Proofgold Signed Transaction

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 b6bd3.. := λ x0 . λ x1 x2 : ι → ι → ι . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x3 . If_i (x3 = 4a7ef..) x0 (If_i (x3 = 4ae4a.. 4a7ef..) (eb53d.. x0 x1) (eb53d.. x0 x2)))
Param f482f.. : ι → ι → ι
Known 52da6.. : ∀ x0 x1 x2 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x4 . If_i (x4 = 4a7ef..) x0 (If_i (x4 = 4ae4a.. 4a7ef..) x1 x2))) 4a7ef.. = x0
Theorem b2295.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . x0 = b6bd3.. x1 x2 x3 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 8ccc5.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . x0 = f482f.. (b6bd3.. x0 x1 x2) 4a7ef.. (proof)
Param e3162.. : ι → ι → ι → ι
Known c2bca.. : ∀ x0 x1 x2 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x4 . If_i (x4 = 4a7ef..) x0 (If_i (x4 = 4ae4a.. 4a7ef..) x1 x2))) (4ae4a.. 4a7ef..) = x1
Known 35054.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ e3162.. (eb53d.. x0 x1) x2 x3 = x1 x2 x3
Theorem 27335.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . x0 = b6bd3.. x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x1 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x2 x4 x5 = e3162.. (f482f.. x0 (4ae4a.. 4a7ef..)) x4 x5 (proof)
Theorem a5628.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x1 x3 x4 = e3162.. (f482f.. (b6bd3.. x0 x1 x2) (4ae4a.. 4a7ef..)) x3 x4 (proof)
Known 11d6d.. : ∀ x0 x1 x2 . f482f.. (0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x4 . If_i (x4 = 4a7ef..) x0 (If_i (x4 = 4ae4a.. 4a7ef..) x1 x2))) (4ae4a.. (4ae4a.. 4a7ef..)) = x2
Theorem 9101a.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . x0 = b6bd3.. x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x1 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x3 x4 x5 = e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x4 x5 (proof)
Theorem f278f.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x2 x3 x4 = e3162.. (f482f.. (b6bd3.. x0 x1 x2) (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4 (proof)
Definition and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Theorem 4f0f8.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι → ι . b6bd3.. x0 x2 x4 = b6bd3.. x1 x3 x5 ⟶ and (and (x0 = x1) (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x2 x6 x7 = x3 x6 x7)) (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x4 x6 x7 = x5 x6 x7) (proof)
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 dcf72.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι → ι . (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x1 x5 x6 = x2 x5 x6) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x3 x5 x6 = x4 x5 x6) ⟶ b6bd3.. x0 x1 x3 = b6bd3.. x0 x2 x4 (proof)
Definition e2219.. := 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) ⟶ x1 (b6bd3.. x2 x3 x4)) ⟶ x1 x0
Theorem c2d7a.. : ∀ 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) ⟶ e2219.. (b6bd3.. x0 x1 x2) (proof)
Theorem 22315.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . e2219.. (b6bd3.. x0 x1 x2) ⟶ ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x1 x3 x4) x0 (proof)
Theorem d90ae.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . e2219.. (b6bd3.. x0 x1 x2) ⟶ ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x2 x3 x4) x0 (proof)
Theorem d40c9.. : ∀ x0 . e2219.. x0 ⟶ x0 = b6bd3.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Definition 81fc5.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → ι . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 4b220.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → ι . ∀ x1 . ∀ x2 x3 : ι → ι → ι . (∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = x4 x5 x6) ⟶ ∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x3 x6 x7 = x5 x6 x7) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ 81fc5.. (b6bd3.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition a70ce.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → ο . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (e3162.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 12a13.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → ο . ∀ x1 . ∀ x2 x3 : ι → ι → ι . (∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = x4 x5 x6) ⟶ ∀ x5 : ι → ι → ι . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ x3 x6 x7 = x5 x6 x7) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ a70ce.. (b6bd3.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition 971d3.. := λ x0 . λ x1 : ι → ι → ι . λ x2 : ι → ι . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x3 . If_i (x3 = 4a7ef..) x0 (If_i (x3 = 4ae4a.. 4a7ef..) (eb53d.. x0 x1) (0fc90.. x0 x2)))
Theorem 96049.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . x0 = 971d3.. x1 x2 x3 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 38865.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . x0 = f482f.. (971d3.. x0 x1 x2) 4a7ef.. (proof)
Theorem 33e2c.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . x0 = 971d3.. x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x1 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x2 x4 x5 = e3162.. (f482f.. x0 (4ae4a.. 4a7ef..)) x4 x5 (proof)
Theorem 252d8.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x1 x3 x4 = e3162.. (f482f.. (971d3.. x0 x1 x2) (4ae4a.. 4a7ef..)) x3 x4 (proof)
Known f22ec.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 x0 ⟶ f482f.. (0fc90.. x0 x1) x2 = x1 x2
Theorem 3d0b4.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . x0 = 971d3.. x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x1 ⟶ x3 x4 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x4 (proof)
Theorem 2b252.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . ∀ x3 . prim1 x3 x0 ⟶ x2 x3 = f482f.. (f482f.. (971d3.. x0 x1 x2) (4ae4a.. (4ae4a.. 4a7ef..))) x3 (proof)
Theorem f3329.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ι . 971d3.. x0 x2 x4 = 971d3.. x1 x3 x5 ⟶ and (and (x0 = x1) (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x2 x6 x7 = x3 x6 x7)) (∀ x6 . prim1 x6 x0 ⟶ x4 x6 = x5 x6) (proof)
Known 4402a.. : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ x1 x3 = x2 x3) ⟶ 0fc90.. x0 x1 = 0fc90.. x0 x2
Theorem 0c169.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ι . (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x1 x5 x6 = x2 x5 x6) ⟶ (∀ x5 . prim1 x5 x0 ⟶ x3 x5 = x4 x5) ⟶ 971d3.. x0 x1 x3 = 971d3.. x0 x2 x4 (proof)
Definition a5e4b.. := 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) ⟶ x1 (971d3.. x2 x3 x4)) ⟶ x1 x0
Theorem d80b7.. : ∀ 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) ⟶ a5e4b.. (971d3.. x0 x1 x2) (proof)
Theorem f462c.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . a5e4b.. (971d3.. x0 x1 x2) ⟶ ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x1 x3 x4) x0 (proof)
Theorem bfd44.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι . a5e4b.. (971d3.. x0 x1 x2) ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x0 (proof)
Theorem 3c7c5.. : ∀ x0 . a5e4b.. x0 ⟶ x0 = 971d3.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Definition ccf31.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι) → ι . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem eff65.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι) → ι . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . (∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = x4 x5 x6) ⟶ ∀ x5 : ι → ι . (∀ x6 . prim1 x6 x1 ⟶ x3 x6 = x5 x6) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ ccf31.. (971d3.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition f4433.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι) → ο . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem e928d.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι) → ο . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι . (∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = x4 x5 x6) ⟶ ∀ x5 : ι → ι . (∀ x6 . prim1 x6 x1 ⟶ x3 x6 = x5 x6) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ f4433.. (971d3.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Param d2155.. : ι → (ι → ι → ο) → ι
Definition b5cc3.. := λ x0 . λ x1 : ι → ι → ι . λ x2 : ι → ι → ο . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x3 . If_i (x3 = 4a7ef..) x0 (If_i (x3 = 4ae4a.. 4a7ef..) (eb53d.. x0 x1) (d2155.. x0 x2)))
Theorem f46b1.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . x0 = b5cc3.. x1 x2 x3 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem b29ea.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 x3 : ι → ι → ο . x3 x0 (f482f.. (b5cc3.. x0 x1 x2) 4a7ef..) ⟶ x3 (f482f.. (b5cc3.. x0 x1 x2) 4a7ef..) x0 (proof)
Theorem b0ed2.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . x0 = b5cc3.. x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x1 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x2 x4 x5 = e3162.. (f482f.. x0 (4ae4a.. 4a7ef..)) x4 x5 (proof)
Theorem 9890e.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x1 x3 x4 = e3162.. (f482f.. (b5cc3.. x0 x1 x2) (4ae4a.. 4a7ef..)) x3 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 0d5f9.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . x0 = b5cc3.. x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x1 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x3 x4 x5 = 2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x4 x5 (proof)
Theorem aa310.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι → ο . ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x2 x3 x4 = 2b2e3.. (f482f.. (b5cc3.. x0 x1 x2) (4ae4a.. (4ae4a.. 4a7ef..))) x3 x4 (proof)
Theorem f1c06.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ι → ο . b5cc3.. x0 x2 x4 = b5cc3.. x1 x3 x5 ⟶ and (and (x0 = x1) (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x2 x6 x7 = x3 x6 x7)) (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x4 x6 x7 = x5 x6 x7) (proof)
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 af357.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ι → ο . (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x1 x5 x6 = x2 x5 x6) ⟶ (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ iff (x3 x5 x6) (x4 x5 x6)) ⟶ b5cc3.. x0 x1 x3 = b5cc3.. x0 x2 x4 (proof)
Definition ec962.. := 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 . x1 (b5cc3.. x2 x3 x4)) ⟶ x1 x0
Theorem 8c7f6.. : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x1 x2 x3) x0) ⟶ ∀ x2 : ι → ι → ο . ec962.. (b5cc3.. x0 x1 x2) (proof)
Theorem 7a5e3.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ι → ο . ec962.. (b5cc3.. x0 x1 x2) ⟶ ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x1 x3 x4) x0 (proof)
Known iff_refl : ∀ x0 : ο . iff x0 x0
Theorem 567a6.. : ∀ x0 . ec962.. x0 ⟶ x0 = b5cc3.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Definition 68fde.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 6588e.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ο) → ι . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . (∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = x4 x5 x6) ⟶ ∀ x5 : ι → ι → ο . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ iff (x3 x6 x7) (x5 x6 x7)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ 68fde.. (b5cc3.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition c65d4.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 117a0.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ο) → ο . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ι → ο . (∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = x4 x5 x6) ⟶ ∀ x5 : ι → ι → ο . (∀ x6 . prim1 x6 x1 ⟶ ∀ x7 . prim1 x7 x1 ⟶ iff (x3 x6 x7) (x5 x6 x7)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ c65d4.. (b5cc3.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Param 1216a.. : ι → (ι → ο) → ι
Definition 33a0d.. := λ x0 . λ x1 : ι → ι → ι . λ x2 : ι → ο . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x3 . If_i (x3 = 4a7ef..) x0 (If_i (x3 = 4ae4a.. 4a7ef..) (eb53d.. x0 x1) (1216a.. x0 x2)))
Theorem 928fb.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . x0 = 33a0d.. x1 x2 x3 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 0d30e.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . x0 = f482f.. (33a0d.. x0 x1 x2) 4a7ef.. (proof)
Theorem 92cc6.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . x0 = 33a0d.. x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x1 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x2 x4 x5 = e3162.. (f482f.. x0 (4ae4a.. 4a7ef..)) x4 x5 (proof)
Theorem b55f5.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x1 x3 x4 = e3162.. (f482f.. (33a0d.. x0 x1 x2) (4ae4a.. 4a7ef..)) x3 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 49df8.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . x0 = 33a0d.. x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x1 ⟶ x3 x4 = decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x4 (proof)
Theorem 82801.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . ∀ x3 . prim1 x3 x0 ⟶ x2 x3 = decode_p (f482f.. (33a0d.. x0 x1 x2) (4ae4a.. (4ae4a.. 4a7ef..))) x3 (proof)
Theorem 3339e.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 : ι → ο . 33a0d.. x0 x2 x4 = 33a0d.. x1 x3 x5 ⟶ and (and (x0 = x1) (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x2 x6 x7 = x3 x6 x7)) (∀ x6 . prim1 x6 x0 ⟶ x4 x6 = x5 x6) (proof)
Known ee7ef.. : ∀ x0 . ∀ x1 x2 : ι → ο . (∀ x3 . prim1 x3 x0 ⟶ iff (x1 x3) (x2 x3)) ⟶ 1216a.. x0 x1 = 1216a.. x0 x2
Theorem 52ca8.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 : ι → ο . (∀ x5 . prim1 x5 x0 ⟶ ∀ x6 . prim1 x6 x0 ⟶ x1 x5 x6 = x2 x5 x6) ⟶ (∀ x5 . prim1 x5 x0 ⟶ iff (x3 x5) (x4 x5)) ⟶ 33a0d.. x0 x1 x3 = 33a0d.. x0 x2 x4 (proof)
Definition 31bda.. := 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 → ο . x1 (33a0d.. x2 x3 x4)) ⟶ x1 x0
Theorem 895e8.. : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x1 x2 x3) x0) ⟶ ∀ x2 : ι → ο . 31bda.. (33a0d.. x0 x1 x2) (proof)
Theorem efe95.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 : ι → ο . 31bda.. (33a0d.. x0 x1 x2) ⟶ ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x1 x3 x4) x0 (proof)
Theorem 93dcd.. : ∀ x0 . 31bda.. x0 ⟶ x0 = 33a0d.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Definition f4846.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ο) → ι . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 4f5a1.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ο) → ι . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . (∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = x4 x5 x6) ⟶ ∀ x5 : ι → ο . (∀ x6 . prim1 x6 x1 ⟶ iff (x3 x6) (x5 x6)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ f4846.. (33a0d.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition dcf97.. := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ο) → ο . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (decode_p (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem a70e6.. : ∀ x0 : ι → (ι → ι → ι) → (ι → ο) → ο . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 : ι → ο . (∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = x4 x5 x6) ⟶ ∀ x5 : ι → ο . (∀ x6 . prim1 x6 x1 ⟶ iff (x3 x6) (x5 x6)) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ dcf97.. (33a0d.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition 96158.. := λ x0 . λ x1 : ι → ι → ι . λ x2 . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x3 . If_i (x3 = 4a7ef..) x0 (If_i (x3 = 4ae4a.. 4a7ef..) (eb53d.. x0 x1) x2))
Theorem 75497.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 . x0 = 96158.. x1 x2 x3 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 7f6d9.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . x0 = f482f.. (96158.. x0 x1 x2) 4a7ef.. (proof)
Theorem 2a2b9.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 . x0 = 96158.. x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x1 ⟶ ∀ x5 . prim1 x5 x1 ⟶ x2 x4 x5 = e3162.. (f482f.. x0 (4ae4a.. 4a7ef..)) x4 x5 (proof)
Theorem 9b563.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x1 x3 x4 = e3162.. (f482f.. (96158.. x0 x1 x2) (4ae4a.. 4a7ef..)) x3 x4 (proof)
Theorem 68801.. : ∀ x0 x1 . ∀ x2 : ι → ι → ι . ∀ x3 . x0 = 96158.. x1 x2 x3 ⟶ x3 = f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)) (proof)
Theorem 546ed.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . x2 = f482f.. (96158.. x0 x1 x2) (4ae4a.. (4ae4a.. 4a7ef..)) (proof)
Theorem 383cb.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . 96158.. x0 x2 x4 = 96158.. x1 x3 x5 ⟶ and (and (x0 = x1) (∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x2 x6 x7 = x3 x6 x7)) (x4 = x5) (proof)
Theorem 86c18.. : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 . (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 . prim1 x5 x0 ⟶ x1 x4 x5 = x2 x4 x5) ⟶ 96158.. x0 x1 x3 = 96158.. x0 x2 x3 (proof)
Definition 42a91.. := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ι . (∀ x4 . prim1 x4 x2 ⟶ ∀ x5 . prim1 x5 x2 ⟶ prim1 (x3 x4 x5) x2) ⟶ ∀ x4 . prim1 x4 x2 ⟶ x1 (96158.. x2 x3 x4)) ⟶ x1 x0
Theorem 174a4.. : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . prim1 x2 x0 ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x1 x2 x3) x0) ⟶ ∀ x2 . prim1 x2 x0 ⟶ 42a91.. (96158.. x0 x1 x2) (proof)
Theorem 5c855.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . 42a91.. (96158.. x0 x1 x2) ⟶ ∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ prim1 (x1 x3 x4) x0 (proof)
Theorem 4e086.. : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . 42a91.. (96158.. x0 x1 x2) ⟶ prim1 x2 x0 (proof)
Theorem 747ef.. : ∀ x0 . 42a91.. x0 ⟶ x0 = 96158.. (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) (proof)
Definition a6bc8.. := λ x0 . λ x1 : ι → (ι → ι → ι) → ι → ι . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))
Theorem 28784.. : ∀ x0 : ι → (ι → ι → ι) → ι → ι . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 . (∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = x4 x5 x6) ⟶ x0 x1 x4 x3 = x0 x1 x2 x3) ⟶ a6bc8.. (96158.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition 86936.. := λ x0 . λ x1 : ι → (ι → ι → ι) → ι → ο . x1 (f482f.. x0 4a7ef..) (e3162.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))
Theorem 54d21.. : ∀ x0 : ι → (ι → ι → ι) → ι → ο . ∀ x1 . ∀ x2 : ι → ι → ι . ∀ x3 . (∀ x4 : ι → ι → ι . (∀ x5 . prim1 x5 x1 ⟶ ∀ x6 . prim1 x6 x1 ⟶ x2 x5 x6 = x4 x5 x6) ⟶ x0 x1 x4 x3 = x0 x1 x2 x3) ⟶ 86936.. (96158.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition 5fdf5.. := λ x0 . λ x1 x2 : ι → ι . 0fc90.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))) (λ x3 . If_i (x3 = 4a7ef..) x0 (If_i (x3 = 4ae4a.. 4a7ef..) (0fc90.. x0 x1) (0fc90.. x0 x2)))
Theorem 8dad2.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . x0 = 5fdf5.. x1 x2 x3 ⟶ x1 = f482f.. x0 4a7ef.. (proof)
Theorem 170ca.. : ∀ x0 . ∀ x1 x2 : ι → ι . x0 = f482f.. (5fdf5.. x0 x1 x2) 4a7ef.. (proof)
Theorem bbd2c.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . x0 = 5fdf5.. x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x1 ⟶ x2 x4 = f482f.. (f482f.. x0 (4ae4a.. 4a7ef..)) x4 (proof)
Theorem bbde7.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 . prim1 x3 x0 ⟶ x1 x3 = f482f.. (f482f.. (5fdf5.. x0 x1 x2) (4ae4a.. 4a7ef..)) x3 (proof)
Theorem 6df1c.. : ∀ x0 x1 . ∀ x2 x3 : ι → ι . x0 = 5fdf5.. x1 x2 x3 ⟶ ∀ x4 . prim1 x4 x1 ⟶ x3 x4 = f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))) x4 (proof)
Theorem b49ff.. : ∀ x0 . ∀ x1 x2 : ι → ι . ∀ x3 . prim1 x3 x0 ⟶ x2 x3 = f482f.. (f482f.. (5fdf5.. x0 x1 x2) (4ae4a.. (4ae4a.. 4a7ef..))) x3 (proof)
Theorem b576e.. : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι . 5fdf5.. x0 x2 x4 = 5fdf5.. x1 x3 x5 ⟶ and (and (x0 = x1) (∀ x6 . prim1 x6 x0 ⟶ x2 x6 = x3 x6)) (∀ x6 . prim1 x6 x0 ⟶ x4 x6 = x5 x6) (proof)
Theorem 06bf1.. : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι . (∀ x5 . prim1 x5 x0 ⟶ x1 x5 = x2 x5) ⟶ (∀ x5 . prim1 x5 x0 ⟶ x3 x5 = x4 x5) ⟶ 5fdf5.. x0 x1 x3 = 5fdf5.. x0 x2 x4 (proof)
Definition a3341.. := 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) ⟶ x1 (5fdf5.. x2 x3 x4)) ⟶ x1 x0
Theorem b12bd.. : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . prim1 x2 x0 ⟶ prim1 (x1 x2) x0) ⟶ ∀ x2 : ι → ι . (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x0) ⟶ a3341.. (5fdf5.. x0 x1 x2) (proof)
Theorem b3a9c.. : ∀ x0 . ∀ x1 x2 : ι → ι . a3341.. (5fdf5.. x0 x1 x2) ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x1 x3) x0 (proof)
Theorem 123df.. : ∀ x0 . ∀ x1 x2 : ι → ι . a3341.. (5fdf5.. x0 x1 x2) ⟶ ∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x0 (proof)
Theorem 881bf.. : ∀ x0 . a3341.. x0 ⟶ x0 = 5fdf5.. (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..)))) (proof)
Definition adadd.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → ι . x1 (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 85aab.. : ∀ x0 : ι → (ι → ι) → (ι → ι) → ι . ∀ x1 . ∀ x2 x3 : ι → ι . (∀ x4 : ι → ι . (∀ x5 . prim1 x5 x1 ⟶ x2 x5 = x4 x5) ⟶ ∀ x5 : ι → ι . (∀ x6 . prim1 x6 x1 ⟶ x3 x6 = x5 x6) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ adadd.. (5fdf5.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)
Definition 3c64d.. := λ x0 . λ x1 : ι → (ι → ι) → (ι → ι) → ο . x1 (f482f.. x0 4a7ef..) (f482f.. (f482f.. x0 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x0 (4ae4a.. (4ae4a.. 4a7ef..))))
Theorem 8aeb9.. : ∀ x0 : ι → (ι → ι) → (ι → ι) → ο . ∀ x1 . ∀ x2 x3 : ι → ι . (∀ x4 : ι → ι . (∀ x5 . prim1 x5 x1 ⟶ x2 x5 = x4 x5) ⟶ ∀ x5 : ι → ι . (∀ x6 . prim1 x6 x1 ⟶ x3 x6 = x5 x6) ⟶ x0 x1 x4 x5 = x0 x1 x2 x3) ⟶ 3c64d.. (5fdf5.. x1 x2 x3) x0 = x0 x1 x2 x3 (proof)