Proofgold Signed Transaction

Param nat_p^nat_p : ι → ο
Param nat_primrec^nat_primrec : ι → (ι → ι → ι) → ι → ι
Param ordsucc^ordsucc : ι → ι
Param mul_nat^mul_nat : ι → ι → ι
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 nat_primrec_0^{nat_primrec_0} : ∀ x0 . ∀ x1 : ι → ι → ι . nat_primrec x0 x1 0 = x0
Known nat_1^nat_1 : nat_p 1
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)
Known mul_nat_p^mul_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (mul_nat x0 x1)
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known ordsuccI1^ordsuccI1 : ∀ x0 . x0 ⊆ ordsucc x0
Theorem 6523a.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 : ι → ι . (∀ x2 . x2 ∈ x0 ⟶ nat_p (x1 x2)) ⟶ nat_p (nat_primrec 1 (λ x2 . mul_nat (x1 x2)) x0) (proof)
Param add_nat^add_nat : ι → ι → ι
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known nat_inv^nat_inv : ∀ x0 . nat_p x0 ⟶ or (x0 = 0) (∀ x1 : ο . (∀ x2 . and (nat_p x2) (x0 = ordsucc x2) ⟶ x1) ⟶ x1)
Known add_nat_0L^add_nat_0L : ∀ x0 . nat_p x0 ⟶ add_nat 0 x0 = x0
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
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)
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known neq_ordsucc_0^{neq_ordsucc_0} : ∀ x0 . ordsucc x0 = 0 ⟶ ∀ x1 : ο . x1
Theorem 93901.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ add_nat x0 x1 = 0 ⟶ and (x0 = 0) (x1 = 0) (proof)
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known mul_nat_SL^mul_nat_SL : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ mul_nat (ordsucc x0) x1 = add_nat (mul_nat x0 x1) x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Theorem f3fbb..^{mul_nat_0_inv} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ mul_nat x0 x1 = 0 ⟶ or (x0 = 0) (x1 = 0) (proof)
Param setminus^setminus : ι → ι → ι
Param omega^omega : ι
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known setminusE^setminusE : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ and (x2 ∈ x0) (nIn x2 x1)
Known setminusI^setminusI : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ nIn x2 x1 ⟶ x2 ∈ setminus x0 x1
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Known In_0_1^In_0_1 : 0 ∈ 1
Known cases_1^cases_1 : ∀ x0 . x0 ∈ 1 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 x0
Theorem a0ed2.. : ∀ x0 . x0 ∈ setminus omega 1 ⟶ ∀ x1 . x1 ∈ setminus omega 1 ⟶ mul_nat x0 x1 ∈ setminus omega 1 (proof)
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 nat_0^nat_0 : nat_p 0
Known mul_nat_0R^mul_nat_0R : ∀ x0 . mul_nat x0 0 = 0
Known add_nat_0R^add_nat_0R : ∀ x0 . add_nat x0 0 = x0
Theorem mul_nat_1R^mul_nat_1R : ∀ x0 . mul_nat x0 1 = x0 (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 f11bb.. : ∀ x0 . nat_p x0 ⟶ mul_nat 1 x0 = x0 (proof)
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 ordinal_ordsucc^{ordinal_ordsucc} : ∀ x0 . ordinal x0 ⟶ ordinal (ordsucc x0)
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Known In_no2cycle^In_no2cycle : ∀ x0 x1 . x0 ∈ x1 ⟶ x1 ∈ x0 ⟶ False
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Theorem f0633.. : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ x0 ⊆ x1 ⟶ ordsucc x0 ⊆ ordsucc x1 (proof)
Known add_nat_SR^add_nat_SR : ∀ x0 x1 . nat_p x1 ⟶ add_nat x0 (ordsucc x1) = ordsucc (add_nat x0 x1)
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known add_nat_p^add_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (add_nat x0 x1)
Theorem 116d8.. : ∀ x0 x1 . nat_p x0 ⟶ nat_p x1 ⟶ x0 ⊆ x1 ⟶ ∀ x2 . nat_p x2 ⟶ add_nat x0 x2 ⊆ add_nat x1 x2 (proof)
Known add_nat_com^add_nat_com : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ add_nat x0 x1 = add_nat x1 x0
Theorem abb9e.. : ∀ x0 x1 . nat_p x0 ⟶ nat_p x1 ⟶ x0 ⊆ x1 ⟶ ∀ x2 . nat_p x2 ⟶ add_nat x2 x0 ⊆ add_nat x2 x1 (proof)
Known Subq_ref^Subq_ref : ∀ x0 . x0 ⊆ x0
Known Subq_tra^Subq_tra : ∀ x0 x1 x2 . x0 ⊆ x1 ⟶ x1 ⊆ x2 ⟶ x0 ⊆ x2
Theorem 4324d..^{mul_nat_Subq_R} : ∀ x0 x1 . nat_p x0 ⟶ nat_p x1 ⟶ x0 ⊆ x1 ⟶ ∀ x2 . nat_p x2 ⟶ mul_nat x0 x2 ⊆ mul_nat x1 x2 (proof)
Theorem 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 (proof)
Theorem 6899f.. : ∀ x0 . x0 ∈ setminus omega 1 ⟶ ∀ x1 : ο . (∀ x2 . x2 ∈ omega ⟶ x0 = ordsucc x2 ⟶ x1) ⟶ x1 (proof)
Known ordinal_ordsucc_In^{ordinal_ordsucc_In} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ∈ ordsucc x0
Known ordinal_1^ordinal_1 : ordinal 1
Known Subq_1_2^Subq_1_2 : 1 ⊆ 2
Theorem ee3ec.. : ∀ x0 . x0 ∈ setminus omega 2 ⟶ ∀ x1 : ο . (∀ x2 . x2 ∈ omega ⟶ x0 = ordsucc (ordsucc x2) ⟶ x1) ⟶ x1 (proof)
Theorem add_nat_Subq_L^{add_nat_Subq_L} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ x0 ⊆ add_nat x0 x1 (proof)
Theorem 3b460.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ setminus omega 1 ⟶ x0 ⊆ mul_nat x0 x1 (proof)
Theorem e5f18.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 : ι → ι . (∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ setminus omega 1) ⟶ nat_primrec 1 (λ x2 . mul_nat (x1 x2)) x0 ∈ setminus omega 1 (proof)
Known setminusE1^setminusE1 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ x2 ∈ x0
Theorem 39880.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ setminus omega 1 ⟶ x0 ⊆ mul_nat x1 x0 (proof)
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Theorem d1a0b.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 : ι → ι . (∀ x2 . x2 ∈ x0 ⟶ x1 x2 ∈ setminus omega 1) ⟶ ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⊆ nat_primrec 1 (λ x3 . mul_nat (x1 x3)) x0 (proof)
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
Theorem 9baed.. : ∀ x0 . x0 ∈ omega ⟶ divides_nat x0 x0 (proof)
Known and3E^and3E : ∀ x0 x1 x2 : ο . and (and x0 x1) x2 ⟶ ∀ x3 : ο . (x0 ⟶ x1 ⟶ x2 ⟶ x3) ⟶ x3
Known mul_nat_asso^mul_nat_asso : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ ∀ x2 . nat_p x2 ⟶ mul_nat (mul_nat x0 x1) x2 = mul_nat x0 (mul_nat x1 x2)
Theorem e4e38..^{divides_nat_tra} : ∀ x0 x1 x2 . divides_nat x0 x1 ⟶ divides_nat x1 x2 ⟶ divides_nat x0 x2 (proof)
Known nat_ordsucc^nat_ordsucc : ∀ x0 . nat_p x0 ⟶ nat_p (ordsucc x0)
Theorem 4c147.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 : ι → ι . (∀ x2 . x2 ∈ x0 ⟶ nat_p (x1 x2)) ⟶ ∀ x2 . x2 ∈ x0 ⟶ divides_nat (x1 x2) (nat_primrec 1 (λ x3 . mul_nat (x1 x3)) x0) (proof)
Theorem 01ac5.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ (∀ x2 . x2 ∈ setminus omega 1 ⟶ x1 x2) ⟶ x1 x0 (proof)
Known cases_2^cases_2 : ∀ x0 . x0 ∈ 2 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 1 ⟶ x1 x0
Known ordsucc_inj_contra^{ordsucc_inj_contra} : ∀ x0 x1 . (x0 = x1 ⟶ ∀ x2 : ο . x2) ⟶ ordsucc x0 = ordsucc x1 ⟶ ∀ x2 : ο . x2
Theorem 3aff2.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 1 ⟶ (∀ x2 . x2 ∈ setminus omega 2 ⟶ x1 x2) ⟶ x1 x0 (proof)
Theorem 6a866.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ or (x1 = 0) (x0 ∈ add_nat x0 x1) (proof)
Theorem 20225.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ or (x1 = 0) (x0 ∈ add_nat x1 x0) (proof)
Known neq_0_1^neq_0_1 : 0 = 1 ⟶ ∀ x0 : ο . x0
Known mul_nat_0L^mul_nat_0L : ∀ x0 . nat_p x0 ⟶ mul_nat 0 x0 = 0
Theorem 7092d.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ mul_nat x0 x1 = 1 ⟶ and (x0 = 1) (x1 = 1) (proof)
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 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 xm^xm : ∀ x0 : ο . or x0 (not x0)
Known setminusE2^setminusE2 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ nIn x2 x1
Known In_0_2^In_0_2 : 0 ∈ 2
Known ordinal_trichotomy_or_impred^{ordinal_trichotomy_or_impred} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ ∀ x2 : ο . (x0 ∈ x1 ⟶ x2) ⟶ (x0 = x1 ⟶ x2) ⟶ (x1 ∈ x0 ⟶ x2) ⟶ x2
Known In_1_2^In_1_2 : 1 ∈ 2
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Theorem b513d.. : ∀ x0 . x0 ∈ setminus omega 2 ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (and (prime_nat x2) (divides_nat x2 x0)) ⟶ x1) ⟶ x1 (proof)
Theorem 4e482.. : ∀ x0 . prime_nat x0 ⟶ x0 ∈ setminus omega 2 (proof)
Theorem f7d6d.. : ∀ x0 x1 . (x1 = 0 ⟶ ∀ x2 : ο . x2) ⟶ divides_nat x0 x1 ⟶ x0 ⊆ x1 (proof)
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Theorem 3ed2c.. : ∀ x0 x1 . (x1 = 0 ⟶ ∀ x2 : ο . x2) ⟶ divides_nat x0 x1 ⟶ or (x0 ∈ x1) (x0 = x1) (proof)
Known omega_ordsucc^{omega_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ ordsucc x0 ∈ omega
Theorem a9ab4.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ omega) (add_nat x1 x3 = x0) ⟶ x2) ⟶ x2 (proof)
Known ordsucc_inj^ordsucc_inj : ∀ x0 x1 . ordsucc x0 = ordsucc x1 ⟶ x0 = x1
Theorem c6e57.. : ∀ x0 x1 . nat_p x0 ⟶ nat_p x1 ⟶ ∀ x2 . nat_p x2 ⟶ add_nat x0 x2 = add_nat x1 x2 ⟶ x0 = x1 (proof)
Theorem 0d850.. : ∀ x0 x1 x2 . nat_p x0 ⟶ nat_p x1 ⟶ nat_p x2 ⟶ add_nat x0 x1 = add_nat x0 x2 ⟶ x1 = x2 (proof)
Known mul_add_nat_distrL^{mul_add_nat_distrL} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ ∀ x2 . nat_p x2 ⟶ mul_nat x0 (add_nat x1 x2) = add_nat (mul_nat x0 x1) (mul_nat x0 x2)
Theorem 6ce5b.. : ∀ x0 . x0 ∈ setminus omega 2 ⟶ ∀ x1 . divides_nat x0 x1 ⟶ not (divides_nat x0 (ordsucc x1)) (proof)
Definition surj^surj := λ x0 x1 . λ x2 : ι → ι . and (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1) (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 : ο . (∀ x5 . and (x5 ∈ x0) (x2 x5 = x3) ⟶ x4) ⟶ x4)
Param Sep^Sep : ι → (ι → ο) → ι
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known nat_ordsucc_in_ordsucc^{nat_ordsucc_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ∈ ordsucc x0
Known nat_0_in_ordsucc^{nat_0_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ 0 ∈ ordsucc x0
Known neq_0_ordsucc^{neq_0_ordsucc} : ∀ x0 . 0 = ordsucc x0 ⟶ ∀ x1 : ο . x1
Known SepE1^SepE1 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x2 ∈ x0
Theorem 3a99b.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 : ι → ι . not (surj x0 (Sep omega prime_nat) x1) (proof)
Param bij^bij : ι → ι → (ι → ι) → ο
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)
Known bij_surj^bij_surj : ∀ x0 x1 . ∀ x2 : ι → ι . bij x0 x1 x2 ⟶ surj x0 x1 x2
Known equip_sym^equip_sym : ∀ x0 x1 . equip x0 x1 ⟶ equip x1 x0
Theorem form100_11_infinite_primes^{form100_11_infinite_primes} : infinite (Sep omega prime_nat) (proof)