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
Definition 73737.. := λ x0 . λ x1 : ι → ι . λ x2 : ι → ι → ο . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x3 . If_i (x3 = 4a7ef..) x0 (If_i (x3 = 4ae4a.. 4a7ef..) (0fc90.. x0 x1) (d2155.. x0 x2)))
Param f482f.. : ι → ι → ι
Known 52da6.. : ∀ x0 x1 x2 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x4 . If_i (x4 = 4a7ef..) x0 (If_i (x4 = 4ae4a.. 4a7ef..) x1 x2))) 4a7ef.. = x0
Theorem 5e270.. : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . x0 = 73737.. x1 x2 x3 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 0717a.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 : ι → ι → ο . x3 x0 (f482f.. (73737.. x0 x1 x2) 4a7ef..) ⟶ x3 (f482f.. (73737.. x0 x1 x2) 4a7ef..) x0 (proof)
Known c2bca.. : ∀ x0 x1 x2 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x4 . If_i (x4 = 4a7ef..) x0 (If_i (x4 = 4ae4a.. 4a7ef..) x1 x2))) (4ae4a.. 4a7ef..) = x1
Known f22ec.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 x0 ⟶ f482f.. (0fc90.. x0 x1) x2 = x1 x2
Theorem b5f96.. : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . x0 = 73737.. x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x1 ⟶ x2 x4 = f482f.. (f482f.. x0 (4ae4a.. 4a7ef..)) x4 (proof)
Theorem 13bf1.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 . prim1 x3 x0 ⟶ x1 x3 = f482f.. (f482f.. (73737.. x0 x1 x2) (4ae4a.. 4a7ef..)) x3 (proof)
Param 2b2e3.. : ι → ι → ι → ο
Known 11d6d.. : ∀ x0 x1 x2 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x4 . If_i (x4 = 4a7ef..) x0 (If_i (x4 = 4ae4a.. 4a7ef..) x1 x2))) (4ae4a.. (4ae4a.. 4a7ef..)) = x2
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 99d82.. : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . x0 = 73737.. x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x1 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x3 x4 x5 = 2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x4 x5 (proof)
Theorem 8a141.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x2 x3 x4 = 2b2e3.. (f482f.. (73737.. x0 x1 x2) (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4 (proof)
Definition and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Theorem 0d55e.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . 73737.. x0 x2 x4 = 73737.. x1 x3 x5 ⟶ and (and (x0 = x1) (∀ x6 . prim1 x6 x0 ⟶ x2 x6 = x3 x6)) (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x4 x6 x7 = x5 x6 x7) (proof)
Param iff : ο → ο → ο
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 a6f1a.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . (∀ x5 . prim1 x5 x0 ⟶ x1 x5 = x2 x5) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ iff (x3 x5 x6) (x4 x5 x6)) ⟶ 73737.. x0 x1 x3 = 73737.. x0 x2 x4 (proof)
Definition e5836.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι . (∀ x4 . prim1 x4 x2 ⟶ prim1 (x3 x4) x2) ⟶ ∀ x4 : ι → ι → ο . x1 (73737.. x2 x3 x4)) ⟶ x1 x0
Theorem a15d6.. : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . prim1 x2 x0 ⟶ prim1 (x1 x2) x0) ⟶ ∀ x2 : ι → ι → ο . e5836.. (73737.. x0 x1 x2) (proof)
Theorem 05946.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ο . e5836.. (73737.. x0 x1 x2) ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x1 x3) x0 (proof)
Known iff_refl : ∀ x0 : ο . iff x0 x0
Theorem 6f612.. : ∀ x0 . e5836.. x0 ⟶ x0 = 73737.. (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Definition c7882.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 0ae8d.. : ∀ x0 : ι → (ι → ι) → (ι → ι → ο) → ι . ∀ x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . (∀ x4 : ι → ι . (∀ x5 . prim1 x5 x1 ⟶ x2 x5 = x4 x5) ⟶ ∀ x5 : ι → ι → ο . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ iff (x3 x6 x7) (x5 x6 x7)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ c7882.. (73737.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition 54aad.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 6036d.. : ∀ x0 : ι → (ι → ι) → (ι → ι → ο) → ο . ∀ x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . (∀ x4 : ι → ι . (∀ x5 . prim1 x5 x1 ⟶ x2 x5 = x4 x5) ⟶ ∀ x5 : ι → ι → ο . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ iff (x3 x6 x7) (x5 x6 x7)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ 54aad.. (73737.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Param 1216a.. : ι → (ι → ο) → ι
Definition da24e.. := λ x0 . λ x1 : ι → ι . λ x2 : ι → ο . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x3 . If_i (x3 = 4a7ef..) x0 (If_i (x3 = 4ae4a.. 4a7ef..) (0fc90.. x0 x1) (1216a.. x0 x2)))
Theorem ab466.. : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ο . x0 = da24e.. x1 x2 x3 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 3b630.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ο . x0 = f482f.. (da24e.. x0 x1 x2) 4a7ef.. (proof)
Theorem 059c3.. : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ο . x0 = da24e.. x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x1 ⟶ x2 x4 = f482f.. (f482f.. x0 (4ae4a.. 4a7ef..)) x4 (proof)
Theorem 81d59.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ο . ∀ x3 . prim1 x3 x0 ⟶ x1 x3 = f482f.. (f482f.. (da24e.. x0 x1 x2) (4ae4a.. 4a7ef..)) x3 (proof)
Param decode_p : ι → ι → ο
Known 931fe.. : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . prim1 x2 x0 ⟶ decode_p (1216a.. x0 x1) x2 = x1 x2
Theorem 5bfd0.. : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ο . x0 = da24e.. x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x1 ⟶ x3 x4 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x4 (proof)
Theorem ba55d.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ο . ∀ x3 . prim1 x3 x0 ⟶ x2 x3 = decode_p (f482f.. (da24e.. x0 x1 x2) (4ae4a.. (4ae4a.. 4a7ef..))) x3 (proof)
Theorem 943ec.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ο . da24e.. x0 x2 x4 = da24e.. x1 x3 x5 ⟶ and (and (x0 = x1) (∀ x6 . prim1 x6 x0 ⟶ x2 x6 = x3 x6)) (∀ x6 . prim1 x6 x0 ⟶ x4 x6 = x5 x6) (proof)
Known ee7ef.. : ∀ x0 . ∀ x1 x2 : ι → ο . (∀ x3 . prim1 x3 x0 ⟶ iff (x1 x3) (x2 x3)) ⟶ 1216a.. x0 x1 = 1216a.. x0 x2
Theorem df1be.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ο . (∀ x5 . prim1 x5 x0 ⟶ x1 x5 = x2 x5) ⟶ (∀ x5 . prim1 x5 x0 ⟶ iff (x3 x5) (x4 x5)) ⟶ da24e.. x0 x1 x3 = da24e.. x0 x2 x4 (proof)
Definition 836b1.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι . (∀ x4 . prim1 x4 x2 ⟶ prim1 (x3 x4) x2) ⟶ ∀ x4 : ι → ο . x1 (da24e.. x2 x3 x4)) ⟶ x1 x0
Theorem 6227c.. : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . prim1 x2 x0 ⟶ prim1 (x1 x2) x0) ⟶ ∀ x2 : ι → ο . 836b1.. (da24e.. x0 x1 x2) (proof)
Theorem 9bb50.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ο . 836b1.. (da24e.. x0 x1 x2) ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x1 x3) x0 (proof)
Theorem c29f4.. : ∀ x0 . 836b1.. x0 ⟶ x0 = da24e.. (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Definition f614a.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem b4c19.. : ∀ x0 : ι → (ι → ι) → (ι → ο) → ι . ∀ x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ο . (∀ x4 : ι → ι . (∀ x5 . prim1 x5 x1 ⟶ x2 x5 = x4 x5) ⟶ ∀ x5 : ι → ο . (∀ x6 . prim1 x6 x1 ⟶ iff (x3 x6) (x5 x6)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ f614a.. (da24e.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition b4400.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 16896.. : ∀ x0 : ι → (ι → ι) → (ι → ο) → ο . ∀ x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ο . (∀ x4 : ι → ι . (∀ x5 . prim1 x5 x1 ⟶ x2 x5 = x4 x5) ⟶ ∀ x5 : ι → ο . (∀ x6 . prim1 x6 x1 ⟶ iff (x3 x6) (x5 x6)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ b4400.. (da24e.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition 8eacd.. := λ x0 . λ x1 : ι → ι . λ x2 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x3 . If_i (x3 = 4a7ef..) x0 (If_i (x3 = 4ae4a.. 4a7ef..) (0fc90.. x0 x1) x2))
Theorem 97d26.. : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 . x0 = 8eacd.. x1 x2 x3 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 30e8e.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x0 = f482f.. (8eacd.. x0 x1 x2) 4a7ef.. (proof)
Theorem f1b26.. : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 . x0 = 8eacd.. x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x1 ⟶ x2 x4 = f482f.. (f482f.. x0 (4ae4a.. 4a7ef..)) x4 (proof)
Theorem 1ddeb.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . prim1 x3 x0 ⟶ x1 x3 = f482f.. (f482f.. (8eacd.. x0 x1 x2) (4ae4a.. 4a7ef..)) x3 (proof)
Theorem 1e3c2.. : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 . x0 = 8eacd.. x1 x2 x3 ⟶ x3 = f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)) (proof)
Theorem a04cc.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 = f482f.. (8eacd.. x0 x1 x2) (4ae4a.. (4ae4a.. 4a7ef..)) (proof)
Theorem b6081.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 . 8eacd.. x0 x2 x4 = 8eacd.. x1 x3 x5 ⟶ and (and (x0 = x1) (∀ x6 . prim1 x6 x0 ⟶ x2 x6 = x3 x6)) (x4 = x5) (proof)
Theorem 5ab49.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 . (∀ x4 . prim1 x4 x0 ⟶ x1 x4 = x2 x4) ⟶ 8eacd.. x0 x1 x3 = 8eacd.. x0 x2 x3 (proof)
Definition d7681.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι . (∀ x4 . prim1 x4 x2 ⟶ prim1 (x3 x4) x2) ⟶ ∀ x4 . prim1 x4 x2 ⟶ x1 (8eacd.. x2 x3 x4)) ⟶ x1 x0
Theorem 0cbda.. : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . prim1 x2 x0 ⟶ prim1 (x1 x2) x0) ⟶ ∀ x2 . prim1 x2 x0 ⟶ d7681.. (8eacd.. x0 x1 x2) (proof)
Theorem c857f.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . d7681.. (8eacd.. x0 x1 x2) ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x1 x3) x0 (proof)
Theorem f77e0.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . d7681.. (8eacd.. x0 x1 x2) ⟶ prim1 x2 x0 (proof)
Theorem b617b.. : ∀ x0 . d7681.. x0 ⟶ x0 = 8eacd.. (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Definition a366a.. := λ x0 . λ x1 : ι → (ι → ι) → ι → ι . x1 (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))
Theorem 8faf4.. : ∀ x0 : ι → (ι → ι) → ι → ι . ∀ x1 . ∀ x2 : ι → ι . ∀ x3 . (∀ x4 : ι → ι . (∀ x5 . prim1 x5 x1 ⟶ x2 x5 = x4 x5) ⟶ x0 x1 x4 x3 = x0 x1 x2 x3) ⟶ a366a.. (8eacd.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition f1574.. := λ x0 . λ x1 : ι → (ι → ι) → ι → ο . x1 (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))
Theorem 87465.. : ∀ x0 : ι → (ι → ι) → ι → ο . ∀ x1 . ∀ x2 : ι → ι . ∀ x3 . (∀ x4 : ι → ι . (∀ x5 . prim1 x5 x1 ⟶ x2 x5 = x4 x5) ⟶ x0 x1 x4 x3 = x0 x1 x2 x3) ⟶ f1574.. (8eacd.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition 08354.. := λ x0 . λ x1 x2 : ι → ι → ο . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x3 . If_i (x3 = 4a7ef..) x0 (If_i (x3 = 4ae4a.. 4a7ef..) (d2155.. x0 x1) (d2155.. x0 x2)))
Theorem 7e1bc.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . x0 = 08354.. x1 x2 x3 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem f7f73.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ο . x3 x0 (f482f.. (08354.. x0 x1 x2) 4a7ef..) ⟶ x3 (f482f.. (08354.. x0 x1 x2) 4a7ef..) x0 (proof)
Theorem dad57.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . x0 = 08354.. x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x1 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x2 x4 x5 = 2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..)) x4 x5 (proof)
Theorem 60aa8.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x1 x3 x4 = 2b2e3.. (f482f.. (08354.. x0 x1 x2) (4ae4a.. 4a7ef..)) x3 x4 (proof)
Theorem 6fb62.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . x0 = 08354.. x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x1 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x3 x4 x5 = 2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x4 x5 (proof)
Theorem 592f1.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x2 x3 x4 = 2b2e3.. (f482f.. (08354.. x0 x1 x2) (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4 (proof)
Theorem 7d9fd.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι → ο . 08354.. x0 x2 x4 = 08354.. x1 x3 x5 ⟶ and (and (x0 = x1) (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x2 x6 x7 = x3 x6 x7)) (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x4 x6 x7 = x5 x6 x7) (proof)
Theorem a26f3.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι → ο . (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ iff (x1 x5 x6) (x2 x5 x6)) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ iff (x3 x5 x6) (x4 x5 x6)) ⟶ 08354.. x0 x1 x3 = 08354.. x0 x2 x4 (proof)
Definition 08eae.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 x4 : ι → ι → ο . x1 (08354.. x2 x3 x4)) ⟶ x1 x0
Theorem b3521.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . 08eae.. (08354.. x0 x1 x2) (proof)
Theorem d1048.. : ∀ x0 . 08eae.. x0 ⟶ x0 = 08354.. (f482f.. x0 4a7ef..) (2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Definition 9663d.. := λ x0 . λ x1 : ι → (ι → ι → ο) → (ι → ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 451a8.. : ∀ x0 : ι → (ι → ι → ο) → (ι → ι → ο) → ι . ∀ x1 . ∀ x2 x3 : ι → ι → ο . (∀ x4 : ι → ι → ο . (∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ iff (x2 x5 x6) (x4 x5 x6)) ⟶ ∀ x5 : ι → ι → ο . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ iff (x3 x6 x7) (x5 x6 x7)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ 9663d.. (08354.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition b5e83.. := λ x0 . λ x1 : ι → (ι → ι → ο) → (ι → ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 0c40b.. : ∀ x0 : ι → (ι → ι → ο) → (ι → ι → ο) → ο . ∀ x1 . ∀ x2 x3 : ι → ι → ο . (∀ x4 : ι → ι → ο . (∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ iff (x2 x5 x6) (x4 x5 x6)) ⟶ ∀ x5 : ι → ι → ο . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ iff (x3 x6 x7) (x5 x6 x7)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ b5e83.. (08354.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition 42836.. := λ x0 . λ x1 : ι → ι → ο . λ x2 : ι → ο . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x3 . If_i (x3 = 4a7ef..) x0 (If_i (x3 = 4ae4a.. 4a7ef..) (d2155.. x0 x1) (1216a.. x0 x2)))
Theorem a7210.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . x0 = 42836.. x1 x2 x3 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 3b2a9.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . x0 = f482f.. (42836.. x0 x1 x2) 4a7ef.. (proof)
Theorem 0ad5e.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . x0 = 42836.. x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x1 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x2 x4 x5 = 2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..)) x4 x5 (proof)
Theorem 4bd1d.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x1 x3 x4 = 2b2e3.. (f482f.. (42836.. x0 x1 x2) (4ae4a.. 4a7ef..)) x3 x4 (proof)
Theorem f4109.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . x0 = 42836.. x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x1 ⟶ x3 x4 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x4 (proof)
Theorem 54b81.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . ∀ x3 . prim1 x3 x0 ⟶ x2 x3 = decode_p (f482f.. (42836.. x0 x1 x2) (4ae4a.. (4ae4a.. 4a7ef..))) x3 (proof)
Theorem ea72b.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 x5 : ι → ο . 42836.. x0 x2 x4 = 42836.. x1 x3 x5 ⟶ and (and (x0 = x1) (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x2 x6 x7 = x3 x6 x7)) (∀ x6 . prim1 x6 x0 ⟶ x4 x6 = x5 x6) (proof)
Theorem 0b00c.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 x4 : ι → ο . (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ iff (x1 x5 x6) (x2 x5 x6)) ⟶ (∀ x5 . prim1 x5 x0 ⟶ iff (x3 x5) (x4 x5)) ⟶ 42836.. x0 x1 x3 = 42836.. x0 x2 x4 (proof)
Definition 7b9a8.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . x1 (42836.. x2 x3 x4)) ⟶ x1 x0
Theorem cdeb8.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 : ι → ο . 7b9a8.. (42836.. x0 x1 x2) (proof)
Theorem b127a.. : ∀ x0 . 7b9a8.. x0 ⟶ x0 = 42836.. (f482f.. x0 4a7ef..) (2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Definition 8547b.. := λ x0 . λ x1 : ι → (ι → ι → ο) → (ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 5946d.. : ∀ x0 : ι → (ι → ι → ο) → (ι → ο) → ι . ∀ x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . (∀ x4 : ι → ι → ο . (∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ iff (x2 x5 x6) (x4 x5 x6)) ⟶ ∀ x5 : ι → ο . (∀ x6 . prim1 x6 x1 ⟶ iff (x3 x6) (x5 x6)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ 8547b.. (42836.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition 24d60.. := λ x0 . λ x1 : ι → (ι → ι → ο) → (ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem b19dd.. : ∀ x0 : ι → (ι → ι → ο) → (ι → ο) → ο . ∀ x1 . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . (∀ x4 : ι → ι → ο . (∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ iff (x2 x5 x6) (x4 x5 x6)) ⟶ ∀ x5 : ι → ο . (∀ x6 . prim1 x6 x1 ⟶ iff (x3 x6) (x5 x6)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ 24d60.. (42836.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition dd3c8.. := λ x0 . λ x1 : ι → ι → ο . λ x2 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x3 . If_i (x3 = 4a7ef..) x0 (If_i (x3 = 4ae4a.. 4a7ef..) (d2155.. x0 x1) x2))
Theorem c7e82.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 . x0 = dd3c8.. x1 x2 x3 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem e0b06.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 . x0 = f482f.. (dd3c8.. x0 x1 x2) 4a7ef.. (proof)
Theorem 7c649.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 . x0 = dd3c8.. x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x1 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x2 x4 x5 = 2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..)) x4 x5 (proof)
Theorem 76e61.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x1 x3 x4 = 2b2e3.. (f482f.. (dd3c8.. x0 x1 x2) (4ae4a.. 4a7ef..)) x3 x4 (proof)
Theorem 72fdb.. : ∀ x0 x1 . ∀ x2 : ι → ι → ο . ∀ x3 . x0 = dd3c8.. x1 x2 x3 ⟶ x3 = f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)) (proof)
Theorem fd239.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 . x2 = f482f.. (dd3c8.. x0 x1 x2) (4ae4a.. (4ae4a.. 4a7ef..)) (proof)
Theorem 9c7d4.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ο . ∀ x4 x5 . dd3c8.. x0 x2 x4 = dd3c8.. x1 x3 x5 ⟶ and (and (x0 = x1) (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x2 x6 x7 = x3 x6 x7)) (x4 = x5) (proof)
Theorem 2098a.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . ∀ x3 . (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ iff (x1 x4 x5) (x2 x4 x5)) ⟶ dd3c8.. x0 x1 x3 = dd3c8.. x0 x2 x3 (proof)
Definition f7c54.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ο . ∀ x4 . prim1 x4 x2 ⟶ x1 (dd3c8.. x2 x3 x4)) ⟶ x1 x0
Theorem e8e7c.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 . prim1 x2 x0 ⟶ f7c54.. (dd3c8.. x0 x1 x2) (proof)
Theorem 4b1e4.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 . f7c54.. (dd3c8.. x0 x1 x2) ⟶ prim1 x2 x0 (proof)
Theorem 014b6.. : ∀ x0 . f7c54.. x0 ⟶ x0 = dd3c8.. (f482f.. x0 4a7ef..) (2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Definition 47d0a.. := λ x0 . λ x1 : ι → (ι → ι → ο) → ι → ι . x1 (f482f.. x0 4a7ef..) (2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))
Theorem 8088b.. : ∀ x0 : ι → (ι → ι → ο) → ι → ι . ∀ x1 . ∀ x2 : ι → ι → ο . ∀ x3 . (∀ x4 : ι → ι → ο . (∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ iff (x2 x5 x6) (x4 x5 x6)) ⟶ x0 x1 x4 x3 = x0 x1 x2 x3) ⟶ 47d0a.. (dd3c8.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition a0dc3.. := λ x0 . λ x1 : ι → (ι → ι → ο) → ι → ο . x1 (f482f.. x0 4a7ef..) (2b2e3.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))
Theorem 53c70.. : ∀ x0 : ι → (ι → ι → ο) → ι → ο . ∀ x1 . ∀ x2 : ι → ι → ο . ∀ x3 . (∀ x4 : ι → ι → ο . (∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ iff (x2 x5 x6) (x4 x5 x6)) ⟶ x0 x1 x4 x3 = x0 x1 x2 x3) ⟶ a0dc3.. (dd3c8.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)