Proofgold Asset

Param 0fc90.. : ι → (ι → ι) → ι
Param 4ae4a.. : ι → ι
Param 4a7ef.. : ι
Param If_i : ο → ι → ι → ι
Param 1216a.. : ι → (ι → ο) → ι
Definition ba600.. := λ x0 . λ x1 x2 : 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..) (0fc90.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. 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 5c6b7.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ο . x0 = ba600.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 34d3b.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ο . x0 = f482f.. (ba600.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
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 f22ec.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 x0 ⟶ f482f.. (0fc90.. x0 x1) x2 = x1 x2
Theorem 38231.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ο . x0 = ba600.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x6 = f482f.. (f482f.. x0 (4ae4a.. 4a7ef..)) x6 (proof)
Theorem 56c1e.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x1 x5 = f482f.. (f482f.. (ba600.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 (proof)
Known fb20c.. : ∀ x0 x1 x2 x3 x4 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x6 . If_i (x6 = 4a7ef..) x0 (If_i (x6 = 4ae4a.. 4a7ef..) x1 (If_i (x6 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 (If_i (x6 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4))))) (4ae4a.. (4ae4a.. 4a7ef..)) = x2
Theorem e2c49.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ο . x0 = ba600.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x6 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 (proof)
Theorem 66d42.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x2 x5 = f482f.. (f482f.. (ba600.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 (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 7a9a7.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ο . x0 = ba600.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x4 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 (proof)
Theorem 2a85a.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x3 x5 = decode_p (f482f.. (ba600.. 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 5e2e5.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ο . x0 = ba600.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x5 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x6 (proof)
Theorem 671d1.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x4 x5 = decode_p (f482f.. (ba600.. 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 730cb.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι . ∀ x6 x7 x8 x9 : ι → ο . ba600.. x0 x2 x4 x6 x8 = ba600.. x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 . prim1 x10 x0 ⟶ x2 x10 = x3 x10)) (∀ x10 . prim1 x10 x0 ⟶ x4 x10 = x5 x10)) (∀ 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 4402a.. : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ x1 x3 = x2 x3) ⟶ 0fc90.. x0 x1 = 0fc90.. x0 x2
Theorem 0a986.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι . ∀ x5 x6 x7 x8 : ι → ο . (∀ x9 . prim1 x9 x0 ⟶ x1 x9 = x2 x9) ⟶ (∀ x9 . prim1 x9 x0 ⟶ x3 x9 = x4 x9) ⟶ (∀ x9 . prim1 x9 x0 ⟶ iff (x5 x9) (x6 x9)) ⟶ (∀ x9 . prim1 x9 x0 ⟶ iff (x7 x9) (x8 x9)) ⟶ ba600.. x0 x1 x3 x5 x7 = ba600.. x0 x2 x4 x6 x8 (proof)
Definition 7aeb9.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι . (∀ x4 . prim1 x4 x2 ⟶ prim1 (x3 x4) x2) ⟶ ∀ x4 : ι → ι . (∀ x5 . prim1 x5 x2 ⟶ prim1 (x4 x5) x2) ⟶ ∀ x5 x6 : ι → ο . x1 (ba600.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem d78b8.. : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . prim1 x2 x0 ⟶ prim1 (x1 x2) x0) ⟶ ∀ x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x0) ⟶ ∀ x3 x4 : ι → ο . 7aeb9.. (ba600.. x0 x1 x2 x3 x4) (proof)
Theorem 86929.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ο . 7aeb9.. (ba600.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x1 x5) x0 (proof)
Theorem 3efb3.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ο . 7aeb9.. (ba600.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x2 x5) x0 (proof)
Known iff_refl : ∀ x0 : ο . iff x0 x0
Theorem 1efa9.. : ∀ x0 . 7aeb9.. x0 ⟶ x0 = ba600.. (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))) (proof)
Definition d8d71.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → (ι → ο) → (ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem b5643.. : ∀ x0 : ι → (ι → ι) → (ι → ι) → (ι → ο) → (ι → ο) → ι . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ο . (∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ x2 x7 = x6 x7) ⟶ ∀ x7 : ι → ι . (∀ x8 . prim1 x8 x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ 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) ⟶ d8d71.. (ba600.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition d0331.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → (ι → ο) → (ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem 97885.. : ∀ x0 : ι → (ι → ι) → (ι → ι) → (ι → ο) → (ι → ο) → ο . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ο . (∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ x2 x7 = x6 x7) ⟶ ∀ x7 : ι → ι . (∀ x8 . prim1 x8 x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ 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) ⟶ d0331.. (ba600.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 7e46e.. := λ x0 . λ x1 x2 : 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..) (0fc90.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (1216a.. x0 x3) x4))))
Theorem c9763.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ο . ∀ x5 . x0 = 7e46e.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 70120.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 . x0 = f482f.. (7e46e.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Theorem 6071f.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ο . ∀ x5 . x0 = 7e46e.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x6 = f482f.. (f482f.. x0 (4ae4a.. 4a7ef..)) x6 (proof)
Theorem b23c6.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . prim1 x5 x0 ⟶ x1 x5 = f482f.. (f482f.. (7e46e.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 (proof)
Theorem 7da76.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ο . ∀ x5 . x0 = 7e46e.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x6 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 (proof)
Theorem 9794d.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . prim1 x5 x0 ⟶ x2 x5 = f482f.. (f482f.. (7e46e.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Theorem d6e0e.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ο . ∀ x5 . x0 = 7e46e.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x4 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 (proof)
Theorem 6d178.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . prim1 x5 x0 ⟶ x3 x5 = decode_p (f482f.. (7e46e.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 (proof)
Theorem cea9c.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ο . ∀ x5 . x0 = 7e46e.. x1 x2 x3 x4 x5 ⟶ x5 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem ad831.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 . x4 = f482f.. (7e46e.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem af02e.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι . ∀ x6 x7 : ι → ο . ∀ x8 x9 . 7e46e.. x0 x2 x4 x6 x8 = 7e46e.. x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 . prim1 x10 x0 ⟶ x2 x10 = x3 x10)) (∀ x10 . prim1 x10 x0 ⟶ x4 x10 = x5 x10)) (∀ x10 . prim1 x10 x0 ⟶ x6 x10 = x7 x10)) (x8 = x9) (proof)
Theorem 6d0ba.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι . ∀ x5 x6 : ι → ο . ∀ x7 . (∀ x8 . prim1 x8 x0 ⟶ x1 x8 = x2 x8) ⟶ (∀ x8 . prim1 x8 x0 ⟶ x3 x8 = x4 x8) ⟶ (∀ x8 . prim1 x8 x0 ⟶ iff (x5 x8) (x6 x8)) ⟶ 7e46e.. x0 x1 x3 x5 x7 = 7e46e.. x0 x2 x4 x6 x7 (proof)
Definition 57517.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι . (∀ x4 . prim1 x4 x2 ⟶ prim1 (x3 x4) x2) ⟶ ∀ x4 : ι → ι . (∀ x5 . prim1 x5 x2 ⟶ prim1 (x4 x5) x2) ⟶ ∀ x5 : ι → ο . ∀ x6 . prim1 x6 x2 ⟶ x1 (7e46e.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem 03342.. : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . prim1 x2 x0 ⟶ prim1 (x1 x2) x0) ⟶ ∀ x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x0) ⟶ ∀ x3 : ι → ο . ∀ x4 . prim1 x4 x0 ⟶ 57517.. (7e46e.. x0 x1 x2 x3 x4) (proof)
Theorem ddaaf.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 . 57517.. (7e46e.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x1 x5) x0 (proof)
Theorem c310c.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 . 57517.. (7e46e.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x2 x5) x0 (proof)
Theorem 75de5.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 . 57517.. (7e46e.. x0 x1 x2 x3 x4) ⟶ prim1 x4 x0 (proof)
Theorem e3820.. : ∀ x0 . 57517.. x0 ⟶ x0 = 7e46e.. (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition 311a4.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → (ι → ο) → ι → ι . x1 (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem e5821.. : ∀ x0 : ι → (ι → ι) → (ι → ι) → (ι → ο) → ι → ι . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ο . ∀ x5 . (∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ x2 x7 = x6 x7) ⟶ ∀ x7 : ι → ι . (∀ x8 . prim1 x8 x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ x8 : ι → ο . (∀ x9 . prim1 x9 x1 ⟶ iff (x4 x9) (x8 x9)) ⟶ x0 x1 x6 x7 x8 x5 = x0 x1 x2 x3 x4 x5) ⟶ 311a4.. (7e46e.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 734b8.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → (ι → ο) → ι → ο . x1 (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem aa6de.. : ∀ x0 : ι → (ι → ι) → (ι → ι) → (ι → ο) → ι → ο . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 : ι → ο . ∀ x5 . (∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ x2 x7 = x6 x7) ⟶ ∀ x7 : ι → ι . (∀ x8 . prim1 x8 x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ x8 : ι → ο . (∀ x9 . prim1 x9 x1 ⟶ iff (x4 x9) (x8 x9)) ⟶ x0 x1 x6 x7 x8 x5 = x0 x1 x2 x3 x4 x5) ⟶ 734b8.. (7e46e.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 264ee.. := λ x0 . λ x1 x2 : ι → ι . λ x3 x4 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (0fc90.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4))))
Theorem b50ff.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 . x0 = 264ee.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem af877.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 . x0 = f482f.. (264ee.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Theorem 9e7c7.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 . x0 = 264ee.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x6 = f482f.. (f482f.. x0 (4ae4a.. 4a7ef..)) x6 (proof)
Theorem 8892a.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 x5 . prim1 x5 x0 ⟶ x1 x5 = f482f.. (f482f.. (264ee.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 (proof)
Theorem b7d3c.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 . x0 = 264ee.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x6 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 (proof)
Theorem e5a18.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 x5 . prim1 x5 x0 ⟶ x2 x5 = f482f.. (f482f.. (264ee.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Theorem bd572.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 . x0 = 264ee.. x1 x2 x3 x4 x5 ⟶ x4 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem 46c2b.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 . x3 = f482f.. (264ee.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem 67806.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 . x0 = 264ee.. x1 x2 x3 x4 x5 ⟶ x5 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem 4eb22.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 . x4 = f482f.. (264ee.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem e3dba.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι . ∀ x6 x7 x8 x9 . 264ee.. x0 x2 x4 x6 x8 = 264ee.. x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 . prim1 x10 x0 ⟶ x2 x10 = x3 x10)) (∀ x10 . prim1 x10 x0 ⟶ x4 x10 = x5 x10)) (x6 = x7)) (x8 = x9) (proof)
Theorem d989f.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι . ∀ x5 x6 . (∀ x7 . prim1 x7 x0 ⟶ x1 x7 = x2 x7) ⟶ (∀ x7 . prim1 x7 x0 ⟶ x3 x7 = x4 x7) ⟶ 264ee.. x0 x1 x3 x5 x6 = 264ee.. x0 x2 x4 x5 x6 (proof)
Definition 14e81.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι . (∀ x4 . prim1 x4 x2 ⟶ prim1 (x3 x4) x2) ⟶ ∀ x4 : ι → ι . (∀ x5 . prim1 x5 x2 ⟶ prim1 (x4 x5) x2) ⟶ ∀ x5 . prim1 x5 x2 ⟶ ∀ x6 . prim1 x6 x2 ⟶ x1 (264ee.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem eac13.. : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . prim1 x2 x0 ⟶ prim1 (x1 x2) x0) ⟶ ∀ x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x0) ⟶ ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ 14e81.. (264ee.. x0 x1 x2 x3 x4) (proof)
Theorem 7833c.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 . 14e81.. (264ee.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x1 x5) x0 (proof)
Theorem 29f0c.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 . 14e81.. (264ee.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x2 x5) x0 (proof)
Theorem 61077.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 . 14e81.. (264ee.. x0 x1 x2 x3 x4) ⟶ prim1 x3 x0 (proof)
Theorem a6148.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 . 14e81.. (264ee.. x0 x1 x2 x3 x4) ⟶ prim1 x4 x0 (proof)
Theorem f870f.. : ∀ x0 . 14e81.. x0 ⟶ x0 = 264ee.. (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition 3be00.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → ι → ι → ι . x1 (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 0c40e.. : ∀ x0 : ι → (ι → ι) → (ι → ι) → ι → ι → ι . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 . (∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ x2 x7 = x6 x7) ⟶ ∀ x7 : ι → ι . (∀ x8 . prim1 x8 x1 ⟶ x3 x8 = x7 x8) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5) ⟶ 3be00.. (264ee.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 882cc.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → ι → ι → ο . x1 (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 08db2.. : ∀ x0 : ι → (ι → ι) → (ι → ι) → ι → ι → ο . ∀ x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 . (∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ x2 x7 = x6 x7) ⟶ ∀ x7 : ι → ι . (∀ x8 . prim1 x8 x1 ⟶ x3 x8 = x7 x8) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5) ⟶ 882cc.. (264ee.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Param d2155.. : ι → (ι → ι → ο) → ι
Definition ae02b.. := λ x0 . λ x1 : ι → ι . λ x2 : ι → ι → ο . λ x3 x4 : ι → ο . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (0fc90.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (d2155.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (1216a.. x0 x3) (1216a.. x0 x4)))))
Theorem cc896.. : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 : ι → ο . x0 = ae02b.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 99e8a.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 x4 : ι → ο . x0 = f482f.. (ae02b.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Theorem cf832.. : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 : ι → ο . x0 = ae02b.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x6 = f482f.. (f482f.. x0 (4ae4a.. 4a7ef..)) x6 (proof)
Theorem 1e440.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x1 x5 = f482f.. (f482f.. (ae02b.. x0 x1 x2 x3 x4) (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 e670a.. : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 : ι → ο . x0 = ae02b.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x3 x6 x7 = 2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 x7 (proof)
Theorem df2ad.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x5 x6 = 2b2e3.. (f482f.. (ae02b.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Theorem 293e1.. : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 : ι → ο . x0 = ae02b.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x4 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 (proof)
Theorem 2473b.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x3 x5 = decode_p (f482f.. (ae02b.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 (proof)
Theorem 5549c.. : ∀ x0 x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 : ι → ο . x0 = ae02b.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x5 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x6 (proof)
Theorem d04af.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x4 x5 = decode_p (f482f.. (ae02b.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x5 (proof)
Theorem 3bc26.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . ∀ x6 x7 x8 x9 : ι → ο . ae02b.. x0 x2 x4 x6 x8 = ae02b.. x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 . prim1 x10 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)
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 0a4ae.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ x5 x6 x7 x8 : ι → ο . (∀ x9 . prim1 x9 x0 ⟶ x1 x9 = x2 x9) ⟶ (∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 x0 ⟶ iff (x3 x9 x10) (x4 x9 x10)) ⟶ (∀ x9 . prim1 x9 x0 ⟶ iff (x5 x9) (x6 x9)) ⟶ (∀ x9 . prim1 x9 x0 ⟶ iff (x7 x9) (x8 x9)) ⟶ ae02b.. x0 x1 x3 x5 x7 = ae02b.. x0 x2 x4 x6 x8 (proof)
Definition a4680.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι . (∀ x4 . prim1 x4 x2 ⟶ prim1 (x3 x4) x2) ⟶ ∀ x4 : ι → ι → ο . ∀ x5 x6 : ι → ο . x1 (ae02b.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem 58929.. : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . prim1 x2 x0 ⟶ prim1 (x1 x2) x0) ⟶ ∀ x2 : ι → ι → ο . ∀ x3 x4 : ι → ο . a4680.. (ae02b.. x0 x1 x2 x3 x4) (proof)
Theorem 0a953.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 x4 : ι → ο . a4680.. (ae02b.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x1 x5) x0 (proof)
Theorem e88d1.. : ∀ x0 . a4680.. x0 ⟶ x0 = ae02b.. (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))) (proof)
Definition 8d403.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι → ο) → (ι → ο) → (ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem 11ab3.. : ∀ x0 : ι → (ι → ι) → (ι → ι → ο) → (ι → ο) → (ι → ο) → ι . ∀ x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 : ι → ο . (∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ x2 x7 = x6 x7) ⟶ ∀ x7 : ι → ι → ο . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ iff (x3 x8 x9) (x7 x8 x9)) ⟶ ∀ x8 : ι → ο . (∀ x9 . prim1 x9 x1 ⟶ iff (x4 x9) (x8 x9)) ⟶ ∀ x9 : ι → ο . (∀ x10 . prim1 x10 x1 ⟶ iff (x5 x10) (x9 x10)) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ 8d403.. (ae02b.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 6d899.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι → ο) → (ι → ο) → (ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem 77bfd.. : ∀ x0 : ι → (ι → ι) → (ι → ι → ο) → (ι → ο) → (ι → ο) → ο . ∀ x1 . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 x5 : ι → ο . (∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ x2 x7 = x6 x7) ⟶ ∀ x7 : ι → ι → ο . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ iff (x3 x8 x9) (x7 x8 x9)) ⟶ ∀ x8 : ι → ο . (∀ x9 . prim1 x9 x1 ⟶ iff (x4 x9) (x8 x9)) ⟶ ∀ x9 : ι → ο . (∀ x10 . prim1 x10 x1 ⟶ iff (x5 x10) (x9 x10)) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ 6d899.. (ae02b.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)