Proofgold Address

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 c42e1.. := λ x0 . λ x1 : (ι → ο) → 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..) (e0e40.. 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 4f989.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ο . ∀ x4 . x0 = c42e1.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 572ce.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ο . ∀ x3 . x0 = f482f.. (c42e1.. x0 x1 x2 x3) 4a7ef.. (proof)
Param decode_c : ι → (ι → ο) → 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 81500.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ο . (∀ x3 . x2 x3 ⟶ prim1 x3 x0) ⟶ decode_c (e0e40.. x0 x1) x2 = x1 x2
Theorem 7f14b.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ο . ∀ x4 . x0 = c42e1.. x1 x2 x3 x4 ⟶ ∀ x5 : ι → ο . (∀ x6 . x5 x6 ⟶ prim1 x6 x1) ⟶ x2 x5 = decode_c (f482f.. x0 (4ae4a.. 4a7ef..)) x5 (proof)
Theorem b579b.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ο . ∀ x3 . ∀ x4 : ι → ο . (∀ x5 . x4 x5 ⟶ prim1 x5 x0) ⟶ x1 x4 = decode_c (f482f.. (c42e1.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 (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 5c88b.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ο . ∀ x4 . x0 = c42e1.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x3 x5 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Theorem 559b0.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ο . ∀ x3 x4 . prim1 x4 x0 ⟶ x2 x4 = decode_p (f482f.. (c42e1.. 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 3896b.. : ∀ x0 x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ο . ∀ x4 . x0 = c42e1.. x1 x2 x3 x4 ⟶ x4 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem 0821e.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ο . ∀ x3 . x3 = f482f.. (c42e1.. 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 429e9.. : ∀ x0 x1 . ∀ x2 x3 : (ι → ο) → ο . ∀ x4 x5 : ι → ο . ∀ x6 x7 . c42e1.. x0 x2 x4 x6 = c42e1.. x1 x3 x5 x7 ⟶ and (and (and (x0 = x1) (∀ x8 : ι → ο . (∀ x9 . x8 x9 ⟶ prim1 x9 x0) ⟶ x2 x8 = x3 x8)) (∀ 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 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 b08f5.. : ∀ x0 . ∀ x1 x2 : (ι → ο) → ο . ∀ x3 x4 : ι → ο . ∀ x5 . (∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 x0) ⟶ iff (x1 x6) (x2 x6)) ⟶ (∀ x6 . prim1 x6 x0 ⟶ iff (x3 x6) (x4 x6)) ⟶ c42e1.. x0 x1 x3 x5 = c42e1.. x0 x2 x4 x5 (proof)
Definition 7f5b4.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : (ι → ο) → ο . ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x2 ⟶ x1 (c42e1.. x2 x3 x4 x5)) ⟶ x1 x0
Theorem c3940.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ο . ∀ x3 . prim1 x3 x0 ⟶ 7f5b4.. (c42e1.. x0 x1 x2 x3) (proof)
Theorem 22eb2.. : ∀ x0 . ∀ x1 : (ι → ο) → ο . ∀ x2 : ι → ο . ∀ x3 . 7f5b4.. (c42e1.. x0 x1 x2 x3) ⟶ prim1 x3 x0 (proof)
Known iff_refl : ∀ x0 : ο . iff x0 x0
Theorem 58abf.. : ∀ x0 . 7f5b4.. x0 ⟶ x0 = c42e1.. (f482f.. x0 4a7ef..) (decode_c (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Definition bbb33.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ο) → 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..))))
Theorem 63f68.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ο) → ι → ι . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ο . ∀ x4 . (∀ x5 : (ι → ο) → ο . (∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 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) ⟶ bbb33.. (c42e1.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 7a422.. := λ x0 . λ x1 : ι → ((ι → ο) → ο) → (ι → ο) → 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..))))
Theorem 65b34.. : ∀ x0 : ι → ((ι → ο) → ο) → (ι → ο) → ι → ο . ∀ x1 . ∀ x2 : (ι → ο) → ο . ∀ x3 : ι → ο . ∀ x4 . (∀ x5 : (ι → ο) → ο . (∀ x6 : ι → ο . (∀ x7 . x6 x7 ⟶ prim1 x7 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) ⟶ 7a422.. (c42e1.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Param eb53d.. : ι → CT2 ι
Definition 7c612.. := λ x0 . λ x1 x2 x3 : ι → ι → ι . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x4 . If_i (x4 = 4a7ef..) x0 (If_i (x4 = 4ae4a.. 4a7ef..) (eb53d.. x0 x1) (If_i (x4 = 4ae4a.. (4ae4a.. 4a7ef..)) (eb53d.. x0 x2) (eb53d.. x0 x3))))
Theorem eb96c.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . x0 = 7c612.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 02f73.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . x0 = f482f.. (7c612.. x0 x1 x2 x3) 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 adf33.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . x0 = 7c612.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = e3162.. (f482f.. x0 (4ae4a.. 4a7ef..)) x5 x6 (proof)
Theorem 72a00.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x1 x4 x5 = e3162.. (f482f.. (7c612.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 x5 (proof)
Theorem 320f0.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . x0 = 7c612.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x5 x6 = e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Theorem 5e96d.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x2 x4 x5 = e3162.. (f482f.. (7c612.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) x4 x5 (proof)
Theorem e20c4.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . x0 = 7c612.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x4 x5 x6 = e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 x6 (proof)
Theorem 8be55.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x3 x4 x5 = e3162.. (f482f.. (7c612.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x4 x5 (proof)
Theorem 55876.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 x6 x7 : ι → ι → ι . 7c612.. x0 x2 x4 x6 = 7c612.. 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 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x4 x8 x9 = x5 x8 x9)) (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x6 x8 x9 = x7 x8 x9) (proof)
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 a5c5d.. : ∀ x0 . ∀ x1 x2 x3 x4 x5 x6 : ι → ι → ι . (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x1 x7 x8 = x2 x7 x8) ⟶ (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x3 x7 x8 = x4 x7 x8) ⟶ (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x5 x7 x8 = x6 x7 x8) ⟶ 7c612.. x0 x1 x3 x5 = 7c612.. x0 x2 x4 x6 (proof)
Definition 56cd3.. := 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) ⟶ x1 (7c612.. x2 x3 x4 x5)) ⟶ x1 x0
Theorem 8b690.. : ∀ 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) ⟶ 56cd3.. (7c612.. x0 x1 x2 x3) (proof)
Theorem a44f4.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . 56cd3.. (7c612.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x1 x4 x5) x0 (proof)
Theorem 12642.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . 56cd3.. (7c612.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x2 x4 x5) x0 (proof)
Theorem f09cb.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . 56cd3.. (7c612.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x3 x4 x5) x0 (proof)
Theorem b1359.. : ∀ x0 . 56cd3.. x0 ⟶ x0 = 7c612.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition 02be0.. := λ x0 . λ x1 : 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..)))))
Theorem 368fb.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ι) → ι . ∀ x1 . ∀ x2 x3 x4 : ι → ι → ι . (∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x3 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ x4 x8 x9 = x7 x8 x9) ⟶ x0 x1 x5 x6 x7 = x0 x1 x2 x3 x4) ⟶ 02be0.. (7c612.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 6a70f.. := λ x0 . λ x1 : 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..)))))
Theorem 668b5.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ι) → ο . ∀ x1 . ∀ x2 x3 x4 : ι → ι → ι . (∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x3 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ x4 x8 x9 = x7 x8 x9) ⟶ x0 x1 x5 x6 x7 = x0 x1 x2 x3 x4) ⟶ 6a70f.. (7c612.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition a255b.. := λ x0 . λ x1 x2 : ι → ι → ι . λ x3 : ι → ι . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x4 . If_i (x4 = 4a7ef..) x0 (If_i (x4 = 4ae4a.. 4a7ef..) (eb53d.. x0 x1) (If_i (x4 = 4ae4a.. (4ae4a.. 4a7ef..)) (eb53d.. x0 x2) (0fc90.. x0 x3))))
Theorem c1c1f.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . x0 = a255b.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 6301b.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . x0 = f482f.. (a255b.. x0 x1 x2 x3) 4a7ef.. (proof)
Theorem 89cf7.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . x0 = a255b.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = e3162.. (f482f.. x0 (4ae4a.. 4a7ef..)) x5 x6 (proof)
Theorem 522c4.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x1 x4 x5 = e3162.. (f482f.. (a255b.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 x5 (proof)
Theorem 2da5e.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . x0 = a255b.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x5 x6 = e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Theorem 8873d.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x2 x4 x5 = e3162.. (f482f.. (a255b.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) x4 x5 (proof)
Known f22ec.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 x0 ⟶ f482f.. (0fc90.. x0 x1) x2 = x1 x2
Theorem 6f41d.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . x0 = a255b.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x4 x5 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 (proof)
Theorem e53a5.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 . prim1 x4 x0 ⟶ x3 x4 = f482f.. (f482f.. (a255b.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x4 (proof)
Theorem dba9c.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι → ι . ∀ x6 x7 : ι → ι . a255b.. x0 x2 x4 x6 = a255b.. 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 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x4 x8 x9 = x5 x8 x9)) (∀ x8 . prim1 x8 x0 ⟶ x6 x8 = x7 x8) (proof)
Known 4402a.. : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ x1 x3 = x2 x3) ⟶ 0fc90.. x0 x1 = 0fc90.. x0 x2
Theorem 378d1.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι → ι . ∀ x5 x6 : ι → ι . (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x1 x7 x8 = x2 x7 x8) ⟶ (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x3 x7 x8 = x4 x7 x8) ⟶ (∀ x7 . prim1 x7 x0 ⟶ x5 x7 = x6 x7) ⟶ a255b.. x0 x1 x3 x5 = a255b.. x0 x2 x4 x6 (proof)
Definition 74f92.. := 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 . (∀ x6 . prim1 x6 x2 ⟶ prim1 (x5 x6) x2) ⟶ x1 (a255b.. x2 x3 x4 x5)) ⟶ x1 x0
Theorem f5883.. : ∀ 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 . (∀ x4 . prim1 x4 x0 ⟶ prim1 (x3 x4) x0) ⟶ 74f92.. (a255b.. x0 x1 x2 x3) (proof)
Theorem 99505.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . 74f92.. (a255b.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x1 x4 x5) x0 (proof)
Theorem 24dbe.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . 74f92.. (a255b.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x2 x4 x5) x0 (proof)
Theorem c282a.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . 74f92.. (a255b.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x3 x4) x0 (proof)
Theorem 931eb.. : ∀ x0 . 74f92.. x0 ⟶ x0 = a255b.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition 98d5c.. := λ x0 . λ x1 : 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..)))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 17fc0.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι) → ι . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . (∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x3 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ι . (∀ x8 . prim1 x8 x1 ⟶ x4 x8 = x7 x8) ⟶ x0 x1 x5 x6 x7 = x0 x1 x2 x3 x4) ⟶ 98d5c.. (a255b.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 824ea.. := λ x0 . λ x1 : 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..)))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem cfdca.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι) → ο . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . (∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x3 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ι . (∀ x8 . prim1 x8 x1 ⟶ x4 x8 = x7 x8) ⟶ x0 x1 x5 x6 x7 = x0 x1 x2 x3 x4) ⟶ 824ea.. (a255b.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Param d2155.. : ι → (ι → ι → ο) → ι
Definition 64302.. := λ x0 . λ x1 x2 : ι → ι → ι . λ x3 : ι → ι → ο . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x4 . If_i (x4 = 4a7ef..) x0 (If_i (x4 = 4ae4a.. 4a7ef..) (eb53d.. x0 x1) (If_i (x4 = 4ae4a.. (4ae4a.. 4a7ef..)) (eb53d.. x0 x2) (d2155.. x0 x3))))
Theorem 59d0c.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . x0 = 64302.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 3541c.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ι → ο . x4 x0 (f482f.. (64302.. x0 x1 x2 x3) 4a7ef..) ⟶ x4 (f482f.. (64302.. x0 x1 x2 x3) 4a7ef..) x0 (proof)
Theorem ac959.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . x0 = 64302.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = e3162.. (f482f.. x0 (4ae4a.. 4a7ef..)) x5 x6 (proof)
Theorem d5534.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x1 x4 x5 = e3162.. (f482f.. (64302.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 x5 (proof)
Theorem 0a008.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . x0 = 64302.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x5 x6 = e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Theorem efc45.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x2 x4 x5 = e3162.. (f482f.. (64302.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) x4 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 93da0.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . x0 = 64302.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x4 x5 x6 = 2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 x6 (proof)
Theorem 4f7a6.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x3 x4 x5 = 2b2e3.. (f482f.. (64302.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x4 x5 (proof)
Theorem a4816.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι → ι . ∀ x6 x7 : ι → ι → ο . 64302.. x0 x2 x4 x6 = 64302.. 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 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x4 x8 x9 = x5 x8 x9)) (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x6 x8 x9 = x7 x8 x9) (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 db7ec.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι → ι . ∀ x5 x6 : ι → ι → ο . (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x1 x7 x8 = x2 x7 x8) ⟶ (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x3 x7 x8 = x4 x7 x8) ⟶ (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ iff (x5 x7 x8) (x6 x7 x8)) ⟶ 64302.. x0 x1 x3 x5 = 64302.. x0 x2 x4 x6 (proof)
Definition 17cf8.. := 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 → ο . x1 (64302.. x2 x3 x4 x5)) ⟶ x1 x0
Theorem c4a94.. : ∀ 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 → ο . 17cf8.. (64302.. x0 x1 x2 x3) (proof)
Theorem 32733.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . 17cf8.. (64302.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x1 x4 x5) x0 (proof)
Theorem 24fe3.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . 17cf8.. (64302.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x2 x4 x5) x0 (proof)
Theorem 3bcdc.. : ∀ x0 . 17cf8.. x0 ⟶ x0 = 64302.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition b2ca5.. := λ x0 . λ x1 : 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..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 4d4c5.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ι . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . (∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x3 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ι → ο . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ iff (x4 x8 x9) (x7 x8 x9)) ⟶ x0 x1 x5 x6 x7 = x0 x1 x2 x3 x4) ⟶ b2ca5.. (64302.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 851fe.. := λ x0 . λ x1 : 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..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 8d675.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ο . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι → ο . (∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x3 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ι → ο . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ iff (x4 x8 x9) (x7 x8 x9)) ⟶ x0 x1 x5 x6 x7 = x0 x1 x2 x3 x4) ⟶ 851fe.. (64302.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 8f64a.. := λ x0 . λ x1 x2 : ι → ι → ι . λ x3 : ι → ο . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (λ x4 . If_i (x4 = 4a7ef..) x0 (If_i (x4 = 4ae4a.. 4a7ef..) (eb53d.. x0 x1) (If_i (x4 = 4ae4a.. (4ae4a.. 4a7ef..)) (eb53d.. x0 x2) (1216a.. x0 x3))))
Theorem 3461b.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ο . x0 = 8f64a.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 0c12c.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ο . x0 = f482f.. (8f64a.. x0 x1 x2 x3) 4a7ef.. (proof)
Theorem 69a7d.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ο . x0 = 8f64a.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = e3162.. (f482f.. x0 (4ae4a.. 4a7ef..)) x5 x6 (proof)
Theorem 7bd1f.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x1 x4 x5 = e3162.. (f482f.. (8f64a.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 x5 (proof)
Theorem 61558.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ο . x0 = 8f64a.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x5 x6 = e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Theorem 4a13b.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x2 x4 x5 = e3162.. (f482f.. (8f64a.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) x4 x5 (proof)
Theorem 3799b.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ο . x0 = 8f64a.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x4 x5 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 (proof)
Theorem 63e29.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 . prim1 x4 x0 ⟶ x3 x4 = decode_p (f482f.. (8f64a.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x4 (proof)
Theorem 61696.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι → ι . ∀ x6 x7 : ι → ο . 8f64a.. x0 x2 x4 x6 = 8f64a.. 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 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x4 x8 x9 = x5 x8 x9)) (∀ x8 . prim1 x8 x0 ⟶ x6 x8 = x7 x8) (proof)
Theorem 7e158.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι → ι . ∀ x5 x6 : ι → ο . (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x1 x7 x8 = x2 x7 x8) ⟶ (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x3 x7 x8 = x4 x7 x8) ⟶ (∀ x7 . prim1 x7 x0 ⟶ iff (x5 x7) (x6 x7)) ⟶ 8f64a.. x0 x1 x3 x5 = 8f64a.. x0 x2 x4 x6 (proof)
Definition 0ee11.. := 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 → ο . x1 (8f64a.. x2 x3 x4 x5)) ⟶ x1 x0
Theorem 7af5a.. : ∀ 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 → ο . 0ee11.. (8f64a.. x0 x1 x2 x3) (proof)
Theorem 0328a.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ο . 0ee11.. (8f64a.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x1 x4 x5) x0 (proof)
Theorem aabb3.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ο . 0ee11.. (8f64a.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x2 x4 x5) x0 (proof)
Theorem d704c.. : ∀ x0 . 0ee11.. x0 ⟶ x0 = 8f64a.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition 41669.. := λ x0 . λ x1 : 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..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 16ffc.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ο) → ι . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ο . (∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x3 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ο . (∀ x8 . prim1 x8 x1 ⟶ iff (x4 x8) (x7 x8)) ⟶ x0 x1 x5 x6 x7 = x0 x1 x2 x3 x4) ⟶ 41669.. (8f64a.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition cc976.. := λ x0 . λ x1 : 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..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 9d144.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ο) → ο . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ο . (∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ x6 : ι → ι → ι . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ x3 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ο . (∀ x8 . prim1 x8 x1 ⟶ iff (x4 x8) (x7 x8)) ⟶ x0 x1 x5 x6 x7 = x0 x1 x2 x3 x4) ⟶ cc976.. (8f64a.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)