Proofgold Signed Transaction

Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition explicit_Ring^{explicit_Ring} := λ x0 x1 . λ x2 x3 : ι → ι → ι . and (and (and (and (and (and (and (and (and (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 x5 ∈ x0) (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x2 x4 (x2 x5 x6) = x2 (x2 x4 x5) x6)) (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 x5 = x2 x5 x4)) (x1 ∈ x0)) (∀ x4 . x4 ∈ x0 ⟶ x2 x1 x4 = x4)) (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 : ο . (∀ x6 . and (x6 ∈ x0) (x2 x4 x6 = x1) ⟶ x5) ⟶ x5)) (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x3 x4 x5 ∈ x0)) (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x4 (x3 x5 x6) = x3 (x3 x4 x5) x6)) (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x4 (x2 x5 x6) = x2 (x3 x4 x5) (x3 x4 x6))) (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 (x2 x4 x5) x6 = x2 (x3 x4 x6) (x3 x5 x6))
Known and4I^and4I : ∀ x0 x1 x2 x3 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ and (and (and x0 x1) x2) x3
Known and7I^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
Theorem explicit_Ring_I^{explicit_Ring_I} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 x5 ∈ x0) ⟶ (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x2 x4 (x2 x5 x6) = x2 (x2 x4 x5) x6) ⟶ (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 x5 = x2 x5 x4) ⟶ x1 ∈ x0 ⟶ (∀ x4 . x4 ∈ x0 ⟶ x2 x1 x4 = x4) ⟶ (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 : ο . (∀ x6 . and (x6 ∈ x0) (x2 x4 x6 = x1) ⟶ x5) ⟶ x5) ⟶ (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x3 x4 x5 ∈ x0) ⟶ (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x4 (x3 x5 x6) = x3 (x3 x4 x5) x6) ⟶ (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x4 (x2 x5 x6) = x2 (x3 x4 x5) (x3 x4 x6)) ⟶ (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 (x2 x4 x5) x6 = x2 (x3 x4 x6) (x3 x5 x6)) ⟶ explicit_Ring x0 x1 x2 x3 (proof)
Known and4E^and4E : ∀ x0 x1 x2 x3 : ο . and (and (and x0 x1) x2) x3 ⟶ ∀ x4 : ο . (x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4) ⟶ x4
Known and7E^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
Theorem explicit_Ring_E^{explicit_Ring_E} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 : ο . (explicit_Ring x0 x1 x2 x3 ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x2 x5 x6 ∈ x0) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x2 x5 (x2 x6 x7) = x2 (x2 x5 x6) x7) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x2 x5 x6 = x2 x6 x5) ⟶ x1 ∈ x0 ⟶ (∀ x5 . x5 ∈ x0 ⟶ x2 x1 x5 = x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 : ο . (∀ x7 . and (x7 ∈ x0) (x2 x5 x7 = x1) ⟶ x6) ⟶ x6) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 ∈ x0) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 (x3 x6 x7) = x3 (x3 x5 x6) x7) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 (x2 x6 x7) = x2 (x3 x5 x6) (x3 x5 x7)) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 (x2 x5 x6) x7 = x2 (x3 x5 x7) (x3 x6 x7)) ⟶ x4) ⟶ explicit_Ring x0 x1 x2 x3 ⟶ x4 (proof)
Definition explicit_Ring_minus^{explicit_Ring_minus} := λ x0 x1 . λ x2 x3 : ι → ι → ι . λ x4 . prim0 (λ x5 . and (x5 ∈ x0) (x2 x4 x5 = x1))
Known Eps_i_ex^Eps_i_ex : ∀ x0 : ι → ο . (∀ x1 : ο . (∀ x2 . x0 x2 ⟶ x1) ⟶ x1) ⟶ x0 (prim0 x0)
Theorem explicit_Ring_minus_prop^{explicit_Ring_minus_prop} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . explicit_Ring x0 x1 x2 x3 ⟶ ∀ x4 . x4 ∈ x0 ⟶ and (explicit_Ring_minus x0 x1 x2 x3 x4 ∈ x0) (x2 x4 (explicit_Ring_minus x0 x1 x2 x3 x4) = x1) (proof)
Theorem explicit_Ring_minus_clos^{explicit_Ring_minus_clos} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . explicit_Ring x0 x1 x2 x3 ⟶ ∀ x4 . x4 ∈ x0 ⟶ explicit_Ring_minus x0 x1 x2 x3 x4 ∈ x0 (proof)
Theorem explicit_Ring_minus_R^{explicit_Ring_minus_R} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . explicit_Ring x0 x1 x2 x3 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x4 (explicit_Ring_minus x0 x1 x2 x3 x4) = x1 (proof)
Theorem explicit_Ring_minus_L^{explicit_Ring_minus_L} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . explicit_Ring x0 x1 x2 x3 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 (explicit_Ring_minus x0 x1 x2 x3 x4) x4 = x1 (proof)
Theorem explicit_Ring_plus_cancelL^{explicit_Ring_plus_cancelL} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . explicit_Ring x0 x1 x2 x3 ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x2 x4 x5 = x2 x4 x6 ⟶ x5 = x6 (proof)
Theorem explicit_Ring_plus_cancelR^{explicit_Ring_plus_cancelR} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . explicit_Ring x0 x1 x2 x3 ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x2 x4 x6 = x2 x5 x6 ⟶ x4 = x5 (proof)
Theorem explicit_Ring_minus_invol^{explicit_Ring_minus_invol} : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . explicit_Ring x0 x1 x2 x3 ⟶ ∀ x4 . x4 ∈ x0 ⟶ explicit_Ring_minus x0 x1 x2 x3 (explicit_Ring_minus x0 x1 x2 x3 x4) = x4 (proof)
Definition explicit_Ring_with_id^{explicit_Ring_with_id} := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . and (and (and (and (and (and (and (and (and (and (and (and (and (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 ∈ x0) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 (x3 x6 x7) = x3 (x3 x5 x6) x7)) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 = x3 x6 x5)) (x1 ∈ x0)) (∀ x5 . x5 ∈ x0 ⟶ x3 x1 x5 = x5)) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 : ο . (∀ x7 . and (x7 ∈ x0) (x3 x5 x7 = x1) ⟶ x6) ⟶ x6)) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 x6 ∈ x0)) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x5 (x4 x6 x7) = x4 (x4 x5 x6) x7)) (x2 ∈ x0)) (x2 = x1 ⟶ ∀ x5 : ο . x5)) (∀ x5 . x5 ∈ x0 ⟶ x4 x2 x5 = x5)) (∀ x5 . x5 ∈ x0 ⟶ x4 x5 x2 = x5)) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x5 (x3 x6 x7) = x3 (x4 x5 x6) (x4 x5 x7))) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 (x3 x5 x6) x7 = x3 (x4 x5 x7) (x4 x6 x7))
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Theorem explicit_Ring_with_id_I^{explicit_Ring_with_id_I} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 ∈ x0) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 (x3 x6 x7) = x3 (x3 x5 x6) x7) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 = x3 x6 x5) ⟶ x1 ∈ x0 ⟶ (∀ x5 . x5 ∈ x0 ⟶ x3 x1 x5 = x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 : ο . (∀ x7 . and (x7 ∈ x0) (x3 x5 x7 = x1) ⟶ x6) ⟶ x6) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 x6 ∈ x0) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x5 (x4 x6 x7) = x4 (x4 x5 x6) x7) ⟶ x2 ∈ x0 ⟶ (x2 = x1 ⟶ ∀ x5 : ο . x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ x4 x2 x5 = x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ x4 x5 x2 = x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x5 (x3 x6 x7) = x3 (x4 x5 x6) (x4 x5 x7)) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 (x3 x5 x6) x7 = x3 (x4 x5 x7) (x4 x6 x7)) ⟶ explicit_Ring_with_id x0 x1 x2 x3 x4 (proof)
Theorem explicit_Ring_with_id_E^{explicit_Ring_with_id_E} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ο . (explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x6 x7 ∈ x0) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x3 x6 (x3 x7 x8) = x3 (x3 x6 x7) x8) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x6 x7 = x3 x7 x6) ⟶ x1 ∈ x0 ⟶ (∀ x6 . x6 ∈ x0 ⟶ x3 x1 x6 = x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 : ο . (∀ x8 . and (x8 ∈ x0) (x3 x6 x8 = x1) ⟶ x7) ⟶ x7) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x6 x7 ∈ x0) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x6 (x4 x7 x8) = x4 (x4 x6 x7) x8) ⟶ x2 ∈ x0 ⟶ (x2 = x1 ⟶ ∀ x6 : ο . x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ x4 x2 x6 = x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ x4 x6 x2 = x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x6 (x3 x7 x8) = x3 (x4 x6 x7) (x4 x6 x8)) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 (x3 x6 x7) x8 = x3 (x4 x6 x8) (x4 x7 x8)) ⟶ x5) ⟶ explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ x5 (proof)
Theorem explicit_Ring_with_id_Ring^{explicit_Ring_with_id_Ring} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ explicit_Ring x0 x1 x3 x4 (proof)
Theorem explicit_Ring_with_id_minus_clos^{explicit_Ring_with_id_minus_clos} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ explicit_Ring_minus x0 x1 x3 x4 x5 ∈ x0 (proof)
Theorem explicit_Ring_with_id_minus_R^{explicit_Ring_with_id_minus_R} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x3 x5 (explicit_Ring_minus x0 x1 x3 x4 x5) = x1 (proof)
Theorem explicit_Ring_with_id_minus_L^{explicit_Ring_with_id_minus_L} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x3 (explicit_Ring_minus x0 x1 x3 x4 x5) x5 = x1 (proof)
Theorem explicit_Ring_with_id_plus_cancelL^{explicit_Ring_with_id_plus_cancelL} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 x6 = x3 x5 x7 ⟶ x6 = x7 (proof)
Theorem explicit_Ring_with_id_plus_cancelR^{explicit_Ring_with_id_plus_cancelR} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 x7 = x3 x6 x7 ⟶ x5 = x6 (proof)
Theorem explicit_Ring_with_id_minus_invol^{explicit_Ring_with_id_minus_invol} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ explicit_Ring_minus x0 x1 x3 x4 (explicit_Ring_minus x0 x1 x3 x4 x5) = x5 (proof)
Theorem explicit_Ring_with_id_minus_one_In^{explicit_Ring_with_id_minus_one_In} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ explicit_Ring_minus x0 x1 x3 x4 x2 ∈ x0 (proof)
Theorem explicit_Ring_with_id_zero_multR^{explicit_Ring_with_id_zero_multR} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x4 x5 x1 = x1 (proof)
Theorem explicit_Ring_with_id_zero_multL^{explicit_Ring_with_id_zero_multL} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x4 x1 x5 = x1 (proof)
Theorem explicit_Ring_with_id_minus_mult^{explicit_Ring_with_id_minus_mult} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ explicit_Ring_minus x0 x1 x3 x4 x5 = x4 (explicit_Ring_minus x0 x1 x3 x4 x2) x5 (proof)
Theorem explicit_Ring_with_id_mult_minus^{explicit_Ring_with_id_mult_minus} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ explicit_Ring_minus x0 x1 x3 x4 x5 = x4 x5 (explicit_Ring_minus x0 x1 x3 x4 x2) (proof)
Theorem explicit_Ring_with_id_minus_one_square^{explicit_Ring_with_id_minus_one_square} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ x4 (explicit_Ring_minus x0 x1 x3 x4 x2) (explicit_Ring_minus x0 x1 x3 x4 x2) = x2 (proof)
Theorem explicit_Ring_with_id_minus_square^{explicit_Ring_with_id_minus_square} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_Ring_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x4 (explicit_Ring_minus x0 x1 x3 x4 x5) (explicit_Ring_minus x0 x1 x3 x4 x5) = x4 x5 x5 (proof)
Param nat_primrec^nat_primrec : ι → (ι → ι → ι) → ι → ι
Definition explicit_Ring_with_id_exp_nat^{explicit_Ring_with_id_exp_nat} := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . λ x5 . nat_primrec x2 (λ x6 . x4 x5)
Definition explicit_CRing_with_id^{explicit_CRing_with_id} := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . and (and (and (and (and (and (and (and (and (and (and (and (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 ∈ x0) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 (x3 x6 x7) = x3 (x3 x5 x6) x7)) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 = x3 x6 x5)) (x1 ∈ x0)) (∀ x5 . x5 ∈ x0 ⟶ x3 x1 x5 = x5)) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 : ο . (∀ x7 . and (x7 ∈ x0) (x3 x5 x7 = x1) ⟶ x6) ⟶ x6)) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 x6 ∈ x0)) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x5 (x4 x6 x7) = x4 (x4 x5 x6) x7)) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 x6 = x4 x6 x5)) (x2 ∈ x0)) (x2 = x1 ⟶ ∀ x5 : ο . x5)) (∀ x5 . x5 ∈ x0 ⟶ x4 x2 x5 = x5)) (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x5 (x3 x6 x7) = x3 (x4 x5 x6) (x4 x5 x7))
Theorem explicit_CRing_with_id_I^{explicit_CRing_with_id_I} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 ∈ x0) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 (x3 x6 x7) = x3 (x3 x5 x6) x7) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x3 x5 x6 = x3 x6 x5) ⟶ x1 ∈ x0 ⟶ (∀ x5 . x5 ∈ x0 ⟶ x3 x1 x5 = x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 : ο . (∀ x7 . and (x7 ∈ x0) (x3 x5 x7 = x1) ⟶ x6) ⟶ x6) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 x6 ∈ x0) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x5 (x4 x6 x7) = x4 (x4 x5 x6) x7) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x4 x5 x6 = x4 x6 x5) ⟶ x2 ∈ x0 ⟶ (x2 = x1 ⟶ ∀ x5 : ο . x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ x4 x2 x5 = x5) ⟶ (∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x5 (x3 x6 x7) = x3 (x4 x5 x6) (x4 x5 x7)) ⟶ explicit_CRing_with_id x0 x1 x2 x3 x4 (proof)
Theorem explicit_CRing_with_id_E^{explicit_CRing_with_id_E} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ο . (explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x6 x7 ∈ x0) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x3 x6 (x3 x7 x8) = x3 (x3 x6 x7) x8) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x6 x7 = x3 x7 x6) ⟶ x1 ∈ x0 ⟶ (∀ x6 . x6 ∈ x0 ⟶ x3 x1 x6 = x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 : ο . (∀ x8 . and (x8 ∈ x0) (x3 x6 x8 = x1) ⟶ x7) ⟶ x7) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x6 x7 ∈ x0) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x6 (x4 x7 x8) = x4 (x4 x6 x7) x8) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x6 x7 = x4 x7 x6) ⟶ x2 ∈ x0 ⟶ (x2 = x1 ⟶ ∀ x6 : ο . x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ x4 x2 x6 = x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x6 (x3 x7 x8) = x3 (x4 x6 x7) (x4 x6 x8)) ⟶ x5) ⟶ explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ x5 (proof)
Theorem explicit_CRing_with_id_Ring_with_id^{explicit_CRing_with_id_Ring_with_id} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ explicit_Ring_with_id x0 x1 x2 x3 x4 (proof)
Theorem explicit_CRing_with_id_Ring^{explicit_CRing_with_id_Ring} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ explicit_Ring x0 x1 x3 x4 (proof)
Theorem explicit_CRing_with_id_minus_clos^{explicit_CRing_with_id_minus_clos} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ explicit_Ring_minus x0 x1 x3 x4 x5 ∈ x0 (proof)
Theorem explicit_CRing_with_id_minus_R^{explicit_CRing_with_id_minus_R} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x3 x5 (explicit_Ring_minus x0 x1 x3 x4 x5) = x1 (proof)
Theorem explicit_CRing_with_id_minus_L^{explicit_CRing_with_id_minus_L} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x3 (explicit_Ring_minus x0 x1 x3 x4 x5) x5 = x1 (proof)
Theorem explicit_CRing_with_id_plus_cancelL^{explicit_CRing_with_id_plus_cancelL} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 x6 = x3 x5 x7 ⟶ x6 = x7 (proof)
Theorem explicit_CRing_with_id_plus_cancelR^{explicit_CRing_with_id_plus_cancelR} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x5 x7 = x3 x6 x7 ⟶ x5 = x6 (proof)
Theorem explicit_CRing_with_id_minus_invol^{explicit_CRing_with_id_minus_invol} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ explicit_Ring_minus x0 x1 x3 x4 (explicit_Ring_minus x0 x1 x3 x4 x5) = x5 (proof)
Theorem explicit_CRing_with_id_minus_one_In^{explicit_CRing_with_id_minus_one_In} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ explicit_Ring_minus x0 x1 x3 x4 x2 ∈ x0 (proof)
Theorem explicit_CRing_with_id_zero_multR^{explicit_CRing_with_id_zero_multR} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x4 x5 x1 = x1 (proof)
Theorem explicit_CRing_with_id_zero_multL^{explicit_CRing_with_id_zero_multL} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x4 x1 x5 = x1 (proof)
Theorem explicit_CRing_with_id_minus_mult^{explicit_CRing_with_id_minus_mult} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ explicit_Ring_minus x0 x1 x3 x4 x5 = x4 (explicit_Ring_minus x0 x1 x3 x4 x2) x5 (proof)
Theorem explicit_CRing_with_id_mult_minus^{explicit_CRing_with_id_mult_minus} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ explicit_Ring_minus x0 x1 x3 x4 x5 = x4 x5 (explicit_Ring_minus x0 x1 x3 x4 x2) (proof)
Theorem explicit_CRing_with_id_minus_one_square^{explicit_CRing_with_id_minus_one_square} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ x4 (explicit_Ring_minus x0 x1 x3 x4 x2) (explicit_Ring_minus x0 x1 x3 x4 x2) = x2 (proof)
Theorem explicit_CRing_with_id_minus_square^{explicit_CRing_with_id_minus_square} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x4 (explicit_Ring_minus x0 x1 x3 x4 x5) (explicit_Ring_minus x0 x1 x3 x4 x5) = x4 x5 x5 (proof)
Param lam^Sigma : ι → (ι → ι) → ι
Param ordsucc^ordsucc : ι → ι
Param If_i^If_i : ο → ι → ι → ι
Param encode_b^encode_b : ι → CT2 ι
Definition pack_b_b_e^pack_b_b_e := λ x0 . λ x1 x2 : ι → ι → ι . λ x3 . lam 4 (λ x4 . If_i (x4 = 0) x0 (If_i (x4 = 1) (encode_b x0 x1) (If_i (x4 = 2) (encode_b x0 x2) x3)))
Param ap^ap : ι → ι → ι
Known pack_b_b_e_0_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 . x0 = pack_b_b_e x1 x2 x3 x4 ⟶ x1 = ap x0 0
Known pack_b_b_e_0_eq2^{pack_b_b_e_0_eq2} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 . x0 = ap (pack_b_b_e x0 x1 x2 x3) 0
Param decode_b^decode_b : ι → ι → ι → ι
Known pack_b_b_e_1_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 . x0 = pack_b_b_e x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x1 ⟶ ∀ x6 . x6 ∈ x1 ⟶ x2 x5 x6 = decode_b (ap x0 1) x5 x6
Known pack_b_b_e_1_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x1 x4 x5 = decode_b (ap (pack_b_b_e x0 x1 x2 x3) 1) x4 x5
Known pack_b_b_e_2_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 . x0 = pack_b_b_e x1 x2 x3 x4 ⟶ ∀ x5 . x5 ∈ x1 ⟶ ∀ x6 . x6 ∈ x1 ⟶ x3 x5 x6 = decode_b (ap x0 2) x5 x6
Known pack_b_b_e_2_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 x5 = decode_b (ap (pack_b_b_e x0 x1 x2 x3) 2) x4 x5
Known pack_b_b_e_3_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 . x0 = pack_b_b_e x1 x2 x3 x4 ⟶ x4 = ap x0 3
Known pack_b_b_e_3_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 . x3 = ap (pack_b_b_e x0 x1 x2 x3) 3
Known pack_b_b_e_inj : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι → ι . ∀ x6 x7 . pack_b_b_e x0 x2 x4 x6 = pack_b_b_e x1 x3 x5 x7 ⟶ and (and (and (x0 = x1) (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x2 x8 x9 = x3 x8 x9)) (∀ x8 . x8 ∈ x0 ⟶ ∀ x9 . x9 ∈ x0 ⟶ x4 x8 x9 = x5 x8 x9)) (x6 = x7)
Known pack_b_b_e_ext : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι → ι . ∀ x5 . (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x1 x6 x7 = x2 x6 x7) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x6 x7 = x4 x6 x7) ⟶ pack_b_b_e x0 x1 x3 x5 = pack_b_b_e x0 x2 x4 x5
Definition struct_b_b_e^struct_b_b_e := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ι . (∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x2 ⟶ x3 x4 x5 ∈ x2) ⟶ ∀ x4 : ι → ι → ι . (∀ x5 . x5 ∈ x2 ⟶ ∀ x6 . x6 ∈ x2 ⟶ x4 x5 x6 ∈ x2) ⟶ ∀ x5 . x5 ∈ x2 ⟶ x1 (pack_b_b_e x2 x3 x4 x5)) ⟶ x1 x0
Known pack_struct_b_b_e_I : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ∈ x0) ⟶ ∀ x2 : ι → ι → ι . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 x4 ∈ x0) ⟶ ∀ x3 . x3 ∈ x0 ⟶ struct_b_b_e (pack_b_b_e x0 x1 x2 x3)
Known pack_struct_b_b_e_E1 : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 . struct_b_b_e (pack_b_b_e x0 x1 x2 x3) ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x1 x4 x5 ∈ x0
Known pack_struct_b_b_e_E2 : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 . struct_b_b_e (pack_b_b_e x0 x1 x2 x3) ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 x5 ∈ x0
Known pack_struct_b_b_e_E3 : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 . struct_b_b_e (pack_b_b_e x0 x1 x2 x3) ⟶ x3 ∈ x0
Known struct_b_b_e_eta : ∀ x0 . struct_b_b_e x0 ⟶ x0 = pack_b_b_e (ap x0 0) (decode_b (ap x0 1)) (decode_b (ap x0 2)) (ap x0 3)
Definition unpack_b_b_e_i^{unpack_b_b_e_i} := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι . x1 (ap x0 0) (decode_b (ap x0 1)) (decode_b (ap x0 2)) (ap x0 3)
Known unpack_b_b_e_i_eq : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 . (∀ x5 : ι → ι → ι . (∀ x6 . x6 ∈ x1 ⟶ ∀ x7 . x7 ∈ x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ x6 : ι → ι → ι . (∀ x7 . x7 ∈ x1 ⟶ ∀ x8 . x8 ∈ x1 ⟶ x3 x7 x8 = x6 x7 x8) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4) ⟶ unpack_b_b_e_i (pack_b_b_e x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4
Definition unpack_b_b_e_o^{unpack_b_b_e_o} := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ο . x1 (ap x0 0) (decode_b (ap x0 1)) (decode_b (ap x0 2)) (ap x0 3)
Known unpack_b_b_e_o_eq^{unpack_b_b_e_o_eq} : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ο . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 . (∀ x5 : ι → ι → ι . (∀ x6 . x6 ∈ x1 ⟶ ∀ x7 . x7 ∈ x1 ⟶ x2 x6 x7 = x5 x6 x7) ⟶ ∀ x6 : ι → ι → ι . (∀ x7 . x7 ∈ x1 ⟶ ∀ x8 . x8 ∈ x1 ⟶ x3 x7 x8 = x6 x7 x8) ⟶ x0 x1 x5 x6 x4 = x0 x1 x2 x3 x4) ⟶ unpack_b_b_e_o (pack_b_b_e x1 x2 x3 x4) x0 = x0 x1 x2 x3 x4
Definition Ring^Ring := λ x0 . and (struct_b_b_e x0) (unpack_b_b_e_o x0 (λ x1 . λ x2 x3 : ι → ι → ι . λ x4 . explicit_Ring x1 x4 x2 x3))
Definition pack_b_b_e_e^pack_b_b_e_e := λ x0 . λ x1 x2 : ι → ι → ι . λ x3 x4 . lam 5 (λ x5 . If_i (x5 = 0) x0 (If_i (x5 = 1) (encode_b x0 x1) (If_i (x5 = 2) (encode_b x0 x2) (If_i (x5 = 3) x3 x4))))
Known pack_b_b_e_e_0_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . x0 = pack_b_b_e_e x1 x2 x3 x4 x5 ⟶ x1 = ap x0 0
Known pack_b_b_e_e_0_eq2^{pack_b_b_e_e_0_eq2} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . x0 = ap (pack_b_b_e_e x0 x1 x2 x3 x4) 0
Known pack_b_b_e_e_1_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . x0 = pack_b_b_e_e x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x1 ⟶ ∀ x7 . x7 ∈ x1 ⟶ x2 x6 x7 = decode_b (ap x0 1) x6 x7
Known pack_b_b_e_e_1_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x1 x5 x6 = decode_b (ap (pack_b_b_e_e x0 x1 x2 x3 x4) 1) x5 x6
Known pack_b_b_e_e_2_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . x0 = pack_b_b_e_e x1 x2 x3 x4 x5 ⟶ ∀ x6 . x6 ∈ x1 ⟶ ∀ x7 . x7 ∈ x1 ⟶ x3 x6 x7 = decode_b (ap x0 2) x6 x7
Known pack_b_b_e_e_2_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x2 x5 x6 = decode_b (ap (pack_b_b_e_e x0 x1 x2 x3 x4) 2) x5 x6
Known pack_b_b_e_e_3_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . x0 = pack_b_b_e_e x1 x2 x3 x4 x5 ⟶ x4 = ap x0 3
Known pack_b_b_e_e_3_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . x3 = ap (pack_b_b_e_e x0 x1 x2 x3 x4) 3
Known pack_b_b_e_e_4_eq : ∀ x0 x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . x0 = pack_b_b_e_e x1 x2 x3 x4 x5 ⟶ x5 = ap x0 4
Known pack_b_b_e_e_4_eq2 : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . x4 = ap (pack_b_b_e_e x0 x1 x2 x3 x4) 4
Known pack_b_b_e_e_inj : ∀ x0 x1 . ∀ x2 x3 x4 x5 : ι → ι → ι . ∀ x6 x7 x8 x9 . pack_b_b_e_e x0 x2 x4 x6 x8 = pack_b_b_e_e x1 x3 x5 x7 x9 ⟶ and (and (and (and (x0 = x1) (∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ x2 x10 x11 = x3 x10 x11)) (∀ x10 . x10 ∈ x0 ⟶ ∀ x11 . x11 ∈ x0 ⟶ x4 x10 x11 = x5 x10 x11)) (x6 = x7)) (x8 = x9)
Known pack_b_b_e_e_ext : ∀ x0 . ∀ x1 x2 x3 x4 : ι → ι → ι . ∀ x5 x6 . (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x1 x7 x8 = x2 x7 x8) ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x3 x7 x8 = x4 x7 x8) ⟶ pack_b_b_e_e x0 x1 x3 x5 x6 = pack_b_b_e_e x0 x2 x4 x5 x6
Definition struct_b_b_e_e^{struct_b_b_e_e} := λ x0 . ∀ x1 : ι → ο . (∀ x2 . ∀ x3 : ι → ι → ι . (∀ x4 . x4 ∈ x2 ⟶ ∀ x5 . x5 ∈ x2 ⟶ x3 x4 x5 ∈ x2) ⟶ ∀ x4 : ι → ι → ι . (∀ x5 . x5 ∈ x2 ⟶ ∀ x6 . x6 ∈ x2 ⟶ x4 x5 x6 ∈ x2) ⟶ ∀ x5 . x5 ∈ x2 ⟶ ∀ x6 . x6 ∈ x2 ⟶ x1 (pack_b_b_e_e x2 x3 x4 x5 x6)) ⟶ x1 x0
Known pack_struct_b_b_e_e_I : ∀ x0 . ∀ x1 : ι → ι → ι . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ x1 x2 x3 ∈ x0) ⟶ ∀ x2 : ι → ι → ι . (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 x4 ∈ x0) ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ struct_b_b_e_e (pack_b_b_e_e x0 x1 x2 x3 x4)
Known pack_struct_b_b_e_e_E1 : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . struct_b_b_e_e (pack_b_b_e_e x0 x1 x2 x3 x4) ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x1 x5 x6 ∈ x0
Known pack_struct_b_b_e_e_E2 : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . struct_b_b_e_e (pack_b_b_e_e x0 x1 x2 x3 x4) ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x2 x5 x6 ∈ x0
Known pack_struct_b_b_e_e_E3 : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . struct_b_b_e_e (pack_b_b_e_e x0 x1 x2 x3 x4) ⟶ x3 ∈ x0
Known pack_struct_b_b_e_e_E4 : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . struct_b_b_e_e (pack_b_b_e_e x0 x1 x2 x3 x4) ⟶ x4 ∈ x0
Known struct_b_b_e_e_eta : ∀ x0 . struct_b_b_e_e x0 ⟶ x0 = pack_b_b_e_e (ap x0 0) (decode_b (ap x0 1)) (decode_b (ap x0 2)) (ap x0 3) (ap x0 4)
Definition unpack_b_b_e_e_i^{unpack_b_b_e_e_i} := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι → ι . x1 (ap x0 0) (decode_b (ap x0 1)) (decode_b (ap x0 2)) (ap x0 3) (ap x0 4)
Known unpack_b_b_e_e_i_eq : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι → ι . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . (∀ x6 : ι → ι → ι . (∀ x7 . x7 ∈ x1 ⟶ ∀ x8 . x8 ∈ x1 ⟶ x2 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . x8 ∈ x1 ⟶ ∀ x9 . x9 ∈ x1 ⟶ x3 x8 x9 = x7 x8 x9) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5) ⟶ unpack_b_b_e_e_i (pack_b_b_e_e x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5
Definition unpack_b_b_e_e_o^{unpack_b_b_e_e_o} := λ x0 . λ x1 : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι → ο . x1 (ap x0 0) (decode_b (ap x0 1)) (decode_b (ap x0 2)) (ap x0 3) (ap x0 4)
Known unpack_b_b_e_e_o_eq^{unpack_b_b_e_e_o_eq} : ∀ x0 : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι → ο . ∀ x1 . ∀ x2 x3 : ι → ι → ι . ∀ x4 x5 . (∀ x6 : ι → ι → ι . (∀ x7 . x7 ∈ x1 ⟶ ∀ x8 . x8 ∈ x1 ⟶ x2 x7 x8 = x6 x7 x8) ⟶ ∀ x7 : ι → ι → ι . (∀ x8 . x8 ∈ x1 ⟶ ∀ x9 . x9 ∈ x1 ⟶ x3 x8 x9 = x7 x8 x9) ⟶ x0 x1 x6 x7 x4 x5 = x0 x1 x2 x3 x4 x5) ⟶ unpack_b_b_e_e_o (pack_b_b_e_e x1 x2 x3 x4 x5) x0 = x0 x1 x2 x3 x4 x5
Definition Ring_with_id^Ring_with_id := λ x0 . and (struct_b_b_e_e x0) (unpack_b_b_e_e_o x0 (λ x1 . λ x2 x3 : ι → ι → ι . λ x4 x5 . explicit_Ring_with_id x1 x4 x5 x2 x3))
Definition CRing_with_id^{CRing_with_id} := λ x0 . and (struct_b_b_e_e x0) (unpack_b_b_e_e_o x0 (λ x1 . λ x2 x3 : ι → ι → ι . λ x4 x5 . explicit_CRing_with_id x1 x4 x5 x2 x3))
Definition Ring_minus^Ring_minus := λ x0 x1 . unpack_b_b_e_i x0 (λ x2 . λ x3 x4 : ι → ι → ι . λ x5 . explicit_Ring_minus x2 x5 x3 x4 x1)
Definition Ring_of_Ring_with_id^{Ring_of_Ring_with_id} := λ x0 . unpack_b_b_e_e_i x0 (λ x1 . λ x2 x3 : ι → ι → ι . λ x4 x5 . pack_b_b_e x1 x2 x3 x4)