4 x6 = 9a89f.. x1 x3 x5 x7 ⟶ and (and (and (x0 = x1) (∀ x8 . prim1 x8 x0 ⟶ x2 x8 = x3 x8)) (∀ x8 . prim1 x8 x0 ⟶ x4 x8 = x5 x8)) (x6 = x7) (proof)
Theorem 54e98.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ο . ∀ x5 . (∀ x6 . prim1 x6 x0 ⟶ iff (x1 x6) (x2 x6)) ⟶ (∀ x6 . prim1 x6 x0 ⟶ iff (x3 x6) (x4 x6)) ⟶ 9a89f.. x0 x1 x3 x5 = 9a89f.. x0 x2 x4 x5 (proof)
Definition ce4b9.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 x4 : ι → ο . ∀ x5 . prim1 x5 x2 ⟶ x1 (9a89f.. x2 x3 x4 x5)) ⟶ x1 x0
Theorem 41c46.. : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 . prim1 x3 x0 ⟶ ce4b9.. (9a89f.. x0 x1 x2 x3) (proof)
Theorem 82284.. : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 . ce4b9.. (9a89f.. x0 x1 x2 x3) ⟶ prim1 x3 x0 (proof)
Theorem 63f7b.. : ∀ x0 . ce4b9.. x0 ⟶ x0 = 9a89f.. (f482f.. x0 4a7ef..) (decode_p (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Definition 447ab.. := λ x0 . λ x1 : ι → (ι → ο) → (ι → ο) → ι → ι . x1 (f482f.. x0 4a7ef..) (decode_p (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem ab844.. : ∀ x0 : ι → (ι → ο) → (ι → ο) → ι → ι . ∀ x1 . ∀ x2 x3 : ι → ο . ∀ x4 . (∀ x5 : ι → ο . (∀ x6 . prim1 x6 x1 ⟶ iff (x2 x6) (x5 x6)) ⟶ ∀ x6 : ι → ο . (∀ x7 . prim1 x7 x1 ⟶ iff (x3 x7) (x6 x7)) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4) ⟶ 447ab.. (9a89f.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 4162f.. := λ x0 . λ x1 : ι → (ι → ο) → (ι → ο) → ι → ο . x1 (f482f.. x0 4a7ef..) (decode_p (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 3c62e.. : ∀ x0 : ι → (ι → ο) → (ι → ο) → ι → ο . ∀ x1 . ∀ x2 x3 : ι → ο . ∀ x4 . (∀ x5 : ι → ο . (∀ x6 . prim1 x6 x1 ⟶ iff (x2 x6) (x5 x6)) ⟶ ∀ x6 : ι → ο . (∀ x7 . prim1 x7 x1 ⟶ iff (x3 x7) (x6 x7)) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4) ⟶ 4162f.. (9a89f.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)