Proofgold Asset

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
Definition e707a.. := λ x0 . λ x1 x2 : ι → ι → ι . λ x3 . 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) 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 dcdae.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 . x0 = e707a.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 53a20.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 . x0 = f482f.. (e707a.. x0 x1 x2 x3) 4a7ef.. (proof)
Param e3162.. : ι → ι → ι → ι
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 35054.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ e3162.. (eb53d.. x0 x1) x2 x3 = x1 x2 x3
Theorem edc55.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 . x0 = e707a.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = e3162.. (f482f.. x0 (4ae4a.. 4a7ef..)) x5 x6 (proof)
Theorem a698e.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x1 x4 x5 = e3162.. (f482f.. (e707a.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 x5 (proof)
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
Theorem 0a774.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 . x0 = e707a.. 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 7bf04.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x2 x4 x5 = e3162.. (f482f.. (e707a.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) x4 x5 (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 f3f77.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 . x0 = e707a.. x1 x2 x3 x4 ⟶ x4 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem 54060.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 . x3 = f482f.. (e707a.. 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 63f04.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι → ι . ∀ x6 x7 . e707a.. x0 x2 x4 x6 = e707a.. 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)) (x6 = x7) (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 abad8.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι → ι . ∀ x5 . (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x1 x6 x7 = x2 x6 x7) ⟶ (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x3 x6 x7 = x4 x6 x7) ⟶ e707a.. x0 x1 x3 x5 = e707a.. x0 x2 x4 x5 (proof)
Definition 06179.. := 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 ⟶ x1 (e707a.. x2 x3 x4 x5)) ⟶ x1 x0
Theorem 0cbd8.. : ∀ 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 ⟶ 06179.. (e707a.. x0 x1 x2 x3) (proof)
Theorem d484b.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 . 06179.. (e707a.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x1 x4 x5) x0 (proof)
Theorem e4f10.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 . 06179.. (e707a.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x2 x4 x5) x0 (proof)
Theorem 02d3f.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 . 06179.. (e707a.. x0 x1 x2 x3) ⟶ prim1 x3 x0 (proof)
Theorem b91ee.. : ∀ x0 . 06179.. x0 ⟶ x0 = e707a.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Definition 677e4.. := λ x0 . λ x1 : 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..))))
Theorem cff9f.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 . (∀ 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) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4) ⟶ 677e4.. (e707a.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 2f4b2.. := λ x0 . λ x1 : 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..))))
Theorem ad938.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ο . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 . (∀ 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) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4) ⟶ 2f4b2.. (e707a.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Param d2155.. : ι → (ι → ι → ο) → ι
Definition d89a7.. := λ x0 . λ x1 : ι → ι → ι . λ x2 : 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..)) (0fc90.. x0 x2) (d2155.. x0 x3))))
Theorem f8286.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . x0 = d89a7.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem f1f07.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 x4 : ι → ι → ο . x4 x0 (f482f.. (d89a7.. x0 x1 x2 x3) 4a7ef..) ⟶ x4 (f482f.. (d89a7.. x0 x1 x2 x3) 4a7ef..) x0 (proof)
Theorem 0bf39.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . x0 = d89a7.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = e3162.. (f482f.. x0 (4ae4a.. 4a7ef..)) x5 x6 (proof)
Theorem 1bd93.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x1 x4 x5 = e3162.. (f482f.. (d89a7.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 x5 (proof)
Known f22ec.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 x0 ⟶ f482f.. (0fc90.. x0 x1) x2 = x1 x2
Theorem 9c9f4.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . x0 = d89a7.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x3 x5 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Theorem faf62.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . prim1 x4 x0 ⟶ x2 x4 = f482f.. (f482f.. (d89a7.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) x4 (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 12451.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . x0 = d89a7.. 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 292e8.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x3 x4 x5 = 2b2e3.. (f482f.. (d89a7.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x4 x5 (proof)
Theorem 866ca.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ι . ∀ x6 x7 : ι → ι → ο . d89a7.. x0 x2 x4 x6 = d89a7.. 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 ⟶ x4 x8 = x5 x8)) (∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x6 x8 x9 = x7 x8 x9) (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 bd041.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : 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 ⟶ x3 x7 = x4 x7) ⟶ (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ iff (x5 x7 x8) (x6 x7 x8)) ⟶ d89a7.. x0 x1 x3 x5 = d89a7.. x0 x2 x4 x6 (proof)
Definition 809f6.. := 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 : ι → ι → ο . x1 (d89a7.. x2 x3 x4 x5)) ⟶ x1 x0
Theorem 0bb39.. : ∀ 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 : ι → ι → ο . 809f6.. (d89a7.. x0 x1 x2 x3) (proof)
Theorem 2ac51.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . 809f6.. (d89a7.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x1 x4 x5) x0 (proof)
Theorem 4110f.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 : ι → ι → ο . 809f6.. (d89a7.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x4) x0 (proof)
Known iff_refl : ∀ x0 : ο . iff x0 x0
Theorem c4dc9.. : ∀ x0 . 809f6.. x0 ⟶ x0 = d89a7.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition cfc82.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι) → (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..)))))
Theorem aa6ea.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι) → (ι → ι → ο) → ι . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : 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 . (∀ x7 . prim1 x7 x1 ⟶ x3 x7 = x6 x7) ⟶ ∀ 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) ⟶ cfc82.. (d89a7.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition b0c30.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι) → (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..)))))
Theorem bbed6.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι) → (ι → ι → ο) → ο . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : 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 . (∀ x7 . prim1 x7 x1 ⟶ x3 x7 = x6 x7) ⟶ ∀ 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) ⟶ b0c30.. (d89a7.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Param 1216a.. : ι → (ι → ο) → ι
Definition 1670d.. := λ x0 . λ x1 : ι → ι → ι . λ x2 : 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..)) (0fc90.. x0 x2) (1216a.. x0 x3))))
Theorem e7b29.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ο . x0 = 1670d.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 702f9.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 : ι → ο . x0 = f482f.. (1670d.. x0 x1 x2 x3) 4a7ef.. (proof)
Theorem c19c9.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ο . x0 = 1670d.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = e3162.. (f482f.. x0 (4ae4a.. 4a7ef..)) x5 x6 (proof)
Theorem 3f9d5.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x1 x4 x5 = e3162.. (f482f.. (1670d.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 x5 (proof)
Theorem 642df.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ο . x0 = 1670d.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x3 x5 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Theorem 223a4.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 . prim1 x4 x0 ⟶ x2 x4 = f482f.. (f482f.. (1670d.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) x4 (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 0943f.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ο . x0 = 1670d.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x4 x5 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 (proof)
Theorem 20213.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 : ι → ο . ∀ x4 . prim1 x4 x0 ⟶ x3 x4 = decode_p (f482f.. (1670d.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x4 (proof)
Theorem 7de3c.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ι . ∀ x6 x7 : ι → ο . 1670d.. x0 x2 x4 x6 = 1670d.. 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 ⟶ x4 x8 = x5 x8)) (∀ x8 . prim1 x8 x0 ⟶ x6 x8 = x7 x8) (proof)
Known ee7ef.. : ∀ x0 . ∀ x1 x2 : ι → ο . (∀ x3 . prim1 x3 x0 ⟶ iff (x1 x3) (x2 x3)) ⟶ 1216a.. x0 x1 = 1216a.. x0 x2
Theorem e5cbd.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ι . ∀ x5 x6 : ι → ο . (∀ x7 . prim1 x7 x0 ⟶ ∀ x8 . prim1 x8 x0 ⟶ x1 x7 x8 = x2 x7 x8) ⟶ (∀ x7 . prim1 x7 x0 ⟶ x3 x7 = x4 x7) ⟶ (∀ x7 . prim1 x7 x0 ⟶ iff (x5 x7) (x6 x7)) ⟶ 1670d.. x0 x1 x3 x5 = 1670d.. x0 x2 x4 x6 (proof)
Definition 0040b.. := 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 : ι → ο . x1 (1670d.. x2 x3 x4 x5)) ⟶ x1 x0
Theorem 70f8f.. : ∀ 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 : ι → ο . 0040b.. (1670d.. x0 x1 x2 x3) (proof)
Theorem 33405.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 : ι → ο . 0040b.. (1670d.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x1 x4 x5) x0 (proof)
Theorem 89267.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 : ι → ο . 0040b.. (1670d.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x4) x0 (proof)
Theorem 985a2.. : ∀ x0 . 0040b.. x0 ⟶ x0 = 1670d.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition e535d.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι) → (ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem cdbe1.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι) → (ι → ο) → ι . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : 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 . (∀ x7 . prim1 x7 x1 ⟶ x3 x7 = x6 x7) ⟶ ∀ x7 : ι → ο . (∀ x8 . prim1 x8 x1 ⟶ iff (x4 x8) (x7 x8)) ⟶ x0 x1 x5 x6 x7 = x0 x1 x2 x3 x4) ⟶ e535d.. (1670d.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 8b59f.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι) → (ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 76f24.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι) → (ι → ο) → ο . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : 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 . (∀ x7 . prim1 x7 x1 ⟶ x3 x7 = x6 x7) ⟶ ∀ x7 : ι → ο . (∀ x8 . prim1 x8 x1 ⟶ iff (x4 x8) (x7 x8)) ⟶ x0 x1 x5 x6 x7 = x0 x1 x2 x3 x4) ⟶ 8b59f.. (1670d.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 48567.. := λ x0 . λ x1 : ι → ι → ι . λ x2 : ι → ι . λ x3 . 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..)) (0fc90.. x0 x2) x3)))
Theorem 302f9.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 . x0 = 48567.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 0c2e8.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 . x0 = f482f.. (48567.. x0 x1 x2 x3) 4a7ef.. (proof)
Theorem 84361.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 . x0 = 48567.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = e3162.. (f482f.. x0 (4ae4a.. 4a7ef..)) x5 x6 (proof)
Theorem 667ed.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x1 x4 x5 = e3162.. (f482f.. (48567.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 x5 (proof)
Theorem b50a5.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 . x0 = 48567.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x3 x5 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x5 (proof)
Theorem 5b4ed.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 x4 . prim1 x4 x0 ⟶ x2 x4 = f482f.. (f482f.. (48567.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) x4 (proof)
Theorem 2a06f.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 . x0 = 48567.. x1 x2 x3 x4 ⟶ x4 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem 7abf4.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 . x3 = f482f.. (48567.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Theorem 480cd.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ι . ∀ x6 x7 . 48567.. x0 x2 x4 x6 = 48567.. 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 ⟶ x4 x8 = x5 x8)) (x6 = x7) (proof)
Theorem 27ade.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ι . ∀ x5 . (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x1 x6 x7 = x2 x6 x7) ⟶ (∀ x6 . prim1 x6 x0 ⟶ x3 x6 = x4 x6) ⟶ 48567.. x0 x1 x3 x5 = 48567.. x0 x2 x4 x5 (proof)
Definition c990c.. := 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 . prim1 x5 x2 ⟶ x1 (48567.. x2 x3 x4 x5)) ⟶ x1 x0
Theorem 7e2a5.. : ∀ 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 . prim1 x3 x0 ⟶ c990c.. (48567.. x0 x1 x2 x3) (proof)
Theorem 88d1a.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 . c990c.. (48567.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x1 x4 x5) x0 (proof)
Theorem 7ffab.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 . c990c.. (48567.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x4) x0 (proof)
Theorem c2bb8.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 . c990c.. (48567.. x0 x1 x2 x3) ⟶ prim1 x3 x0 (proof)
Theorem 0a36d.. : ∀ x0 . c990c.. x0 ⟶ x0 = 48567.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Definition 27f38.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι) → ι → ι . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 6e387.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι) → ι → ι . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 . (∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ x3 x7 = x6 x7) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4) ⟶ 27f38.. (48567.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition cd21d.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι) → ι → ο . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 05f1b.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι) → ι → ο . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 . (∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ x6 : ι → ι . (∀ x7 . prim1 x7 x1 ⟶ x3 x7 = x6 x7) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4) ⟶ cd21d.. (48567.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 0fd05.. := λ x0 . λ x1 : ι → ι → ι . λ x2 : ι → ι → ο . 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..)) (d2155.. x0 x2) (1216a.. x0 x3))))
Theorem 7e005.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . x0 = 0fd05.. x1 x2 x3 x4 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 79ecf.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . x0 = f482f.. (0fd05.. x0 x1 x2 x3) 4a7ef.. (proof)
Theorem 6603f.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . x0 = 0fd05.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = e3162.. (f482f.. x0 (4ae4a.. 4a7ef..)) x5 x6 (proof)
Theorem a3c76.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x1 x4 x5 = e3162.. (f482f.. (0fd05.. x0 x1 x2 x3) (4ae4a.. 4a7ef..)) x4 x5 (proof)
Theorem e31f7.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . x0 = 0fd05.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x3 x5 x6 = 2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Theorem d4ea6.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x2 x4 x5 = 2b2e3.. (f482f.. (0fd05.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. 4a7ef..))) x4 x5 (proof)
Theorem 42dda.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . x0 = 0fd05.. x1 x2 x3 x4 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x4 x5 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 (proof)
Theorem 3f154.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . ∀ x4 . prim1 x4 x0 ⟶ x3 x4 = decode_p (f482f.. (0fd05.. x0 x1 x2 x3) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x4 (proof)
Theorem 6feb4.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ι → ο . ∀ x6 x7 : ι → ο . 0fd05.. x0 x2 x4 x6 = 0fd05.. 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 fee39.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ι → ο . ∀ 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 ⟶ iff (x3 x7 x8) (x4 x7 x8)) ⟶ (∀ x7 . prim1 x7 x0 ⟶ iff (x5 x7) (x6 x7)) ⟶ 0fd05.. x0 x1 x3 x5 = 0fd05.. x0 x2 x4 x6 (proof)
Definition 896af.. := 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 → ο . ∀ x5 : ι → ο . x1 (0fd05.. x2 x3 x4 x5)) ⟶ x1 x0
Theorem ecc62.. : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x1 x2 x3) x0) ⟶ ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . 896af.. (0fd05.. x0 x1 x2 x3) (proof)
Theorem 4e4ad.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 : ι → ο . 896af.. (0fd05.. x0 x1 x2 x3) ⟶ ∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x1 x4 x5) x0 (proof)
Theorem 2967c.. : ∀ x0 . 896af.. x0 ⟶ x0 = 0fd05.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition 69fc9.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ο) → (ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 635d1.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ο) → (ι → ο) → ι . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . (∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ x6 : ι → ι → ο . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ iff (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) ⟶ 69fc9.. (0fd05.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)
Definition 80022.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ο) → (ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 3b05e.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ο) → (ι → ο) → ο . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . ∀ x4 : ι → ο . (∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ x6 : ι → ι → ο . (∀ x7 . prim1 x7 x1 ⟶ ∀ x8 . prim1 x8 x1 ⟶ iff (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) ⟶ 80022.. (0fd05.. x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4 (proof)