Proofgold Address

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 3d68f.. := λ x0 . λ x1 x2 : ι → ι → ο . λ x3 : ι → ο . λ x4 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (d2155.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (d2155.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (1216a.. x0 x3) x4))))
Param f482f.. : ι → ι → ι
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 20346.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . x0 = 3d68f.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 440ae.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 . x0 = f482f.. (3d68f.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Param 2b2e3.. : ι → ι → ι → 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 67416.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ 2b2e3.. (d2155.. x0 x1) x2 x3 = x1 x2 x3
Theorem c523c.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . x0 = 3d68f.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = 2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..)) x6 x7 (proof)
Theorem e8070.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x1 x5 x6 = 2b2e3.. (f482f.. (3d68f.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 x6 (proof)
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
Theorem 38ec6.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . x0 = 3d68f.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x3 x6 x7 = 2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 x7 (proof)
Theorem 6af8a.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x5 x6 = 2b2e3.. (f482f.. (3d68f.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Param decode_p : ι → ι → ο
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 931fe.. : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . prim1 x2 x0 ⟶ decode_p (1216a.. x0 x1) x2 = x1 x2
Theorem bab90.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . x0 = 3d68f.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x4 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 (proof)
Theorem 07fdf.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 x5 . prim1 x5 x0 ⟶ x3 x5 = decode_p (f482f.. (3d68f.. 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 ead03.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . x0 = 3d68f.. x1 x2 x3 x4 x5 ⟶ x5 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem db6e8.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 . x4 = f482f.. (3d68f.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Definition and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known and5I : ∀ x0 x1 x2 x3 x4 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ and (and (and (and x0 x1) x2) x3) x4
Theorem ba288.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι → ο . ∀ x6 x7 : ι → ο . ∀ x8 x9 . 3d68f.. x0 x2 x4 x6 x8 = 3d68f.. x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x2 x10 x11 = x3 x10 x11)) (∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x4 x10 x11 = x5 x10 x11)) (∀ x10 . prim1 x10 x0 ⟶ x6 x10 = x7 x10)) (x8 = x9) (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 d075f.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι → ο . ∀ x5 x6 : ι → ο . ∀ x7 . (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ iff (x1 x8 x9) (x2 x8 x9)) ⟶ (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ iff (x3 x8 x9) (x4 x8 x9)) ⟶ (∀ x8 . prim1 x8 x0 ⟶ iff (x5 x8) (x6 x8)) ⟶ 3d68f.. x0 x1 x3 x5 x7 = 3d68f.. x0 x2 x4 x6 x7 (proof)
Definition d469f.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 x4 : ι → ι → ο . ∀ x5 : ι → ο . ∀ x6 . prim1 x6 x2 ⟶ x1 (3d68f.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem bb5c0.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 . prim1 x4 x0 ⟶ d469f.. (3d68f.. x0 x1 x2 x3 x4) (proof)
Theorem 02e09.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 . d469f.. (3d68f.. x0 x1 x2 x3 x4) ⟶ prim1 x4 x0 (proof)
Known iff_refl : ∀ x0 : ο . iff x0 x0
Theorem f7bea.. : ∀ x0 . d469f.. x0 ⟶ x0 = 3d68f.. (f482f.. x0 4a7ef..) (2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition cc817.. := λ x0 . λ x1 : ι → (ι → ι → ο) → (ι → ι → ο) → (ι → ο) → ι → ι . x1 (f482f.. x0 4a7ef..) (2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 31e00.. : ∀ x0 : ι → (ι → ι → ο) → (ι → ι → ο) → (ι → ο) → ι → ι . ∀ x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . (∀ x6 : ι → ι → ο . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ iff (x2 x7 x8) (x6 x7 x8)) ⟶ ∀ x7 : ι → ι → ο . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ iff (x3 x8 x9) (x7 x8 x9)) ⟶ ∀ x8 : ι → ο . (∀ x9 . prim1 x9 x1 ⟶ iff (x4 x9) (x8 x9)) ⟶ x0 x1 x6 x7 x8 x5 = x0 x1 x2 x3 x4 x5) ⟶ cc817.. (3d68f.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition d768a.. := λ x0 . λ x1 : ι → (ι → ι → ο) → (ι → ι → ο) → (ι → ο) → ι → ο . x1 (f482f.. x0 4a7ef..) (2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 2349c.. : ∀ x0 : ι → (ι → ι → ο) → (ι → ι → ο) → (ι → ο) → ι → ο . ∀ x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . (∀ x6 : ι → ι → ο . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ iff (x2 x7 x8) (x6 x7 x8)) ⟶ ∀ x7 : ι → ι → ο . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ iff (x3 x8 x9) (x7 x8 x9)) ⟶ ∀ x8 : ι → ο . (∀ x9 . prim1 x9 x1 ⟶ iff (x4 x9) (x8 x9)) ⟶ x0 x1 x6 x7 x8 x5 = x0 x1 x2 x3 x4 x5) ⟶ d768a.. (3d68f.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 0d9e7.. := λ x0 . λ x1 x2 : ι → ι → ο . λ x3 x4 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (d2155.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (d2155.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4))))
Theorem e0b1d.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 x5 . x0 = 0d9e7.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem fffe9.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 x4 . x0 = f482f.. (0d9e7.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Theorem 6e35b.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 x5 . x0 = 0d9e7.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = 2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..)) x6 x7 (proof)
Theorem c8f7f.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 x4 x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x1 x5 x6 = 2b2e3.. (f482f.. (0d9e7.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 x6 (proof)
Theorem 94c28.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 x5 . x0 = 0d9e7.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x3 x6 x7 = 2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 x7 (proof)
Theorem 4b5da.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 x4 x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x5 x6 = 2b2e3.. (f482f.. (0d9e7.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Theorem f1f54.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 x5 . x0 = 0d9e7.. x1 x2 x3 x4 x5 ⟶ x4 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem 7e4a4.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 x4 . x3 = f482f.. (0d9e7.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem 6e26a.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 x5 . x0 = 0d9e7.. x1 x2 x3 x4 x5 ⟶ x5 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem 8569e.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 x4 . x4 = f482f.. (0d9e7.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem eb01a.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι → ο . ∀ x6 x7 x8 x9 . 0d9e7.. x0 x2 x4 x6 x8 = 0d9e7.. x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x2 x10 x11 = x3 x10 x11)) (∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x4 x10 x11 = x5 x10 x11)) (x6 = x7)) (x8 = x9) (proof)
Theorem 44294.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι → ο . ∀ x5 x6 . (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ iff (x1 x7 x8) (x2 x7 x8)) ⟶ (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ iff (x3 x7 x8) (x4 x7 x8)) ⟶ 0d9e7.. x0 x1 x3 x5 x6 = 0d9e7.. x0 x2 x4 x5 x6 (proof)
Definition 8f2ce.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 x4 : ι → ι → ο . ∀ x5 . prim1 x5 x2 ⟶ ∀ x6 . prim1 x6 x2 ⟶ x1 (0d9e7.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem 71832.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ 8f2ce.. (0d9e7.. x0 x1 x2 x3 x4) (proof)
Theorem 67912.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 x4 . 8f2ce.. (0d9e7.. x0 x1 x2 x3 x4) ⟶ prim1 x3 x0 (proof)
Theorem 4108c.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 x4 . 8f2ce.. (0d9e7.. x0 x1 x2 x3 x4) ⟶ prim1 x4 x0 (proof)
Theorem ff222.. : ∀ x0 . 8f2ce.. x0 ⟶ x0 = 0d9e7.. (f482f.. x0 4a7ef..) (2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition d6ebe.. := λ x0 . λ x1 : ι → (ι → ι → ο) → (ι → ι → ο) → ι → ι → ι . x1 (f482f.. x0 4a7ef..) (2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 7a018.. : ∀ x0 : ι → (ι → ι → ο) → (ι → ι → ο) → ι → ι → ι . ∀ x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 x5 . (∀ x6 : ι → ι → ο . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ iff (x2 x7 x8) (x6 x7 x8)) ⟶ ∀ x7 : ι → ι → ο . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ iff (x3 x8 x9) (x7 x8 x9)) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5) ⟶ d6ebe.. (0d9e7.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 45090.. := λ x0 . λ x1 : ι → (ι → ι → ο) → (ι → ι → ο) → ι → ι → ο . x1 (f482f.. x0 4a7ef..) (2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 1283d.. : ∀ x0 : ι → (ι → ι → ο) → (ι → ι → ο) → ι → ι → ο . ∀ x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 x5 . (∀ x6 : ι → ι → ο . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ iff (x2 x7 x8) (x6 x7 x8)) ⟶ ∀ x7 : ι → ι → ο . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ iff (x3 x8 x9) (x7 x8 x9)) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5) ⟶ 45090.. (0d9e7.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 363b9.. := λ x0 . λ x1 : ι → ι → ο . λ x2 : ι → ο . λ x3 x4 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (d2155.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (1216a.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4))))
Theorem 16b9f.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 x5 . x0 = 363b9.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 07bb5.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 x4 . x0 = f482f.. (363b9.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Theorem d5f6d.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 x5 . x0 = 363b9.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = 2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..)) x6 x7 (proof)
Theorem b9b42.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 x4 x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x1 x5 x6 = 2b2e3.. (f482f.. (363b9.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 x6 (proof)
Theorem e66ce.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 x5 . x0 = 363b9.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 (proof)
Theorem ee1d2.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 x4 x5 . prim1 x5 x0 ⟶ x2 x5 = decode_p (f482f.. (363b9.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Theorem 3c050.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 x5 . x0 = 363b9.. x1 x2 x3 x4 x5 ⟶ x4 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem 5b944.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 x4 . x3 = f482f.. (363b9.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem fb682.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 x5 . x0 = 363b9.. x1 x2 x3 x4 x5 ⟶ x5 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem 7a66d.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 x4 . x4 = f482f.. (363b9.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem c33e3.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 x5 : ι → ο . ∀ x6 x7 x8 x9 . 363b9.. x0 x2 x4 x6 x8 = 363b9.. x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x2 x10 x11 = x3 x10 x11)) (∀ x10 . prim1 x10 x0 ⟶ x4 x10 = x5 x10)) (x6 = x7)) (x8 = x9) (proof)
Theorem 1c47d.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 x4 : ι → ο . ∀ x5 x6 . (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ iff (x1 x7 x8) (x2 x7 x8)) ⟶ (∀ x7 . prim1 x7 x0 ⟶ iff (x3 x7) (x4 x7)) ⟶ 363b9.. x0 x1 x3 x5 x6 = 363b9.. x0 x2 x4 x5 x6 (proof)
Definition b0aa9.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x2 ⟶ ∀ x6 . prim1 x6 x2 ⟶ x1 (363b9.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem 57c76.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ b0aa9.. (363b9.. x0 x1 x2 x3 x4) (proof)
Theorem f788e.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 x4 . b0aa9.. (363b9.. x0 x1 x2 x3 x4) ⟶ prim1 x3 x0 (proof)
Theorem 6c77d.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 x4 . b0aa9.. (363b9.. x0 x1 x2 x3 x4) ⟶ prim1 x4 x0 (proof)
Theorem ed400.. : ∀ x0 . b0aa9.. x0 ⟶ x0 = 363b9.. (f482f.. x0 4a7ef..) (2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition 6cae4.. := λ x0 . λ x1 : ι → (ι → ι → ο) → (ι → ο) → ι → ι → ι . x1 (f482f.. x0 4a7ef..) (2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem dccff.. : ∀ x0 : ι → (ι → ι → ο) → (ι → ο) → ι → ι → ι . ∀ x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 x5 . (∀ x6 : ι → ι → ο . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ iff (x2 x7 x8) (x6 x7 x8)) ⟶ ∀ x7 : ι → ο . (∀ x8 . prim1 x8 x1 ⟶ iff (x3 x8) (x7 x8)) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5) ⟶ 6cae4.. (363b9.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition f5144.. := λ x0 . λ x1 : ι → (ι → ι → ο) → (ι → ο) → ι → ι → ο . x1 (f482f.. x0 4a7ef..) (2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 00522.. : ∀ x0 : ι → (ι → ι → ο) → (ι → ο) → ι → ι → ο . ∀ x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 x5 . (∀ x6 : ι → ι → ο . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ iff (x2 x7 x8) (x6 x7 x8)) ⟶ ∀ x7 : ι → ο . (∀ x8 . prim1 x8 x1 ⟶ iff (x3 x8) (x7 x8)) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5) ⟶ f5144.. (363b9.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 97f57.. := λ x0 . λ x1 x2 : ι → ο . λ x3 x4 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (1216a.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (1216a.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4))))
Theorem 6ab76.. : ∀ x0 x1 . ∀ x2 x3 : ι → ο . ∀ x4 x5 . x0 = 97f57.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 9c391.. : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 x4 . x0 = f482f.. (97f57.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Theorem e2b9c.. : ∀ x0 x1 . ∀ x2 x3 : ι → ο . ∀ x4 x5 . x0 = 97f57.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x6 = decode_p (f482f.. x0 (4ae4a.. 4a7ef..)) x6 (proof)
Theorem 91639.. : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 x4 x5 . prim1 x5 x0 ⟶ x1 x5 = decode_p (f482f.. (97f57.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 (proof)
Theorem 87536.. : ∀ x0 x1 . ∀ x2 x3 : ι → ο . ∀ x4 x5 . x0 = 97f57.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 (proof)
Theorem ffd4f.. : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 x4 x5 . prim1 x5 x0 ⟶ x2 x5 = decode_p (f482f.. (97f57.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Theorem 7317d.. : ∀ x0 x1 . ∀ x2 x3 : ι → ο . ∀ x4 x5 . x0 = 97f57.. x1 x2 x3 x4 x5 ⟶ x4 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem eda5f.. : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 x4 . x3 = f482f.. (97f57.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem 09398.. : ∀ x0 x1 . ∀ x2 x3 : ι → ο . ∀ x4 x5 . x0 = 97f57.. x1 x2 x3 x4 x5 ⟶ x5 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem ee4fc.. : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 x4 . x4 = f482f.. (97f57.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem 121e6.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ο . ∀ x6 x7 x8 x9 . 97f57.. x0 x2 x4 x6 x8 = 97f57.. x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 . prim1 x10 x0 ⟶ x2 x10 = x3 x10)) (∀ x10 . prim1 x10 x0 ⟶ x4 x10 = x5 x10)) (x6 = x7)) (x8 = x9) (proof)
Theorem 2b7cd.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ο . ∀ x5 x6 . (∀ x7 . prim1 x7 x0 ⟶ iff (x1 x7) (x2 x7)) ⟶ (∀ x7 . prim1 x7 x0 ⟶ iff (x3 x7) (x4 x7)) ⟶ 97f57.. x0 x1 x3 x5 x6 = 97f57.. x0 x2 x4 x5 x6 (proof)
Definition c19ad.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 x4 : ι → ο . ∀ x5 . prim1 x5 x2 ⟶ ∀ x6 . prim1 x6 x2 ⟶ x1 (97f57.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem 86de4.. : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ c19ad.. (97f57.. x0 x1 x2 x3 x4) (proof)
Theorem f20e8.. : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 x4 . c19ad.. (97f57.. x0 x1 x2 x3 x4) ⟶ prim1 x3 x0 (proof)
Theorem 50e1c.. : ∀ x0 . ∀ x1 x2 : ι → ο . ∀ x3 x4 . c19ad.. (97f57.. x0 x1 x2 x3 x4) ⟶ prim1 x4 x0 (proof)
Theorem c309c.. : ∀ x0 . c19ad.. x0 ⟶ x0 = 97f57.. (f482f.. x0 4a7ef..) (decode_p (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition 0ca1b.. := λ x0 . λ x1 : ι → (ι → ο) → (ι → ο) → ι → 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..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 9119e.. : ∀ x0 : ι → (ι → ο) → (ι → ο) → ι → ι → ι . ∀ x1 . ∀ x2 x3 : ι → ο . ∀ x4 x5 . (∀ x6 : ι → ο . (∀ x7 . prim1 x7 x1 ⟶ iff (x2 x7) (x6 x7)) ⟶ ∀ x7 : ι → ο . (∀ x8 . prim1 x8 x1 ⟶ iff (x3 x8) (x7 x8)) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5) ⟶ 0ca1b.. (97f57.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 008b9.. := λ x0 . λ x1 : ι → (ι → ο) → (ι → ο) → ι → 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..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem e3f76.. : ∀ x0 : ι → (ι → ο) → (ι → ο) → ι → ι → ο . ∀ x1 . ∀ x2 x3 : ι → ο . ∀ x4 x5 . (∀ x6 : ι → ο . (∀ x7 . prim1 x7 x1 ⟶ iff (x2 x7) (x6 x7)) ⟶ ∀ x7 : ι → ο . (∀ x8 . prim1 x8 x1 ⟶ iff (x3 x8) (x7 x8)) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5) ⟶ 008b9.. (97f57.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)