Proofgold Signed Transaction

Definition and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition bij := λ x0 x1 . λ x2 : ι → ι . and (and (∀ x3 . prim1 x3 x0 ⟶ prim1 (x2 x3) x1) (∀ x3 . prim1 x3 x0 ⟶ ∀ x4 . prim1 x4 x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4)) (∀ x3 . prim1 x3 x1 ⟶ ∀ x4 : ο . (∀ x5 . and (prim1 x5 x0) (x2 x5 = x3) ⟶ x4) ⟶ x4)
Definition 91630.. := λ x0 . prim2 x0 x0
Known e7a4c.. : ∀ x0 . prim1 x0 (91630.. x0)
Known fead7.. : ∀ x0 x1 . prim1 x1 (91630.. x0) ⟶ x1 = x0
Theorem b9ebd.. : ∀ x0 . 91630.. x0 = prim2 x0 x0 (proof)
Known andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Theorem 9ffeb.. : ∀ x0 x1 . 91630.. x0 = x1 ⟶ and (prim1 x0 x1) (∀ x2 . prim1 x2 x1 ⟶ x2 = x0) (proof)
Definition 7ee77.. := λ x0 x1 . prim2 (prim2 x0 x1) (91630.. x0)
Definition or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known 2532b.. : ∀ x0 x1 x2 . prim1 x0 (prim2 x1 x2) ⟶ or (x0 = x1) (x0 = x2)
Known 67787.. : ∀ x0 x1 . prim1 x0 (prim2 x0 x1)
Known 5a932.. : ∀ x0 x1 . prim1 x1 (prim2 x0 x1)
Theorem 6c815.. : ∀ x0 x1 x2 x3 . 7ee77.. x0 x1 = 7ee77.. x2 x3 ⟶ x0 = x2 (proof)
Theorem 9480e.. : ∀ x0 x1 x2 x3 . 7ee77.. x0 x1 = 7ee77.. x2 x3 ⟶ x1 = x3 (proof)
Theorem 76a5f.. : ∀ x0 x1 x2 x3 . 7ee77.. x0 x1 = 7ee77.. x2 x3 ⟶ and (x0 = x2) (x1 = x3) (proof)
Definition iff := λ x0 x1 : ο . and (x0 ⟶ x1) (x1 ⟶ x0)
Definition c2e41.. := λ x0 x1 . ∀ x2 : ο . (∀ x3 . and (and (∀ x4 . prim1 x4 x0 ⟶ ∀ x5 : ο . (∀ x6 . and (prim1 x6 x1) (prim1 (7ee77.. x4 x6) x3) ⟶ x5) ⟶ x5) (∀ x4 . prim1 x4 x1 ⟶ ∀ x5 : ο . (∀ x6 . and (prim1 x6 x0) (prim1 (7ee77.. x6 x4) x3) ⟶ x5) ⟶ x5)) (∀ x4 x5 x6 x7 . prim1 (7ee77.. x4 x5) x3 ⟶ prim1 (7ee77.. x6 x7) x3 ⟶ iff (x4 = x6) (x5 = x7)) ⟶ x2) ⟶ x2
Param 94f9e.. : ι → (ι → ι) → ι
Known and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known 696c0.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 x0 ⟶ prim1 (x1 x2) (94f9e.. x0 x1)
Known 8c6f6.. : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . prim1 x2 (94f9e.. x0 x1) ⟶ ∀ x3 : ο . (∀ x4 . prim1 x4 x0 ⟶ x2 = x1 x4 ⟶ x3) ⟶ x3
Known iffI : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ iff x0 x1
Theorem c6ad4.. : ∀ x0 x1 . ∀ x2 : ι → ι . bij x0 x1 x2 ⟶ c2e41.. x0 x1 (proof)
Known Eps_i_ax : ∀ x0 : ι → ο . ∀ x1 . x0 x1 ⟶ x0 (prim0 x0)
Theorem 64d57.. : ∀ x0 x1 . c2e41.. x0 x1 ⟶ ∀ x2 : ο . (∀ x3 : ι → ι . bij x0 x1 x3 ⟶ x2) ⟶ x2 (proof)
Known and4I : ∀ x0 x1 x2 x3 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ and (and (and x0 x1) x2) x3
Theorem and5I : ∀ x0 x1 x2 x3 x4 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ and (and (and (and x0 x1) x2) x3) x4 (proof)
Known and4E : ∀ x0 x1 x2 x3 : ο . and (and (and x0 x1) x2) x3 ⟶ ∀ x4 : ο . (x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4) ⟶ x4
Theorem and5E : ∀ x0 x1 x2 x3 x4 : ο . and (and (and (and x0 x1) x2) x3) x4 ⟶ ∀ x5 : ο . (x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ x5) ⟶ x5 (proof)
Known orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known or4I1 : ∀ x0 x1 x2 x3 : ο . x0 ⟶ or (or (or x0 x1) x2) x3
Theorem or5I1 : ∀ x0 x1 x2 x3 x4 : ο . x0 ⟶ or (or (or (or x0 x1) x2) x3) x4 (proof)
Known or4I2 : ∀ x0 x1 x2 x3 : ο . x1 ⟶ or (or (or x0 x1) x2) x3
Theorem or5I2 : ∀ x0 x1 x2 x3 x4 : ο . x1 ⟶ or (or (or (or x0 x1) x2) x3) x4 (proof)
Known or4I3 : ∀ x0 x1 x2 x3 : ο . x2 ⟶ or (or (or x0 x1) x2) x3
Theorem or5I3 : ∀ x0 x1 x2 x3 x4 : ο . x2 ⟶ or (or (or (or x0 x1) x2) x3) x4 (proof)
Known or4I4 : ∀ x0 x1 x2 x3 : ο . x3 ⟶ or (or (or x0 x1) x2) x3
Theorem or5I4 : ∀ x0 x1 x2 x3 x4 : ο . x3 ⟶ or (or (or (or x0 x1) x2) x3) x4 (proof)
Known orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Theorem or5I5 : ∀ x0 x1 x2 x3 x4 : ο . x4 ⟶ or (or (or (or x0 x1) x2) x3) x4 (proof)
Known or4E : ∀ x0 x1 x2 x3 : ο . or (or (or x0 x1) x2) x3 ⟶ ∀ x4 : ο . (x0 ⟶ x4) ⟶ (x1 ⟶ x4) ⟶ (x2 ⟶ x4) ⟶ (x3 ⟶ x4) ⟶ x4
Theorem or5E : ∀ x0 x1 x2 x3 x4 : ο . or (or (or (or x0 x1) x2) x3) x4 ⟶ ∀ x5 : ο . (x0 ⟶ x5) ⟶ (x1 ⟶ x5) ⟶ (x2 ⟶ x5) ⟶ (x3 ⟶ x5) ⟶ (x4 ⟶ x5) ⟶ x5 (proof)
Theorem and6I : ∀ x0 x1 x2 x3 x4 x5 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ x5 ⟶ and (and (and (and (and x0 x1) x2) x3) x4) x5 (proof)
Theorem and6E : ∀ x0 x1 x2 x3 x4 x5 : ο . and (and (and (and (and x0 x1) x2) x3) x4) x5 ⟶ ∀ x6 : ο . (x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ x5 ⟶ x6) ⟶ x6 (proof)
Theorem and7I : ∀ x0 x1 x2 x3 x4 x5 x6 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ x5 ⟶ x6 ⟶ and (and (and (and (and (and x0 x1) x2) x3) x4) x5) x6 (proof)
Theorem and7E : ∀ x0 x1 x2 x3 x4 x5 x6 : ο . and (and (and (and (and (and x0 x1) x2) x3) x4) x5) x6 ⟶ ∀ x7 : ο . (x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ x5 ⟶ x6 ⟶ x7) ⟶ x7 (proof)