Proofgold Signed Transaction

Param omega^omega : ι
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param mul_nat^mul_nat : ι → ι → ι
Definition divides_nat^divides_nat := λ x0 x1 . and (and (x0 ∈ omega) (x1 ∈ omega)) (∀ x2 : ο . (∀ x3 . and (x3 ∈ omega) (mul_nat x0 x3 = x1) ⟶ x2) ⟶ x2)
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Param nat_p^nat_p : ι → ο
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known mul_nat_p^mul_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (mul_nat x0 x1)
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Theorem divides_nat_mulR^{divides_nat_mulR} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ divides_nat x0 (mul_nat x0 x1) (proof)
Known mul_nat_com^mul_nat_com : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ mul_nat x0 x1 = mul_nat x1 x0
Theorem divides_nat_mulL^{divides_nat_mulL} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ divides_nat x1 (mul_nat x0 x1) (proof)
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param ordsucc^ordsucc : ι → ι
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known nat_inv_impred^{nat_inv_impred} : ∀ x0 : ι → ο . x0 0 ⟶ (∀ x1 . nat_p x1 ⟶ x0 (ordsucc x1)) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Known nat_ordsucc_in_ordsucc^{nat_ordsucc_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ∈ ordsucc x0
Known nat_ordsucc^nat_ordsucc : ∀ x0 . nat_p x0 ⟶ nat_p (ordsucc x0)
Known nat_0_in_ordsucc^{nat_0_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ 0 ∈ ordsucc x0
Theorem nat_le1_cases^{nat_le1_cases} : ∀ x0 . nat_p x0 ⟶ x0 ⊆ 1 ⟶ or (x0 = 0) (x0 = 1) (proof)
Definition composite_nat^{composite_nat} := λ x0 . and (x0 ∈ omega) (∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (∀ x3 : ο . (∀ x4 . and (x4 ∈ omega) (and (and (1 ∈ x2) (1 ∈ x4)) (mul_nat x2 x4 = x0)) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1)
Definition prime_nat^prime_nat := λ x0 . and (and (x0 ∈ omega) (1 ∈ x0)) (∀ x1 . x1 ∈ omega ⟶ divides_nat x1 x0 ⟶ or (x1 = 1) (x1 = x0))
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Param ordinal^ordinal : ι → ο
Known ordinal_In_Or_Subq^{ordinal_In_Or_Subq} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ or (x0 ∈ x1) (x1 ⊆ x0)
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known nat_1^nat_1 : nat_p 1
Known In_no2cycle^In_no2cycle : ∀ x0 x1 . x0 ∈ x1 ⟶ x1 ∈ x0 ⟶ False
Known In_0_1^In_0_1 : 0 ∈ 1
Known mul_nat_0L^mul_nat_0L : ∀ x0 . nat_p x0 ⟶ mul_nat 0 x0 = 0
Known mul_nat_0R^mul_nat_0R : ∀ x0 . mul_nat x0 0 = 0
Known mul_nat_1R^mul_nat_1R : ∀ x0 . mul_nat x0 1 = x0
Theorem prime_nat_or_composite_nat^{prime_nat_or_composite_nat} : ∀ x0 . x0 ∈ omega ⟶ 1 ∈ x0 ⟶ or (prime_nat x0) (composite_nat x0) (proof)
Known nat_complete_ind^{nat_complete_ind} : ∀ x0 : ι → ο . (∀ x1 . nat_p x1 ⟶ (∀ x2 . x2 ∈ x1 ⟶ x0 x2) ⟶ x0 x1) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1
Known 91381..^{divides_nat_ref} : ∀ x0 . nat_p x0 ⟶ divides_nat x0 x0
Known 58a75.. : ∀ x0 x1 . nat_p x0 ⟶ nat_p x1 ⟶ 0 ∈ x0 ⟶ 1 ∈ x1 ⟶ x0 ∈ mul_nat x0 x1
Known nat_trans^nat_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Known e4e38..^{divides_nat_tra} : ∀ x0 x1 x2 . divides_nat x0 x1 ⟶ divides_nat x1 x2 ⟶ divides_nat x0 x2
Theorem prime_nat_divisor_ex^{prime_nat_divisor_ex} : ∀ x0 . nat_p x0 ⟶ 1 ∈ x0 ⟶ ∀ x1 : ο . (∀ x2 . and (prime_nat x2) (divides_nat x2 x0) ⟶ x1) ⟶ x1 (proof)
Param add_nat^add_nat : ι → ι → ι
Known mul_nat_SR^mul_nat_SR : ∀ x0 x1 . nat_p x1 ⟶ mul_nat x0 (ordsucc x1) = add_nat x0 (mul_nat x0 x1)
Known df9cc..^{mul_nat_Subq_L} : ∀ x0 x1 . nat_p x0 ⟶ nat_p x1 ⟶ x0 ⊆ x1 ⟶ ∀ x2 . nat_p x2 ⟶ mul_nat x2 x0 ⊆ mul_nat x2 x1
Known ordinal_ordsucc_In_Subq^{ordinal_ordsucc_In_Subq} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ⊆ x0
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Known add_nat_SL^add_nat_SL : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ add_nat (ordsucc x0) x1 = ordsucc (add_nat x0 x1)
Known nat_0^nat_0 : nat_p 0
Known add_nat_0L^add_nat_0L : ∀ x0 . nat_p x0 ⟶ add_nat 0 x0 = x0
Known 9cc32.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ ∀ x2 . nat_p x2 ⟶ x1 ∈ x2 ⟶ add_nat x1 x0 ∈ add_nat x2 x0
Theorem nat_1In_not_divides_ordsucc^{nat_1In_not_divides_ordsucc} : ∀ x0 x1 . 1 ∈ x0 ⟶ divides_nat x0 x1 ⟶ not (divides_nat x0 (ordsucc x1)) (proof)
Definition bij^bij := λ x0 x1 . λ x2 : ι → ι . and (and (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1) (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4)) (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 : ο . (∀ x5 . and (x5 ∈ x0) (x2 x5 = x3) ⟶ x4) ⟶ x4)
Definition equip^equip := λ x0 x1 . ∀ x2 : ο . (∀ x3 : ι → ι . bij x0 x1 x3 ⟶ x2) ⟶ x2
Definition finite^finite := λ x0 . ∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (equip x0 x2) ⟶ x1) ⟶ x1
Definition infinite^infinite := λ x0 . not (finite x0)
Param Sep^Sep : ι → (ι → ο) → ι
Definition primes^primes := Sep omega prime_nat
Param nat_primrec^nat_primrec : ι → (ι → ι → ι) → ι → ι
Definition Pi_nat^Pi_nat := λ x0 : ι → ι . nat_primrec 1 (λ x1 x2 . mul_nat x2 (x0 x1))
Known nat_primrec_0^{nat_primrec_0} : ∀ x0 . ∀ x1 : ι → ι → ι . nat_primrec x0 x1 0 = x0
Theorem Pi_nat_0^Pi_nat_0 : ∀ x0 : ι → ι . Pi_nat x0 0 = 1 (proof)
Known nat_primrec_S^{nat_primrec_S} : ∀ x0 . ∀ x1 : ι → ι → ι . ∀ x2 . nat_p x2 ⟶ nat_primrec x0 x1 (ordsucc x2) = x1 x2 (nat_primrec x0 x1 x2)
Theorem Pi_nat_S^Pi_nat_S : ∀ x0 : ι → ι . ∀ x1 . nat_p x1 ⟶ Pi_nat x0 (ordsucc x1) = mul_nat (Pi_nat x0 x1) (x0 x1) (proof)
Known nat_ind^nat_ind : ∀ x0 : ι → ο . x0 0 ⟶ (∀ x1 . nat_p x1 ⟶ x0 x1 ⟶ x0 (ordsucc x1)) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1
Known ordsuccI1^ordsuccI1 : ∀ x0 . x0 ⊆ ordsucc x0
Theorem Pi_nat_p^Pi_nat_p : ∀ x0 : ι → ι . ∀ x1 . nat_p x1 ⟶ (∀ x2 . x2 ∈ x1 ⟶ nat_p (x0 x2)) ⟶ nat_p (Pi_nat x0 x1) (proof)
Known neq_1_0^neq_1_0 : 1 = 0 ⟶ ∀ x0 : ο . x0
Known mul_nat_0_inv^{mul_nat_0_inv} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_nat x0 x1 = 0 ⟶ or (x0 = 0) (x1 = 0)
Theorem Pi_nat_0_inv^Pi_nat_0_inv : ∀ x0 : ι → ι . ∀ x1 . nat_p x1 ⟶ (∀ x2 . x2 ∈ x1 ⟶ nat_p (x0 x2)) ⟶ Pi_nat x0 x1 = 0 ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ x1) (x0 x3 = 0) ⟶ x2) ⟶ x2 (proof)
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Theorem Pi_nat_divides^{Pi_nat_divides} : ∀ x0 : ι → ι . ∀ x1 . nat_p x1 ⟶ (∀ x2 . x2 ∈ x1 ⟶ nat_p (x0 x2)) ⟶ ∀ x2 . x2 ∈ x1 ⟶ divides_nat (x0 x2) (Pi_nat x0 x1) (proof)
Known equip_sym^equip_sym : ∀ x0 x1 . equip x0 x1 ⟶ equip x1 x0
Known ordinal_Empty^{ordinal_Empty} : ordinal 0
Known Empty_Subq_eq^{Empty_Subq_eq} : ∀ x0 . x0 ⊆ 0 ⟶ x0 = 0
Known SepE2^SepE2 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x1 x2
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Known SepE1^SepE1 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x2 ∈ x0
Theorem form100_11_infinite_primes^{form100_11_infinite_primes} : infinite primes (proof)