Proofgold Address

Param 0fc90.. : ι → (ι → ι) → ι
Param 4ae4a.. : ι → ι
Param 4a7ef.. : ι
Param If_i : ο → ι → ι → ι
Param eb53d.. : ι → CT2 ι
Param 1216a.. : ι → (ι → ο) → ι
Definition f7962.. := λ x0 . λ x1 x2 : ι → ι → ι . λ x3 x4 : ι → ο . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (eb53d.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (eb53d.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (1216a.. x0 x3) (1216a.. x0 x4)))))
Param f482f.. : ι → ι → ι
Known 7d2e2.. : ∀ x0 x1 x2 x3 x4 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x6 . If_i (x6 = 4a7ef..) x0 (If_i (x6 = 4ae4a.. 4a7ef..) x1 (If_i (x6 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 (If_i (x6 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4))))) 4a7ef.. = x0
Theorem f88de.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ο . x0 = f7962.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 4e3d3.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ο . x0 = f482f.. (f7962.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Param e3162.. : ι → ι → ι → ι
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 35054.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ e3162.. (eb53d.. x0 x1) x2 x3 = x1 x2 x3
Theorem afa00.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ο . x0 = f7962.. 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 99c13.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x1 x5 x6 = e3162.. (f482f.. (f7962.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 x6 (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 33c92.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ο . x0 = f7962.. 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 bc41a.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x5 x6 = e3162.. (f482f.. (f7962.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Param decode_p : ι → ι → ο
Known 431f3.. : ∀ x0 x1 x2 x3 x4 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x6 . If_i (x6 = 4a7ef..) x0 (If_i (x6 = 4ae4a.. 4a7ef..) x1 (If_i (x6 = 4ae4a.. (4ae4a.. 4a7ef..)) x2 (If_i (x6 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4))))) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) = x3
Known 931fe.. : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . prim1 x2 x0 ⟶ decode_p (1216a.. x0 x1) x2 = x1 x2
Theorem 1ed3c.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ο . x0 = f7962.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x4 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 (proof)
Theorem 4e2bc.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x3 x5 = decode_p (f482f.. (f7962.. 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 797e1.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ο . x0 = f7962.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x5 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x6 (proof)
Theorem 90e82.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x4 x5 = decode_p (f482f.. (f7962.. 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 1c75e.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι → ι . ∀ x6 x7 x8 x9 : ι → ο . f7962.. x0 x2 x4 x6 x8 = f7962.. 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 ⟶ x6 x10 = x7 x10)) (∀ x10 . prim1 x10 x0 ⟶ x8 x10 = x9 x10) (proof)
Param iff : ο → ο → ο
Known ee7ef.. : ∀ x0 . ∀ x1 x2 : ι → ο . (∀ x3 . prim1 x3 x0 ⟶ iff (x1 x3) (x2 x3)) ⟶ 1216a.. x0 x1 = 1216a.. x0 x2
Known 8fdaf.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x1 x3 x4 = x2 x3 x4) ⟶ eb53d.. x0 x1 = eb53d.. x0 x2
Theorem a27e7.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι → ι . ∀ x5 x6 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 ⟶ iff (x5 x9) (x6 x9)) ⟶ (∀ x9 . prim1 x9 x0 ⟶ iff (x7 x9) (x8 x9)) ⟶ f7962.. x0 x1 x3 x5 x7 = f7962.. x0 x2 x4 x6 x8 (proof)
Definition 33bf8.. := 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 x6 : ι → ο . x1 (f7962.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem 65946.. : ∀ 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 x4 : ι → ο . 33bf8.. (f7962.. x0 x1 x2 x3 x4) (proof)
Theorem ac2a2.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ο . 33bf8.. (f7962.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x1 x5 x6) x0 (proof)
Theorem 451b8.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ο . 33bf8.. (f7962.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x2 x5 x6) x0 (proof)
Known iff_refl : ∀ x0 : ο . iff x0 x0
Theorem fa488.. : ∀ x0 . 33bf8.. x0 ⟶ x0 = f7962.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))) (proof)
Definition 9bdf0.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ο) → (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..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem 38d44.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ο) → (ι → ο) → ι . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 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 → 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) ⟶ 9bdf0.. (f7962.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 866d7.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ο) → (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..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem e68e5.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ο) → (ι → ο) → ο . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 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 → 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) ⟶ 866d7.. (f7962.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 1e782.. := λ x0 . λ x1 x2 : ι → ι → ι . λ x3 : ι → ο . λ x4 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (eb53d.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (eb53d.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (1216a.. x0 x3) x4))))
Theorem bbc04.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 . x0 = 1e782.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem bce93.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 . x0 = f482f.. (1e782.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Theorem e3b7c.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 . x0 = 1e782.. 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 e6bc5.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x1 x5 x6 = e3162.. (f482f.. (1e782.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 x6 (proof)
Theorem 993d3.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 . x0 = 1e782.. 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 d0dc0.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x5 x6 = e3162.. (f482f.. (1e782.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Theorem f8edd.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 . x0 = 1e782.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x4 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 (proof)
Theorem addbb.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 x5 . prim1 x5 x0 ⟶ x3 x5 = decode_p (f482f.. (1e782.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 (proof)
Theorem ffd6d.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 . x0 = 1e782.. x1 x2 x3 x4 x5 ⟶ x5 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem 4ba72.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 . x4 = f482f.. (1e782.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem fae3b.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι → ι . ∀ x6 x7 : ι → ο . ∀ x8 x9 . 1e782.. x0 x2 x4 x6 x8 = 1e782.. 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 ⟶ x6 x10 = x7 x10)) (x8 = x9) (proof)
Theorem f7e1f.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι → ι . ∀ x5 x6 : ι → ο . ∀ x7 . (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x1 x8 x9 = x2 x8 x9) ⟶ (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x3 x8 x9 = x4 x8 x9) ⟶ (∀ x8 . prim1 x8 x0 ⟶ iff (x5 x8) (x6 x8)) ⟶ 1e782.. x0 x1 x3 x5 x7 = 1e782.. x0 x2 x4 x6 x7 (proof)
Definition 160e6.. := 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 . ∀ x6 . prim1 x6 x2 ⟶ x1 (1e782.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem 50df3.. : ∀ 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 . ∀ x4 . prim1 x4 x0 ⟶ 160e6.. (1e782.. x0 x1 x2 x3 x4) (proof)
Theorem 8699c.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 . 160e6.. (1e782.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x1 x5 x6) x0 (proof)
Theorem a6b66.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 . 160e6.. (1e782.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x2 x5 x6) x0 (proof)
Theorem c34a4.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ο . ∀ x4 . 160e6.. (1e782.. x0 x1 x2 x3 x4) ⟶ prim1 x4 x0 (proof)
Theorem 1a9ad.. : ∀ x0 . 160e6.. x0 ⟶ x0 = 1e782.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition c3c22.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ο) → ι → ι . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 2148c.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ο) → ι → ι . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 . (∀ 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 → ο . (∀ x9 . prim1 x9 x1 ⟶ iff (x4 x9) (x8 x9)) ⟶ x0 x1 x6 x7 x8 x5 = x0 x1 x2 x3 x4 x5) ⟶ c3c22.. (1e782.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 32202.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ο) → ι → ο . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 78584.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ο) → ι → ο . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ο . ∀ x5 . (∀ 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 → ο . (∀ x9 . prim1 x9 x1 ⟶ iff (x4 x9) (x8 x9)) ⟶ x0 x1 x6 x7 x8 x5 = x0 x1 x2 x3 x4 x5) ⟶ 32202.. (1e782.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition c77b5.. := λ x0 . λ x1 x2 : ι → ι → ι . λ x3 x4 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (eb53d.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (eb53d.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4))))
Theorem 90132.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . x0 = c77b5.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem cb757.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . x0 = f482f.. (c77b5.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Theorem 1b277.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . x0 = c77b5.. 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 bdaba.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x1 x5 x6 = e3162.. (f482f.. (c77b5.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 x6 (proof)
Theorem 45d05.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . x0 = c77b5.. 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 74356.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x5 x6 = e3162.. (f482f.. (c77b5.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Theorem 8307f.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . x0 = c77b5.. x1 x2 x3 x4 x5 ⟶ x4 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem 73020.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . x3 = f482f.. (c77b5.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem 85b9b.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . x0 = c77b5.. x1 x2 x3 x4 x5 ⟶ x5 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem e32d8.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . x4 = f482f.. (c77b5.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem d0777.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι → ι . ∀ x6 x7 x8 x9 . c77b5.. x0 x2 x4 x6 x8 = c77b5.. 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)) (x6 = x7)) (x8 = x9) (proof)
Theorem 836a9.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι → ι . ∀ x5 x6 . (∀ 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) ⟶ c77b5.. x0 x1 x3 x5 x6 = c77b5.. x0 x2 x4 x5 x6 (proof)
Definition 3f0d0.. := 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 . prim1 x5 x2 ⟶ ∀ x6 . prim1 x6 x2 ⟶ x1 (c77b5.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem 9b39d.. : ∀ 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 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ 3f0d0.. (c77b5.. x0 x1 x2 x3 x4) (proof)
Theorem 38e2b.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . 3f0d0.. (c77b5.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x1 x5 x6) x0 (proof)
Theorem 2ca4b.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . 3f0d0.. (c77b5.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x2 x5 x6) x0 (proof)
Theorem f7980.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . 3f0d0.. (c77b5.. x0 x1 x2 x3 x4) ⟶ prim1 x3 x0 (proof)
Theorem 619d5.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . 3f0d0.. (c77b5.. x0 x1 x2 x3 x4) ⟶ prim1 x4 x0 (proof)
Theorem a296b.. : ∀ x0 . 3f0d0.. x0 ⟶ x0 = c77b5.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition 92512.. := 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.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 242ff.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι → ι . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . (∀ 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) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5) ⟶ 92512.. (c77b5.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition c3510.. := 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.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 24f4f.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι → ο . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . (∀ 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) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5) ⟶ c3510.. (c77b5.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Param d2155.. : ι → (ι → ι → ο) → ι
Definition d0dcf.. := λ x0 . λ x1 : ι → ι → ι . λ x2 : ι → ι . λ x3 x4 : ι → ι → ο . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (λ x5 . If_i (x5 = 4a7ef..) x0 (If_i (x5 = 4ae4a.. 4a7ef..) (eb53d.. x0 x1) (If_i (x5 = 4ae4a.. (4ae4a.. 4a7ef..)) (0fc90.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (d2155.. x0 x3) (d2155.. x0 x4)))))
Theorem 7c19d.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = d0dcf.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem d2b30.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 x4 x5 : ι → ι → ο . x5 x0 (f482f.. (d0dcf.. x0 x1 x2 x3 x4) 4a7ef..) ⟶ x5 (f482f.. (d0dcf.. x0 x1 x2 x3 x4) 4a7ef..) x0 (proof)
Theorem 7b79f.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = d0dcf.. 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 e5b54.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x1 x5 x6 = e3162.. (f482f.. (d0dcf.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 x6 (proof)
Known f22ec.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 x0 ⟶ f482f.. (0fc90.. x0 x1) x2 = x1 x2
Theorem 89e9d.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = d0dcf.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x6 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x6 (proof)
Theorem 4d4e3.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x2 x5 = f482f.. (f482f.. (d0dcf.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Param 2b2e3.. : ι → ι → ι → ο
Known 67416.. : ∀ x0 . ∀ x1 : ι → ι → ο . ∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ 2b2e3.. (d2155.. x0 x1) x2 x3 = x1 x2 x3
Theorem c4598.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = d0dcf.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x4 x6 x7 = 2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 x7 (proof)
Theorem 5d78f.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 x5 x6 = 2b2e3.. (f482f.. (d0dcf.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 x6 (proof)
Theorem e8d05.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . x0 = d0dcf.. 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 a809c.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x4 x5 x6 = 2b2e3.. (f482f.. (d0dcf.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x5 x6 (proof)
Theorem b94ff.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ι . ∀ x6 x7 x8 x9 : ι → ι → ο . d0dcf.. x0 x2 x4 x6 x8 = d0dcf.. 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 ⟶ x4 x10 = x5 x10)) (∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x6 x10 x11 = x7 x10 x11)) (∀ x10 . prim1 x10 x0 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x8 x10 x11 = x9 x10 x11) (proof)
Known 62ef7.. : ∀ x0 . ∀ x1 x2 : ι → ι → ο . (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ iff (x1 x3 x4) (x2 x3 x4)) ⟶ d2155.. x0 x1 = d2155.. x0 x2
Known 4402a.. : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ x1 x3 = x2 x3) ⟶ 0fc90.. x0 x1 = 0fc90.. x0 x2
Theorem 1e035.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ι . ∀ x5 x6 x7 x8 : ι → ι → ο . (∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 x0 ⟶ x1 x9 x10 = x2 x9 x10) ⟶ (∀ x9 . prim1 x9 x0 ⟶ x3 x9 = x4 x9) ⟶ (∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 x0 ⟶ iff (x5 x9 x10) (x6 x9 x10)) ⟶ (∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 x0 ⟶ iff (x7 x9 x10) (x8 x9 x10)) ⟶ d0dcf.. x0 x1 x3 x5 x7 = d0dcf.. x0 x2 x4 x6 x8 (proof)
Definition 76ccd.. := 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 . (∀ x5 . prim1 x5 x2 ⟶ prim1 (x4 x5) x2) ⟶ ∀ x5 x6 : ι → ι → ο . x1 (d0dcf.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem 1bc9e.. : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x1 x2 x3) x0) ⟶ ∀ x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x0) ⟶ ∀ x3 x4 : ι → ι → ο . 76ccd.. (d0dcf.. x0 x1 x2 x3 x4) (proof)
Theorem 73173.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . 76ccd.. (d0dcf.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x1 x5 x6) x0 (proof)
Theorem 3f145.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . 76ccd.. (d0dcf.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x2 x5) x0 (proof)
Theorem d468d.. : ∀ x0 . 76ccd.. x0 ⟶ x0 = d0dcf.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))) (proof)
Definition af09b.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι) → (ι → ι → ο) → (ι → ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem ddc02.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι) → (ι → ι → ο) → (ι → ι → ο) → ι . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 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 . (∀ x8 . prim1 x8 x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ x8 : ι → ι → ο . (∀ x9 . prim1 x9 x1 ⟶ ∀ x10 . prim1 x10 x1 ⟶ iff (x4 x9 x10) (x8 x9 x10)) ⟶ ∀ x9 : ι → ι → ο . (∀ x10 . prim1 x10 x1 ⟶ ∀ x11 . prim1 x11 x1 ⟶ iff (x5 x10 x11) (x9 x10 x11)) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ af09b.. (d0dcf.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 4e608.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι) → (ι → ι → ο) → (ι → ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem 422f0.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι) → (ι → ι → ο) → (ι → ι → ο) → ο . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 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 . (∀ x8 . prim1 x8 x1 ⟶ x3 x8 = x7 x8) ⟶ ∀ x8 : ι → ι → ο . (∀ x9 . prim1 x9 x1 ⟶ ∀ x10 . prim1 x10 x1 ⟶ iff (x4 x9 x10) (x8 x9 x10)) ⟶ ∀ x9 : ι → ι → ο . (∀ x10 . prim1 x10 x1 ⟶ ∀ x11 . prim1 x11 x1 ⟶ iff (x5 x10 x11) (x9 x10 x11)) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ 4e608.. (d0dcf.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)