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 1216a.. : ι → (ι → ο) → ι
Definition 98165.. := λ x0 . λ x1 : (ι → ο) → 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..) (e0e40.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (1216a.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) 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 9833d.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ο . ∀ x4 x5 . x0 = 98165.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem b1f5f.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ο . ∀ x3 x4 . x0 = f482f.. (98165.. 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 54425.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ο . ∀ x4 x5 . x0 = 98165.. 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 aa1f5.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ο . ∀ x3 x4 . ∀ x5 : ι → ο . (∀ x6 . x5 x6 ⟶ prim1 x6 x0) ⟶ x1 x5 = decode_c (f482f.. (98165.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 (proof)
Param decode_p : ι → ι → ο
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 931fe.. : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . prim1 x2 x0 ⟶ decode_p (1216a.. x0 x1) x2 = x1 x2
Theorem 16264.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ο . ∀ x4 x5 . x0 = 98165.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 (proof)
Theorem 19106.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ο . ∀ x3 x4 x5 . prim1 x5 x0 ⟶ x2 x5 = decode_p (f482f.. (98165.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 (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
Theorem 99347.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ο . ∀ x4 x5 . x0 = 98165.. x1 x2 x3 x4 x5 ⟶ x4 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem 25edf.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ο . ∀ x3 x4 . x3 = f482f.. (98165.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (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 36f32.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ο . ∀ x4 x5 . x0 = 98165.. x1 x2 x3 x4 x5 ⟶ x5 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem 84335.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ο . ∀ x3 x4 . x4 = f482f.. (98165.. 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 e176b.. : ∀ x0 x1 . ∀ x2 x3 : (ι → ο) → ο . ∀ x4 x5 : ι → ο . ∀ x6 x7 x8 x9 . 98165.. x0 x2 x4 x6 x8 = 98165.. 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)) (x6 = x7)) (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 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 01f39.. : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . ∀ x3 x4 : ι → ο . ∀ x5 x6 . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 x0) ⟶ iff (x1 x7) (x2 x7)) ⟶ (∀ x7 . prim1 x7 x0 ⟶ iff (x3 x7) (x4 x7)) ⟶ 98165.. x0 x1 x3 x5 x6 = 98165.. x0 x2 x4 x5 x6 (proof)
Definition 56056.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : (ι → ο) → ο . ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x2 ⟶ ∀ x6 . prim1 x6 x2 ⟶ x1 (98165.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem ce3ec.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ο . ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ 56056.. (98165.. x0 x1 x2 x3 x4) (proof)
Theorem 7d48e.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ο . ∀ x3 x4 . 56056.. (98165.. x0 x1 x2 x3 x4) ⟶ prim1 x3 x0 (proof)
Theorem 270c1.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ο . ∀ x3 x4 . 56056.. (98165.. x0 x1 x2 x3 x4) ⟶ prim1 x4 x0 (proof)
Known iff_refl : ∀ x0 : ο . iff x0 x0
Theorem f08de.. : ∀ x0 . 56056.. x0 ⟶ x0 = 98165.. (f482f.. x0 4a7ef..) (decode_c (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 8ac7d.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ο) → ι → ι → ι . x1 (f482f.. x0 4a7ef..) (decode_c (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 fba5f.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ο) → ι → ι → ι . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ο . ∀ x4 x5 . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 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) ⟶ 8ac7d.. (98165.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 4aeb5.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ο) → ι → ι → ο . x1 (f482f.. x0 4a7ef..) (decode_c (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 48205.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ο) → ι → ι → ο . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ο . ∀ x4 x5 . (∀ x6 : (ι → ο) → ο . (∀ x7 : ι → ο . (∀ x8 . x7 x8 ⟶ prim1 x8 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) ⟶ 4aeb5.. (98165.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Param eb53d.. : ι → CT2 ι
Definition 39199.. := λ x0 . λ x1 x2 x3 : ι → ι → ι . λ x4 : ι → ι . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (eb53d.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (eb53d.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (eb53d.. x0 x3) (0fc90.. x0 x4)))))
Theorem f5284.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 : ι → ι . x0 = 39199.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 48ea9.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . x0 = f482f.. (39199.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Param e3162.. : ι → ι → ι → ι
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 d6a83.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 : ι → ι . x0 = 39199.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = e3162.. (f482f.. x0 (4ae4a.. 4a7ef..)) x6 x7 (proof)
Theorem 10c2e.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x1 x5 x6 = e3162.. (f482f.. (39199.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 x6 (proof)
Theorem 48f4e.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 : ι → ι . x0 = 39199.. 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 2bb37.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x5 x6 = e3162.. (f482f.. (39199.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Theorem 1c20d.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 : ι → ι . x0 = 39199.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x4 x6 x7 = e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 x7 (proof)
Theorem 5508b.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 x5 x6 = e3162.. (f482f.. (39199.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 x6 (proof)
Known f22ec.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 x0 ⟶ f482f.. (0fc90.. x0 x1) x2 = x1 x2
Theorem 98709.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 : ι → ι . x0 = 39199.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x5 x6 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x6 (proof)
Theorem ff25b.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ x5 . prim1 x5 x0 ⟶ x4 x5 = f482f.. (f482f.. (39199.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x5 (proof)
Theorem 9dfac.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 x6 x7 : ι → ι → ι . ∀ x8 x9 : ι → ι . 39199.. x0 x2 x4 x6 x8 = 39199.. 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 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x6 x10 x11 = x7 x10 x11)) (∀ x10 . prim1 x10 x0 ⟶ x8 x10 = x9 x10) (proof)
Known 4402a.. : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ x1 x3 = x2 x3) ⟶ 0fc90.. x0 x1 = 0fc90.. 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
Theorem 365e9.. : ∀ x0 . ∀ x1 x2 x3 x4 x5 x6 : ι → ι → ι . ∀ x7 x8 : ι → ι . (∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 x0 ⟶ x1 x9 x10 = x2 x9 x10) ⟶ (∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 x0 ⟶ x3 x9 x10 = x4 x9 x10) ⟶ (∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 x0 ⟶ x5 x9 x10 = x6 x9 x10) ⟶ (∀ x9 . prim1 x9 x0 ⟶ x7 x9 = x8 x9) ⟶ 39199.. x0 x1 x3 x5 x7 = 39199.. x0 x2 x4 x6 x8 (proof)
Definition 38635.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ι . (∀ x4 . prim1 x4 x2 ⟶ ∀ x5 . prim1 x5 x2 ⟶ prim1 (x3 x4 x5) x2) ⟶ ∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x2 ⟶ ∀ x6 . prim1 x6 x2 ⟶ prim1 (x4 x5 x6) x2) ⟶ ∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x2 ⟶ ∀ x7 . prim1 x7 x2 ⟶ prim1 (x5 x6 x7) x2) ⟶ ∀ x6 : ι → ι . (∀ x7 . prim1 x7 x2 ⟶ prim1 (x6 x7) x2) ⟶ x1 (39199.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem abda6.. : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x1 x2 x3) x0) ⟶ ∀ x2 : ι → ι → ι . (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x3 x4) x0) ⟶ ∀ x3 : ι → ι → ι . (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x3 x4 x5) x0) ⟶ ∀ x4 : ι → ι . (∀ x5 . prim1 x5 x0 ⟶ prim1 (x4 x5) x0) ⟶ 38635.. (39199.. x0 x1 x2 x3 x4) (proof)
Theorem babf6.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . 38635.. (39199.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x1 x5 x6) x0 (proof)
Theorem 2092e.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . 38635.. (39199.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x2 x5 x6) x0 (proof)
Theorem 03cda.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . 38635.. (39199.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x3 x5 x6) x0 (proof)
Theorem 9e42b.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . 38635.. (39199.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x4 x5) x0 (proof)
Theorem b1824.. : ∀ x0 . 38635.. x0 ⟶ x0 = 39199.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))) (proof)
Definition 8695a.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ι) → (ι → ι) → ι . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem bcc0c.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ι) → (ι → ι) → ι . ∀ x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 : ι → ι . (∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x2 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ x3 x8 x9 = x7 x8 x9) ⟶ ∀ x8 : ι → ι → ι . (∀ x9 . prim1 x9 x1 ⟶ ∀ x10 . prim1 x10 x1 ⟶ x4 x9 x10 = x8 x9 x10) ⟶ ∀ x9 : ι → ι . (∀ x10 . prim1 x10 x1 ⟶ x5 x10 = x9 x10) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ 8695a.. (39199.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 8c11b.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ι) → (ι → ι) → ο . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem 223da.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ι) → (ι → ι) → ο . ∀ x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 : ι → ι . (∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x2 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ x3 x8 x9 = x7 x8 x9) ⟶ ∀ x8 : ι → ι → ι . (∀ x9 . prim1 x9 x1 ⟶ ∀ x10 . prim1 x10 x1 ⟶ x4 x9 x10 = x8 x9 x10) ⟶ ∀ x9 : ι → ι . (∀ x10 . prim1 x10 x1 ⟶ x5 x10 = x9 x10) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ 8c11b.. (39199.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Param d2155.. : ι → (ι → ι → ο) → ι
Definition 1eafe.. := λ x0 . λ x1 x2 x3 : ι → ι → ι . λ x4 : ι → ι → ο . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (eb53d.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (eb53d.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (eb53d.. x0 x3) (d2155.. x0 x4)))))
Theorem 223ee.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . x0 = 1eafe.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 95ea3.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ι → ο . x5 x0 (f482f.. (1eafe.. x0 x1 x2 x3 x4) 4a7ef..) ⟶ x5 (f482f.. (1eafe.. x0 x1 x2 x3 x4) 4a7ef..) x0 (proof)
Theorem 9fd74.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . x0 = 1eafe.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = e3162.. (f482f.. x0 (4ae4a.. 4a7ef..)) x6 x7 (proof)
Theorem 05c42.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x1 x5 x6 = e3162.. (f482f.. (1eafe.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 x6 (proof)
Theorem d43b4.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . x0 = 1eafe.. 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 05f51.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x5 x6 = e3162.. (f482f.. (1eafe.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Theorem f1a5f.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . x0 = 1eafe.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x4 x6 x7 = e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 x7 (proof)
Theorem db2c8.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 x5 x6 = e3162.. (f482f.. (1eafe.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 x6 (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 5aadd.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . x0 = 1eafe.. 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 35d4e.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x4 x5 x6 = 2b2e3.. (f482f.. (1eafe.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x5 x6 (proof)
Theorem 4a788.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 x6 x7 : ι → ι → ι . ∀ x8 x9 : ι → ι → ο . 1eafe.. x0 x2 x4 x6 x8 = 1eafe.. 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 ⟶ ∀ 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
Theorem 0a9b2.. : ∀ x0 . ∀ x1 x2 x3 x4 x5 x6 : ι → ι → ι . ∀ x7 x8 : ι → ι → ο . (∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 x0 ⟶ x1 x9 x10 = x2 x9 x10) ⟶ (∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 x0 ⟶ x3 x9 x10 = x4 x9 x10) ⟶ (∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 x0 ⟶ x5 x9 x10 = x6 x9 x10) ⟶ (∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 x0 ⟶ iff (x7 x9 x10) (x8 x9 x10)) ⟶ 1eafe.. x0 x1 x3 x5 x7 = 1eafe.. x0 x2 x4 x6 x8 (proof)
Definition 4e60e.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ι . (∀ x4 . prim1 x4 x2 ⟶ ∀ x5 . prim1 x5 x2 ⟶ prim1 (x3 x4 x5) x2) ⟶ ∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x2 ⟶ ∀ x6 . prim1 x6 x2 ⟶ prim1 (x4 x5 x6) x2) ⟶ ∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x2 ⟶ ∀ x7 . prim1 x7 x2 ⟶ prim1 (x5 x6 x7) x2) ⟶ ∀ x6 : ι → ι → ο . x1 (1eafe.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem 36098.. : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x1 x2 x3) x0) ⟶ ∀ x2 : ι → ι → ι . (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x3 x4) x0) ⟶ ∀ x3 : ι → ι → ι . (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x3 x4 x5) x0) ⟶ ∀ x4 : ι → ι → ο . 4e60e.. (1eafe.. x0 x1 x2 x3 x4) (proof)
Theorem 68c0b.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . 4e60e.. (1eafe.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x1 x5 x6) x0 (proof)
Theorem 08be0.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . 4e60e.. (1eafe.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x2 x5 x6) x0 (proof)
Theorem 3a31c.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . 4e60e.. (1eafe.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x3 x5 x6) x0 (proof)
Theorem 4c198.. : ∀ x0 . 4e60e.. x0 ⟶ x0 = 1eafe.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))) (proof)
Definition a5367.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem 2f766.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ι . ∀ x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . (∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x2 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ x3 x8 x9 = x7 x8 x9) ⟶ ∀ x8 : ι → ι → ι . (∀ x9 . prim1 x9 x1 ⟶ ∀ x10 . prim1 x10 x1 ⟶ 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) ⟶ a5367.. (1eafe.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 83aaa.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem 25c57.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ο . ∀ x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . (∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x2 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ x3 x8 x9 = x7 x8 x9) ⟶ ∀ x8 : ι → ι → ι . (∀ x9 . prim1 x9 x1 ⟶ ∀ x10 . prim1 x10 x1 ⟶ 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) ⟶ 83aaa.. (1eafe.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 94613.. := λ x0 . λ x1 x2 x3 : ι → ι → ι . λ x4 : ι → ο . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (eb53d.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (eb53d.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (eb53d.. x0 x3) (1216a.. x0 x4)))))
Theorem 063b4.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 : ι → ο . x0 = 94613.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 2ab54.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ο . x0 = f482f.. (94613.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Theorem f6f08.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 : ι → ο . x0 = 94613.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = e3162.. (f482f.. x0 (4ae4a.. 4a7ef..)) x6 x7 (proof)
Theorem 22e75.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x1 x5 x6 = e3162.. (f482f.. (94613.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 x6 (proof)
Theorem d6c77.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 : ι → ο . x0 = 94613.. 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 bf883.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x5 x6 = e3162.. (f482f.. (94613.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Theorem 5da7d.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 : ι → ο . x0 = 94613.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x4 x6 x7 = e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 x7 (proof)
Theorem eaad1.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 x5 x6 = e3162.. (f482f.. (94613.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 x6 (proof)
Theorem c4fb2.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 : ι → ο . x0 = 94613.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x5 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x6 (proof)
Theorem 8c421.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x4 x5 = decode_p (f482f.. (94613.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x5 (proof)
Theorem 5cc8d.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 x6 x7 : ι → ι → ι . ∀ x8 x9 : ι → ο . 94613.. x0 x2 x4 x6 x8 = 94613.. 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 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x6 x10 x11 = x7 x10 x11)) (∀ x10 . prim1 x10 x0 ⟶ x8 x10 = x9 x10) (proof)
Theorem 0dcb9.. : ∀ x0 . ∀ x1 x2 x3 x4 x5 x6 : ι → ι → ι . ∀ x7 x8 : ι → ο . (∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 x0 ⟶ x1 x9 x10 = x2 x9 x10) ⟶ (∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 x0 ⟶ x3 x9 x10 = x4 x9 x10) ⟶ (∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 x0 ⟶ x5 x9 x10 = x6 x9 x10) ⟶ (∀ x9 . prim1 x9 x0 ⟶ iff (x7 x9) (x8 x9)) ⟶ 94613.. x0 x1 x3 x5 x7 = 94613.. x0 x2 x4 x6 x8 (proof)
Definition 7c481.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ι . (∀ x4 . prim1 x4 x2 ⟶ ∀ x5 . prim1 x5 x2 ⟶ prim1 (x3 x4 x5) x2) ⟶ ∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x2 ⟶ ∀ x6 . prim1 x6 x2 ⟶ prim1 (x4 x5 x6) x2) ⟶ ∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x2 ⟶ ∀ x7 . prim1 x7 x2 ⟶ prim1 (x5 x6 x7) x2) ⟶ ∀ x6 : ι → ο . x1 (94613.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem 5c4b8.. : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x1 x2 x3) x0) ⟶ ∀ x2 : ι → ι → ι . (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x3 x4) x0) ⟶ ∀ x3 : ι → ι → ι . (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x3 x4 x5) x0) ⟶ ∀ x4 : ι → ο . 7c481.. (94613.. x0 x1 x2 x3 x4) (proof)
Theorem 10c49.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ο . 7c481.. (94613.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x1 x5 x6) x0 (proof)
Theorem fedac.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ο . 7c481.. (94613.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x2 x5 x6) x0 (proof)
Theorem 14447.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 : ι → ο . 7c481.. (94613.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x3 x5 x6) x0 (proof)
Theorem 290eb.. : ∀ x0 . 7c481.. x0 ⟶ x0 = 94613.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))) (proof)
Definition 9ad8d.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ι) → (ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem ffddd.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ι) → (ι → ο) → ι . ∀ x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 : ι → ο . (∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x2 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ x3 x8 x9 = x7 x8 x9) ⟶ ∀ x8 : ι → ι → ι . (∀ x9 . prim1 x9 x1 ⟶ ∀ x10 . prim1 x10 x1 ⟶ x4 x9 x10 = x8 x9 x10) ⟶ ∀ x9 : ι → ο . (∀ x10 . prim1 x10 x1 ⟶ iff (x5 x10) (x9 x10)) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ 9ad8d.. (94613.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 0499f.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ι) → (ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem 61142.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ι) → (ι → ο) → ο . ∀ x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 : ι → ο . (∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x2 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ x3 x8 x9 = x7 x8 x9) ⟶ ∀ x8 : ι → ι → ι . (∀ x9 . prim1 x9 x1 ⟶ ∀ x10 . prim1 x10 x1 ⟶ x4 x9 x10 = x8 x9 x10) ⟶ ∀ x9 : ι → ο . (∀ x10 . prim1 x10 x1 ⟶ iff (x5 x10) (x9 x10)) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ 0499f.. (94613.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)