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 71338.. := λ x0 . λ x1 x2 x3 : ι → ι → ι . 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..))) (eb53d.. x0 x3) 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 62bfa.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 . x0 = 71338.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 5bb38.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . x0 = f482f.. (71338.. 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 1046e.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 . x0 = 71338.. 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 d1558.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x1 x5 x6 = e3162.. (f482f.. (71338.. 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 81f1e.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 . x0 = 71338.. 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 d4b1d.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x5 x6 = e3162.. (f482f.. (71338.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
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
Theorem 8a20d.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 . x0 = 71338.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x4 x6 x7 = e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 x7 (proof)
Theorem 9bed1.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 x5 x6 = e3162.. (f482f.. (71338.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 x6 (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 9d568.. : ∀ x0 x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ x5 . x0 = 71338.. x1 x2 x3 x4 x5 ⟶ x5 = f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Theorem bc3d6.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . x4 = f482f.. (71338.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) (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 82ac9.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 x6 x7 : ι → ι → ι . ∀ x8 x9 . 71338.. x0 x2 x4 x6 x8 = 71338.. 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 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x6 x10 x11 = x7 x10 x11)) (x8 = x9) (proof)
Known 8fdaf.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x1 x3 x4 = x2 x3 x4) ⟶ eb53d.. x0 x1 = eb53d.. x0 x2
Theorem 9da8a.. : ∀ x0 . ∀ x1 x2 x3 x4 x5 x6 : ι → ι → ι . ∀ 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 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x5 x8 x9 = x6 x8 x9) ⟶ 71338.. x0 x1 x3 x5 x7 = 71338.. x0 x2 x4 x6 x7 (proof)
Definition 23f55.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ι . (∀ x4 . prim1 x4 x2 ⟶ ∀ x5 . prim1 x5 x2 ⟶ prim1 (x3 x4 x5) x2) ⟶ ∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x2 ⟶ ∀ x6 . prim1 x6 x2 ⟶ prim1 (x4 x5 x6) x2) ⟶ ∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x2 ⟶ ∀ x7 . prim1 x7 x2 ⟶ prim1 (x5 x6 x7) x2) ⟶ ∀ x6 . prim1 x6 x2 ⟶ x1 (71338.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem 5991f.. : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x1 x2 x3) x0) ⟶ ∀ x2 : ι → ι → ι . (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x3 x4) x0) ⟶ ∀ x3 : ι → ι → ι . (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x3 x4 x5) x0) ⟶ ∀ x4 . prim1 x4 x0 ⟶ 23f55.. (71338.. x0 x1 x2 x3 x4) (proof)
Theorem 21112.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . 23f55.. (71338.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x1 x5 x6) x0 (proof)
Theorem ae691.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . 23f55.. (71338.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x2 x5 x6) x0 (proof)
Theorem d8dcc.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . 23f55.. (71338.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x3 x5 x6) x0 (proof)
Theorem 53049.. : ∀ x0 . ∀ x1 x2 x3 : ι → ι → ι . ∀ x4 . 23f55.. (71338.. x0 x1 x2 x3 x4) ⟶ prim1 x4 x0 (proof)
Theorem 8b133.. : ∀ x0 . 23f55.. x0 ⟶ x0 = 71338.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (proof)
Definition 9fbe5.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ι) → ι → ι . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 9aabc.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ι) → ι → ι . ∀ x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ 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 → ι → ι . (∀ x9 . prim1 x9 x1 ⟶ ∀ x10 . prim1 x10 x1 ⟶ x4 x9 x10 = x8 x9 x10) ⟶ x0 x1 x6 x7 x8 x5 = x0 x1 x2 x3 x4 x5) ⟶ 9fbe5.. (71338.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 7765f.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ι) → ι → ο . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))
Theorem 10d52.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ι) → ι → ο . ∀ x1 . ∀ x2 x3 x4 : ι → ι → ι . ∀ 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 → ι → ι . (∀ x9 . prim1 x9 x1 ⟶ ∀ x10 . prim1 x10 x1 ⟶ x4 x9 x10 = x8 x9 x10) ⟶ x0 x1 x6 x7 x8 x5 = x0 x1 x2 x3 x4 x5) ⟶ 7765f.. (71338.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 37dbe.. := λ x0 . λ x1 x2 : ι → ι → ι . λ x3 x4 : ι → ι . 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..))) (0fc90.. x0 x3) (0fc90.. x0 x4)))))
Theorem 5dcde.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ι . x0 = 37dbe.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 8d6c4.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ι . x0 = f482f.. (37dbe.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Theorem d246f.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ι . x0 = 37dbe.. 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 61aee.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ι . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x1 x5 x6 = e3162.. (f482f.. (37dbe.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 x6 (proof)
Theorem e008e.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ι . x0 = 37dbe.. 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 eea63.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ι . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x5 x6 = e3162.. (f482f.. (37dbe.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Known f22ec.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 x0 ⟶ f482f.. (0fc90.. x0 x1) x2 = x1 x2
Theorem 9c862.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ι . x0 = 37dbe.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x4 x6 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 (proof)
Theorem c1142.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ι . ∀ x5 . prim1 x5 x0 ⟶ x3 x5 = f482f.. (f482f.. (37dbe.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 (proof)
Theorem 42f04.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ι . x0 = 37dbe.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x5 x6 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x6 (proof)
Theorem 3b308.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ι . ∀ x5 . prim1 x5 x0 ⟶ x4 x5 = f482f.. (f482f.. (37dbe.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x5 (proof)
Theorem c5fe9.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι → ι . ∀ x6 x7 x8 x9 : ι → ι . 37dbe.. x0 x2 x4 x6 x8 = 37dbe.. 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)
Known 4402a.. : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ x1 x3 = x2 x3) ⟶ 0fc90.. x0 x1 = 0fc90.. x0 x2
Theorem 3967a.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι → ι . ∀ x5 x6 x7 x8 : ι → ι . (∀ 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 ⟶ x5 x9 = x6 x9) ⟶ (∀ x9 . prim1 x9 x0 ⟶ x7 x9 = x8 x9) ⟶ 37dbe.. x0 x1 x3 x5 x7 = 37dbe.. x0 x2 x4 x6 x8 (proof)
Definition 04451.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ι . (∀ x4 . prim1 x4 x2 ⟶ ∀ x5 . prim1 x5 x2 ⟶ prim1 (x3 x4 x5) x2) ⟶ ∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x2 ⟶ ∀ x6 . prim1 x6 x2 ⟶ prim1 (x4 x5 x6) x2) ⟶ ∀ x5 : ι → ι . (∀ x6 . prim1 x6 x2 ⟶ prim1 (x5 x6) x2) ⟶ ∀ x6 : ι → ι . (∀ x7 . prim1 x7 x2 ⟶ prim1 (x6 x7) x2) ⟶ x1 (37dbe.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem 3c27b.. : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x1 x2 x3) x0) ⟶ ∀ x2 : ι → ι → ι . (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x3 x4) x0) ⟶ ∀ x3 : ι → ι . (∀ x4 . prim1 x4 x0 ⟶ prim1 (x3 x4) x0) ⟶ ∀ x4 : ι → ι . (∀ x5 . prim1 x5 x0 ⟶ prim1 (x4 x5) x0) ⟶ 04451.. (37dbe.. x0 x1 x2 x3 x4) (proof)
Theorem e6800.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ι . 04451.. (37dbe.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x1 x5 x6) x0 (proof)
Theorem 4223b.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ι . 04451.. (37dbe.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x2 x5 x6) x0 (proof)
Theorem 114ab.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ι . 04451.. (37dbe.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x3 x5) x0 (proof)
Theorem 6599b.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ι . 04451.. (37dbe.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x4 x5) x0 (proof)
Theorem 51def.. : ∀ x0 . 04451.. x0 ⟶ x0 = 37dbe.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))) (proof)
Definition a2e89.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι) → (ι → ι) → ι . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem 53862.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι) → (ι → ι) → ι . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ι . (∀ 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 → ι . (∀ x9 . prim1 x9 x1 ⟶ x4 x9 = x8 x9) ⟶ ∀ x9 : ι → ι . (∀ x10 . prim1 x10 x1 ⟶ x5 x10 = x9 x10) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ a2e89.. (37dbe.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 4314e.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι) → (ι → ι) → ο . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem b091d.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι) → (ι → ι) → ο . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ι . (∀ 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 → ι . (∀ x9 . prim1 x9 x1 ⟶ x4 x9 = x8 x9) ⟶ ∀ x9 : ι → ι . (∀ x10 . prim1 x10 x1 ⟶ x5 x10 = x9 x10) ⟶ x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5) ⟶ 4314e.. (37dbe.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Param d2155.. : ι → (ι → ι → ο) → ι
Definition b1713.. := λ x0 . λ x1 x2 : ι → ι → ι . λ x3 : ι → ι . λ 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..)) (eb53d.. x0 x2) (If_i (x5 = 4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (0fc90.. x0 x3) (d2155.. x0 x4)))))
Theorem 2847f.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ x5 : ι → ι → ο . x0 = b1713.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 284cf.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 x5 : ι → ι → ο . x5 x0 (f482f.. (b1713.. x0 x1 x2 x3 x4) 4a7ef..) ⟶ x5 (f482f.. (b1713.. x0 x1 x2 x3 x4) 4a7ef..) x0 (proof)
Theorem dee9e.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ x5 : ι → ι → ο . x0 = b1713.. 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 da4cc.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x1 x5 x6 = e3162.. (f482f.. (b1713.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 x6 (proof)
Theorem 64660.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ x5 : ι → ι → ο . x0 = b1713.. 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 d6f9b.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x5 x6 = e3162.. (f482f.. (b1713.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Theorem ba63c.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ x5 : ι → ι → ο . x0 = b1713.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x4 x6 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 (proof)
Theorem c41bd.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x3 x5 = f482f.. (f482f.. (b1713.. x0 x1 x2 x3 x4) (4ae4a.. (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 c42c4.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ x5 : ι → ι → ο . x0 = b1713.. 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 a55dd.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x4 x5 x6 = 2b2e3.. (f482f.. (b1713.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x5 x6 (proof)
Theorem 309f9.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι → ι . ∀ x6 x7 : ι → ι . ∀ x8 x9 : ι → ι → ο . b1713.. x0 x2 x4 x6 x8 = b1713.. 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 ⟶ ∀ x11 . prim1 x11 x0 ⟶ x8 x10 x11 = x9 x10 x11) (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
Theorem 839ff.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι → ι . ∀ x5 x6 : ι → ι . ∀ x7 x8 : ι → 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 ⟶ x5 x9 = x6 x9) ⟶ (∀ x9 . prim1 x9 x0 ⟶ ∀ x10 . prim1 x10 x0 ⟶ iff (x7 x9 x10) (x8 x9 x10)) ⟶ b1713.. x0 x1 x3 x5 x7 = b1713.. x0 x2 x4 x6 x8 (proof)
Definition d51b5.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ι . (∀ x4 . prim1 x4 x2 ⟶ ∀ x5 . prim1 x5 x2 ⟶ prim1 (x3 x4 x5) x2) ⟶ ∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x2 ⟶ ∀ x6 . prim1 x6 x2 ⟶ prim1 (x4 x5 x6) x2) ⟶ ∀ x5 : ι → ι . (∀ x6 . prim1 x6 x2 ⟶ prim1 (x5 x6) x2) ⟶ ∀ x6 : ι → ι → ο . x1 (b1713.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem e0ddc.. : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x1 x2 x3) x0) ⟶ ∀ x2 : ι → ι → ι . (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x3 x4) x0) ⟶ ∀ x3 : ι → ι . (∀ x4 . prim1 x4 x0 ⟶ prim1 (x3 x4) x0) ⟶ ∀ x4 : ι → ι → ο . d51b5.. (b1713.. x0 x1 x2 x3 x4) (proof)
Theorem 2a5bd.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . d51b5.. (b1713.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x1 x5 x6) x0 (proof)
Theorem b5aaf.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . d51b5.. (b1713.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x2 x5 x6) x0 (proof)
Theorem 1d099.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ι → ο . d51b5.. (b1713.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x3 x5) x0 (proof)
Known iff_refl : ∀ x0 : ο . iff x0 x0
Theorem 1f70a.. : ∀ x0 . d51b5.. x0 ⟶ x0 = b1713.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))) (proof)
Definition 1f4f1.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι) → (ι → ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem c1aec.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι) → (ι → ι → ο) → ι . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ 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 → ι . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ x3 x8 x9 = x7 x8 x9) ⟶ ∀ x8 : ι → ι . (∀ x9 . prim1 x9 x1 ⟶ x4 x9 = x8 x9) ⟶ ∀ 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) ⟶ 1f4f1.. (b1713.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition c83ef.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι) → (ι → ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem 6a5ab.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι) → (ι → ι → ο) → ο . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ 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 → ι . (∀ x8 . prim1 x8 x1 ⟶ ∀ x9 . prim1 x9 x1 ⟶ x3 x8 x9 = x7 x8 x9) ⟶ ∀ x8 : ι → ι . (∀ x9 . prim1 x9 x1 ⟶ x4 x9 = x8 x9) ⟶ ∀ 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) ⟶ c83ef.. (b1713.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Param 1216a.. : ι → (ι → ο) → ι
Definition de6e8.. := λ x0 . λ x1 x2 : ι → ι → ι . λ x3 : ι → ι . λ 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..))) (0fc90.. x0 x3) (1216a.. x0 x4)))))
Theorem f97c5.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ x5 : ι → ο . x0 = de6e8.. x1 x2 x3 x4 x5 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 96cbf.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ο . x0 = f482f.. (de6e8.. x0 x1 x2 x3 x4) 4a7ef.. (proof)
Theorem 10e9c.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ x5 : ι → ο . x0 = de6e8.. 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 7d032.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x1 x5 x6 = e3162.. (f482f.. (de6e8.. x0 x1 x2 x3 x4) (4ae4a.. 4a7ef..)) x5 x6 (proof)
Theorem 9b430.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ x5 : ι → ο . x0 = de6e8.. 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 a411f.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x2 x5 x6 = e3162.. (f482f.. (de6e8.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. 4a7ef..))) x5 x6 (proof)
Theorem 48cf3.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ x5 : ι → ο . x0 = de6e8.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x4 x6 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x6 (proof)
Theorem 41435.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x3 x5 = f482f.. (f482f.. (de6e8.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x5 (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 71c44.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ x5 : ι → ο . x0 = de6e8.. x1 x2 x3 x4 x5 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x5 x6 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x6 (proof)
Theorem cb2b8.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ο . ∀ x5 . prim1 x5 x0 ⟶ x4 x5 = decode_p (f482f.. (de6e8.. x0 x1 x2 x3 x4) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x5 (proof)
Theorem 28853.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι → ι . ∀ x6 x7 : ι → ι . ∀ x8 x9 : ι → ο . de6e8.. x0 x2 x4 x6 x8 = de6e8.. 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)
Known ee7ef.. : ∀ x0 . ∀ x1 x2 : ι → ο . (∀ x3 . prim1 x3 x0 ⟶ iff (x1 x3) (x2 x3)) ⟶ 1216a.. x0 x1 = 1216a.. x0 x2
Theorem c3164.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι → ι . ∀ x5 x6 : ι → ι . ∀ 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 ⟶ x5 x9 = x6 x9) ⟶ (∀ x9 . prim1 x9 x0 ⟶ iff (x7 x9) (x8 x9)) ⟶ de6e8.. x0 x1 x3 x5 x7 = de6e8.. x0 x2 x4 x6 x8 (proof)
Definition c754f.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ι . (∀ x4 . prim1 x4 x2 ⟶ ∀ x5 . prim1 x5 x2 ⟶ prim1 (x3 x4 x5) x2) ⟶ ∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x2 ⟶ ∀ x6 . prim1 x6 x2 ⟶ prim1 (x4 x5 x6) x2) ⟶ ∀ x5 : ι → ι . (∀ x6 . prim1 x6 x2 ⟶ prim1 (x5 x6) x2) ⟶ ∀ x6 : ι → ο . x1 (de6e8.. x2 x3 x4 x5 x6)) ⟶ x1 x0
Theorem 7d63f.. : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x1 x2 x3) x0) ⟶ ∀ x2 : ι → ι → ι . (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x3 x4) x0) ⟶ ∀ x3 : ι → ι . (∀ x4 . prim1 x4 x0 ⟶ prim1 (x3 x4) x0) ⟶ ∀ x4 : ι → ο . c754f.. (de6e8.. x0 x1 x2 x3 x4) (proof)
Theorem 33ba0.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ο . c754f.. (de6e8.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x1 x5 x6) x0 (proof)
Theorem e7ad6.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ο . c754f.. (de6e8.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ prim1 (x2 x5 x6) x0 (proof)
Theorem c7ce6.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 : ι → ι . ∀ x4 : ι → ο . c754f.. (de6e8.. x0 x1 x2 x3 x4) ⟶ ∀ x5 . prim1 x5 x0 ⟶ prim1 (x3 x5) x0 (proof)
Theorem 90377.. : ∀ x0 . c754f.. x0 ⟶ x0 = de6e8.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))))) (proof)
Definition 0a647.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι) → (ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem 7c0bd.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι) → (ι → ο) → ι . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ 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 → ι . (∀ x9 . prim1 x9 x1 ⟶ 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) ⟶ 0a647.. (de6e8.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)
Definition 09b8b.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι) → (ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))))
Theorem e4b93.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι) → (ι → ο) → ο . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ι → ι . ∀ 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 → ι . (∀ x9 . prim1 x9 x1 ⟶ 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) ⟶ 09b8b.. (de6e8.. x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5 (proof)