Proofgold Asset

Param 0fc90.. : ι → (ι → ι) → ι
Param 4ae4a.. : ι → ι
Param 4a7ef.. : ι
Param If_i : ο → ι → ι → ι
Param e0e40.. : ι → ((ι → ο) → ο) → ι
Param eb53d.. : ι → CT2 ι
Param 1216a.. : ι → (ι → ο) → ι
Definition 783bc.. := λ x0 . λ x1 : (ι → ο) → ο . λ x2 : ι → ι → ι . λ x3 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..))) (1216a.. x0 x3) (1216a.. x0 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 0edec.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 x5 : ι → ο . x0 = 783bc.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem e1561.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 x4 : ι → ο . x0 = f482f.. (783bc.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Param decode_c : ι → (ι → ο) → 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 x3 ⟶ prim1 x3 x0) ⟶ decode_c (e0e40.. x0 x1) x2 = x1 x2
Theorem ec944.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 x5 : ι → ο . x0 = 783bc.. x1 x2 x3 x4 x5 ⟶ ∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x1) ⟶ x2 x6 = decode_c (f482f.. x0 (4ae4a.. 4a7ef..)) x6 (proof)
Theorem 6ffdb.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 x4 x5 : ι → ο . (∀ x6 . x5 x6 ⟶ prim1 x6 x0) ⟶ x1 x5 = decode_c (f482f.. (783bc.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 (proof)
Param e3162.. : ι → ι → ι → ι
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 x0 ⟶ e3162.. (eb53d.. x0 x1) x2 x3 = x1 x2 x3
Theorem 1d7a9.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 x5 : ι → ο . x0 = 783bc.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x3 x6 x7 = e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 x7 (proof)
Theorem 8844f.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x5 x6 = e3162.. (f482f.. (783bc.. 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 e347f.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 x5 : ι → ο . x0 = 783bc.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x4 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 (proof)
Theorem 46d59.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x3 x5 = decode_p (f482f.. (783bc.. 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 77353.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 x5 : ι → ο . x0 = 783bc.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x5 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x6 (proof)
Theorem 6944d.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x4 x5 = decode_p (f482f.. (783bc.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x5 (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 b1e5b.. : ∀ x0 x1 . ∀ x2 x3 : (ι → ο) → ο . ∀ x4 x5 : ι → ι → ι . ∀ x6 x7 x8 x9 : ι → ο . 783bc.. x0 x2 x4 x6 x8 = 783bc.. x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 : ι → ο . (∀ x11 . x10 x11 ⟶ prim1 x11 x0) ⟶ x2 x10 = x3 x10)) (∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x4 x10 x11 = x5 x10 x11)) (∀ x10 . prim1 x10 x0 ⟶ x6 x10 = x7 x10)) (∀ x10 . prim1 x10 x0 ⟶ x8 x10 = x9 x10) (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 8fdaf.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x1 x3 x4 = x2 x3 x4) ⟶ eb53d.. x0 x1 = eb53d.. x0 x2
Known fe043.. : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . (∀ x3 : ι → ο . (∀ x4 . x3 x4 ⟶ prim1 x4 x0) ⟶ iff (x1 x3) (x2 x3)) ⟶ e0e40.. x0 x1 = e0e40.. x0 x2
Theorem a7310.. : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . ∀ x3 x4 : ι → ι → ι . ∀ x5 x6 x7 x8 : ι → ο . (∀ x9 : ι → ο . (∀ x10 . x9 x10 ⟶ prim1 x10 x0) ⟶ iff (x1 x9) (x2 x9)) ⟶ (∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 x0 ⟶ x3 x9 x10 = x4 x9 x10) ⟶ (∀ x9 . prim1 x9 x0 ⟶ iff (x5 x9) (x6 x9)) ⟶ (∀ x9 . prim1 x9 x0 ⟶ iff (x7 x9) (x8 x9)) ⟶ 783bc.. x0 x1 x3 x5 x7 = 783bc.. x0 x2 x4 x6 x8 (proof)
Definition c9ba2.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : (ι → ο) → ο . ∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x2 ⟶ ∀ x6 . prim1 x6 x2 ⟶ prim1 (x4 x5 x6) x2) ⟶ ∀ x5 x6 : ι → ο . x1 (783bc.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem 4344e.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x3 x4) x0) ⟶ ∀ x3 x4 : ι → ο . c9ba2.. (783bc.. x0 x1 x2 x3 x4) (proof)
Theorem 0d70c.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 x4 : ι → ο . c9ba2.. (783bc.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x2 x5 x6) x0 (proof)
Known iff_refl : ∀ x0 : ο . iff x0 x0
Theorem 75cc6.. : ∀ x0 . c9ba2.. x0 ⟶ x0 = 783bc.. (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))) (proof)
Definition 64ca8.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι → ι) → (ι → ο) → (ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem 7046d.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι → ι) → (ι → ο) → (ι → ο) → ι . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 x5 : ι → ο . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x1) ⟶ iff (x2 x7) (x6 x7)) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ x3 x8 x9 = x7 x8 x9) ⟶ ∀ x8 : ι → ο . (∀ x9 . prim1 x9 x1 ⟶ iff (x4 x9) (x8 x9)) ⟶ ∀ x9 : ι → ο . (∀ x10 . prim1 x10 x1 ⟶ iff (x5 x10) (x9 x10)) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ 64ca8.. (783bc.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 5d578.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι → ι) → (ι → ο) → (ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem 3a7bb.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι → ι) → (ι → ο) → (ι → ο) → ο . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 x5 : ι → ο . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x1) ⟶ iff (x2 x7) (x6 x7)) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ x3 x8 x9 = x7 x8 x9) ⟶ ∀ x8 : ι → ο . (∀ x9 . prim1 x9 x1 ⟶ iff (x4 x9) (x8 x9)) ⟶ ∀ x9 : ι → ο . (∀ x10 . prim1 x10 x1 ⟶ iff (x5 x10) (x9 x10)) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ 5d578.. (783bc.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition bebf6.. := λ x0 . λ x1 : (ι → ο) → ο . λ x2 : ι → ι → ι . 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..) (e0e40.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (eb53d.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (1216a.. x0 x3) x4))))
Theorem e31dc.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 . x0 = bebf6.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 4d890.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 . x0 = f482f.. (bebf6.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Theorem 6bf38.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 . x0 = bebf6.. x1 x2 x3 x4 x5 ⟶ ∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x1) ⟶ x2 x6 = decode_c (f482f.. x0 (4ae4a.. 4a7ef..)) x6 (proof)
Theorem 2f459.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 . ∀ x5 : ι → ο . (∀ x6 . x5 x6 ⟶ prim1 x6 x0) ⟶ x1 x5 = decode_c (f482f.. (bebf6.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 (proof)
Theorem 8d565.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 . x0 = bebf6.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x3 x6 x7 = e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 x7 (proof)
Theorem 546c0.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x5 x6 = e3162.. (f482f.. (bebf6.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Theorem c5e5b.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 . x0 = bebf6.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x4 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 (proof)
Theorem bedba.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . prim1 x5 x0 ⟶ x3 x5 = decode_p (f482f.. (bebf6.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 (proof)
Theorem afc32.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 . x0 = bebf6.. x1 x2 x3 x4 x5 ⟶ x5 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem 7bb35.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 . x4 = f482f.. (bebf6.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem 4f115.. : ∀ x0 x1 . ∀ x2 x3 : (ι → ο) → ο . ∀ x4 x5 : ι → ι → ι . ∀ x6 x7 : ι → ο . ∀ x8 x9 . bebf6.. x0 x2 x4 x6 x8 = bebf6.. x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 : ι → ο . (∀ x11 . x10 x11 ⟶ prim1 x11 x0) ⟶ x2 x10 = x3 x10)) (∀ 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)
Theorem d3e57.. : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . ∀ x3 x4 : ι → ι → ι . ∀ x5 x6 : ι → ο . ∀ x7 . (∀ x8 : ι → ο . (∀ x9 . x8 x9 ⟶ prim1 x9 x0) ⟶ iff (x1 x8) (x2 x8)) ⟶ (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x3 x8 x9 = x4 x8 x9) ⟶ (∀ x8 . prim1 x8 x0 ⟶ iff (x5 x8) (x6 x8)) ⟶ bebf6.. x0 x1 x3 x5 x7 = bebf6.. x0 x2 x4 x6 x7 (proof)
Definition ab832.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : (ι → ο) → ο . ∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x2 ⟶ ∀ x6 . prim1 x6 x2 ⟶ prim1 (x4 x5 x6) x2) ⟶ ∀ x5 : ι → ο . ∀ x6 . prim1 x6 x2 ⟶ x1 (bebf6.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem 18431.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x3 x4) x0) ⟶ ∀ x3 : ι → ο . ∀ x4 . prim1 x4 x0 ⟶ ab832.. (bebf6.. x0 x1 x2 x3 x4) (proof)
Theorem 04af0.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 . ab832.. (bebf6.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x2 x5 x6) x0 (proof)
Theorem a85d0.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 . ab832.. (bebf6.. x0 x1 x2 x3 x4) ⟶ prim1 x4 x0 (proof)
Theorem 5da3a.. : ∀ x0 . ab832.. x0 ⟶ x0 = bebf6.. (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition f52fd.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι → ι) → (ι → ο) → ι → ι . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem d409f.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι → ι) → (ι → ο) → ι → ι . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x1) ⟶ iff (x2 x7) (x6 x7)) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ 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) ⟶ f52fd.. (bebf6.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 971dc.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι → ι) → (ι → ο) → ι → ο . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 13766.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι → ι) → (ι → ο) → ι → ο . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x1) ⟶ iff (x2 x7) (x6 x7)) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ 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) ⟶ 971dc.. (bebf6.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 6b0a5.. := λ x0 . λ x1 : (ι → ο) → ο . λ x2 : ι → ι → ι . λ x3 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..))) x3 x4))))
Theorem 3c3d8.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 x5 . x0 = 6b0a5.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 4bb01.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 x4 . x0 = f482f.. (6b0a5.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Theorem 96968.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 x5 . x0 = 6b0a5.. x1 x2 x3 x4 x5 ⟶ ∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x1) ⟶ x2 x6 = decode_c (f482f.. x0 (4ae4a.. 4a7ef..)) x6 (proof)
Theorem cd382.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 x4 . ∀ x5 : ι → ο . (∀ x6 . x5 x6 ⟶ prim1 x6 x0) ⟶ x1 x5 = decode_c (f482f.. (6b0a5.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 (proof)
Theorem b21e7.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 x5 . x0 = 6b0a5.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x3 x6 x7 = e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 x7 (proof)
Theorem 1c594.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 x4 x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x5 x6 = e3162.. (f482f.. (6b0a5.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Theorem 8d5fd.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 x5 . x0 = 6b0a5.. x1 x2 x3 x4 x5 ⟶ x4 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem c7eb2.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 x4 . x3 = f482f.. (6b0a5.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem 56b62.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 x5 . x0 = 6b0a5.. x1 x2 x3 x4 x5 ⟶ x5 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem 5753b.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 x4 . x4 = f482f.. (6b0a5.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem 24d22.. : ∀ x0 x1 . ∀ x2 x3 : (ι → ο) → ο . ∀ x4 x5 : ι → ι → ι . ∀ x6 x7 x8 x9 . 6b0a5.. x0 x2 x4 x6 x8 = 6b0a5.. x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 : ι → ο . (∀ x11 . x10 x11 ⟶ prim1 x11 x0) ⟶ x2 x10 = x3 x10)) (∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x4 x10 x11 = x5 x10 x11)) (x6 = x7)) (x8 = x9) (proof)
Theorem a72a6.. : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . ∀ x3 x4 : ι → ι → ι . ∀ x5 x6 . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x0) ⟶ iff (x1 x7) (x2 x7)) ⟶ (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x3 x7 x8 = x4 x7 x8) ⟶ 6b0a5.. x0 x1 x3 x5 x6 = 6b0a5.. x0 x2 x4 x5 x6 (proof)
Definition 8b540.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : (ι → ο) → ο . ∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x2 ⟶ ∀ x6 . prim1 x6 x2 ⟶ prim1 (x4 x5 x6) x2) ⟶ ∀ x5 . prim1 x5 x2 ⟶ ∀ x6 . prim1 x6 x2 ⟶ x1 (6b0a5.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem 08cf8.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x3 x4) x0) ⟶ ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ 8b540.. (6b0a5.. x0 x1 x2 x3 x4) (proof)
Theorem da0e2.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 x4 . 8b540.. (6b0a5.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x2 x5 x6) x0 (proof)
Theorem 402fa.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 x4 . 8b540.. (6b0a5.. x0 x1 x2 x3 x4) ⟶ prim1 x3 x0 (proof)
Theorem 6c39d.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι → ι . ∀ x3 x4 . 8b540.. (6b0a5.. x0 x1 x2 x3 x4) ⟶ prim1 x4 x0 (proof)
Theorem ae43e.. : ∀ x0 . 8b540.. x0 ⟶ x0 = 6b0a5.. (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition d0646.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι → ι) → ι → ι → ι . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 5ba00.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι → ι) → ι → ι → ι . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 x5 . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x1) ⟶ iff (x2 x7) (x6 x7)) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ x3 x8 x9 = x7 x8 x9) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5) ⟶ d0646.. (6b0a5.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 3b3d8.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι → ι) → ι → ι → ο . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 1eb75.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι → ι) → ι → ι → ο . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι → ι . ∀ x4 x5 . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x1) ⟶ iff (x2 x7) (x6 x7)) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ x3 x8 x9 = x7 x8 x9) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5) ⟶ 3b3d8.. (6b0a5.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Param d2155.. : ι → (ι → ι → ο) → ι
Definition 34992.. := λ x0 . λ x1 : (ι → ο) → ο . λ x2 : 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..)) (0fc90.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (d2155.. x0 x3) (d2155.. x0 x4)))))
Theorem 6998e.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = 34992.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem af941.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 x4 x5 : ι → ι → ο . x5 x0 (f482f.. (34992.. x0 x1 x2 x3 x4) 4a7ef..) ⟶ x5 (f482f.. (34992.. x0 x1 x2 x3 x4) 4a7ef..) x0 (proof)
Theorem 9c9ff.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = 34992.. x1 x2 x3 x4 x5 ⟶ ∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x1) ⟶ x2 x6 = decode_c (f482f.. x0 (4ae4a.. 4a7ef..)) x6 (proof)
Theorem 0a284.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ x5 : ι → ο . (∀ x6 . x5 x6 ⟶ prim1 x6 x0) ⟶ x1 x5 = decode_c (f482f.. (34992.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 (proof)
Known f22ec.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 x0 ⟶ f482f.. (0fc90.. x0 x1) x2 = x1 x2
Theorem 6d328.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = 34992.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x6 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 (proof)
Theorem 0f0c4.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x2 x5 = f482f.. (f482f.. (34992.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Param 2b2e3.. : ι → ι → ι → ο
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 7173f.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = 34992.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x4 x6 x7 = 2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 x7 (proof)
Theorem fa07e.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 x5 x6 = 2b2e3.. (f482f.. (34992.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 x6 (proof)
Theorem bcf71.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = 34992.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x5 x6 x7 = 2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x6 x7 (proof)
Theorem c22ae.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x4 x5 x6 = 2b2e3.. (f482f.. (34992.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x5 x6 (proof)
Theorem c0c2b.. : ∀ x0 x1 . ∀ x2 x3 : (ι → ο) → ο . ∀ x4 x5 : ι → ι . ∀ x6 x7 x8 x9 : ι → ι → ο . 34992.. x0 x2 x4 x6 x8 = 34992.. x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 : ι → ο . (∀ x11 . x10 x11 ⟶ prim1 x11 x0) ⟶ x2 x10 = x3 x10)) (∀ x10 . prim1 x10 x0 ⟶ x4 x10 = x5 x10)) (∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x6 x10 x11 = x7 x10 x11)) (∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x8 x10 x11 = x9 x10 x11) (proof)
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
Known 4402a.. : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ x1 x3 = x2 x3) ⟶ 0fc90.. x0 x1 = 0fc90.. x0 x2
Theorem 2f911.. : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . ∀ x3 x4 : ι → ι . ∀ x5 x6 x7 x8 : ι → ι → ο . (∀ x9 : ι → ο . (∀ x10 . x9 x10 ⟶ prim1 x10 x0) ⟶ iff (x1 x9) (x2 x9)) ⟶ (∀ x9 . prim1 x9 x0 ⟶ x3 x9 = x4 x9) ⟶ (∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 x0 ⟶ iff (x5 x9 x10) (x6 x9 x10)) ⟶ (∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 x0 ⟶ iff (x7 x9 x10) (x8 x9 x10)) ⟶ 34992.. x0 x1 x3 x5 x7 = 34992.. x0 x2 x4 x6 x8 (proof)
Definition 49161.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : (ι → ο) → ο . ∀ x4 : ι → ι . (∀ x5 . prim1 x5 x2 ⟶ prim1 (x4 x5) x2) ⟶ ∀ x5 x6 : ι → ι → ο . x1 (34992.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem 30c66.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x0) ⟶ ∀ x3 x4 : ι → ι → ο . 49161.. (34992.. x0 x1 x2 x3 x4) (proof)
Theorem 58b23.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . 49161.. (34992.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x2 x5) x0 (proof)
Theorem 3f026.. : ∀ x0 . 49161.. x0 ⟶ x0 = 34992.. (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))) (proof)
Definition cd973.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ι → ο) → (ι → ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem 9e0ff.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ι → ο) → (ι → ι → ο) → ι . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x1) ⟶ iff (x2 x7) (x6 x7)) ⟶ ∀ x7 : ι → ι . (∀ x8 . prim1 x8 x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ x8 : ι → ι → ο . (∀ x9 . prim1 x9 x1 ⟶ ∀ x10 . prim1 x10 x1 ⟶ iff (x4 x9 x10) (x8 x9 x10)) ⟶ ∀ x9 : ι → ι → ο . (∀ x10 . prim1 x10 x1 ⟶ ∀ x11 . prim1 x11 x1 ⟶ iff (x5 x10 x11) (x9 x10 x11)) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ cd973.. (34992.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 0d56c.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ι → ο) → (ι → ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem caeb2.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ι) → (ι → ι → ο) → (ι → ι → ο) → ο . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x1) ⟶ iff (x2 x7) (x6 x7)) ⟶ ∀ x7 : ι → ι . (∀ x8 . prim1 x8 x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ x8 : ι → ι → ο . (∀ x9 . prim1 x9 x1 ⟶ ∀ x10 . prim1 x10 x1 ⟶ iff (x4 x9 x10) (x8 x9 x10)) ⟶ ∀ x9 : ι → ι → ο . (∀ x10 . prim1 x10 x1 ⟶ ∀ x11 . prim1 x11 x1 ⟶ iff (x5 x10 x11) (x9 x10 x11)) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ 0d56c.. (34992.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)