Proofgold Address

Known 60e4a.. : ∀ x0 : ι → ο . ∀ x1 : ι → ι → ι . (∀ x2 x3 . x0 x2 ⟶ x0 x3 ⟶ x0 (x1 x2 x3)) ⟶ (∀ x2 x3 x4 . x0 x2 ⟶ x0 x3 ⟶ x0 x4 ⟶ x1 x2 (x1 x3 x4) = x1 (x1 x2 x3) x4) ⟶ (∀ x2 x3 . x0 x2 ⟶ x0 x3 ⟶ x1 x2 x3 = x1 x3 x2) ⟶ ∀ x2 x3 x4 x5 . x0 x2 ⟶ x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x1 x2 (x1 x3 (x1 x4 x5)) = x1 x3 (x1 x4 (x1 x2 x5))
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param omega^omega : ι
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Param add_nat^add_nat : ι → ι → ι
Param mul_nat^mul_nat : ι → ι → ι
Definition b3e62..^{equiv_nat_mod} := λ x0 x1 x2 . and (and (and (x0 ∈ omega) (x1 ∈ omega)) (x2 ∈ omega)) (or (∀ x3 : ο . (∀ x4 . and (x4 ∈ omega) (add_nat x0 (mul_nat x4 x2) = x1) ⟶ x3) ⟶ x3) (∀ x3 : ο . (∀ x4 . and (x4 ∈ omega) (add_nat x1 (mul_nat x4 x2) = x0) ⟶ x3) ⟶ x3))
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Param ordsucc^ordsucc : 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)
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))
Param nat_p^nat_p : ι → ο
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known nat_2^nat_2 : nat_p 2
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known nat_4^nat_4 : nat_p 4
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known c3da7.. : mul_nat 2 2 = 4
Known neq_2_1^neq_2_1 : 2 = 1 ⟶ ∀ x0 : ο . x0
Known neq_4_2^neq_4_2 : 4 = 2 ⟶ ∀ x0 : ο . x0
Theorem c14f2.. : not (prime_nat 4) (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 add_nat_0R^add_nat_0R : ∀ x0 . add_nat x0 0 = x0
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Known add_nat_SR^add_nat_SR : ∀ x0 x1 . nat_p x1 ⟶ add_nat x0 (ordsucc x1) = ordsucc (add_nat x0 x1)
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known ordsuccI1^ordsuccI1 : ∀ x0 . x0 ⊆ ordsucc x0
Theorem d8085.. : ∀ x0 x1 . nat_p x1 ⟶ x0 ∈ ordsucc (add_nat x0 x1) (proof)
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 FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Theorem 46720.. : ∀ x0 x1 . nat_p x0 ⟶ nat_p x1 ⟶ add_nat x0 x1 ⊆ x0 ⟶ x1 = 0 (proof)
Known Subq_ref^Subq_ref : ∀ x0 . x0 ⊆ x0
Theorem ab486.. : ∀ x0 x1 . nat_p x0 ⟶ nat_p x1 ⟶ x0 = add_nat x0 x1 ⟶ x1 = 0 (proof)
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known f3fbb..^{mul_nat_0_inv} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ mul_nat x0 x1 = 0 ⟶ or (x0 = 0) (x1 = 0)
Known mul_nat_p^mul_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (mul_nat x0 x1)
Known ordsucc_inj^ordsucc_inj : ∀ x0 x1 . ordsucc x0 = ordsucc x1 ⟶ 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)
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 EmptyE^EmptyE : ∀ x0 . nIn x0 0
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Theorem 92637.. : ∀ x0 . nat_p x0 ⟶ mul_nat x0 x0 = x0 ⟶ or (x0 = 0) (x0 = 1) (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
Param lam^Sigma : ι → (ι → ι) → ι
Definition setprod^setprod := λ x0 x1 . lam x0 (λ x2 . x1)
Known 63881.. : ∀ x0 x1 x2 x3 . nat_p x0 ⟶ nat_p x1 ⟶ equip x0 x2 ⟶ equip x1 x3 ⟶ equip (mul_nat x0 x1) (setprod x2 x3)
Param Pi^Pi : ι → (ι → ι) → ι
Definition setexp^setexp := λ x0 x1 . Pi x1 (λ x2 . x0)
Known bijI^bijI : ∀ x0 x1 . ∀ x2 : ι → ι . (∀ 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) ⟶ bij x0 x1 x2
Known In_0_1^In_0_1 : 0 ∈ 1
Param ap^ap : ι → ι → ι
Known Pi_ext^Pi_ext : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ Pi x0 x1 ⟶ ∀ x3 . x3 ∈ Pi x0 x1 ⟶ (∀ x4 . x4 ∈ x0 ⟶ ap x2 x4 = ap x3 x4) ⟶ x2 = x3
Param pair_p^pair_p : ι → ο
Known PiI^PiI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . (∀ x3 . x3 ∈ x2 ⟶ and (pair_p x3) (ap x3 0 ∈ x0)) ⟶ (∀ x3 . x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3) ⟶ x2 ∈ Pi x0 x1
Known cases_1^cases_1 : ∀ x0 . x0 ∈ 1 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 x0
Theorem 1032b.. : ∀ x0 . equip (setexp x0 0) 1 (proof)
Param exp_nat^exp_nat : ι → ι → ι
Known 856b4..^exp_nat_0 : ∀ x0 . exp_nat x0 0 = 1
Known equip_sym^equip_sym : ∀ x0 x1 . equip x0 x1 ⟶ equip x1 x0
Known caaf4..^exp_nat_S : ∀ x0 x1 . nat_p x1 ⟶ exp_nat x0 (ordsucc x1) = mul_nat x0 (exp_nat x0 x1)
Known equip_tra^equip_tra : ∀ x0 x1 x2 . equip x0 x1 ⟶ equip x1 x2 ⟶ equip x0 x2
Known 4f402..^exp_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (exp_nat x0 x1)
Known equip_ref^equip_ref : ∀ x0 . equip x0 x0
Param If_i^If_i : ο → ι → ι → ι
Known tuple_2_setprod^{tuple_2_setprod} : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x1 ⟶ lam 2 (λ x4 . If_i (x4 = 0) x2 x3) ∈ setprod x0 x1
Known ap_Pi^ap_Pi : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ Pi x0 x1 ⟶ x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3
Known lam_Pi^lam_Pi : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1 x3) ⟶ lam x0 x2 ∈ Pi x0 x1
Known tuple_2_inj^tuple_2_inj : ∀ x0 x1 x2 x3 . lam 2 (λ x5 . If_i (x5 = 0) x0 x1) = lam 2 (λ x5 . If_i (x5 = 0) x2 x3) ⟶ and (x0 = x2) (x1 = x3)
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Known beta^beta : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ ap (lam x0 x1) x2 = x1 x2
Param inv^inv : ι → (ι → ι) → ι → ι
Known surj_rinv^surj_rinv : ∀ x0 x1 . ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x1 ⟶ ∀ x4 : ο . (∀ x5 . and (x5 ∈ x0) (x2 x5 = x3) ⟶ x4) ⟶ x4) ⟶ ∀ x3 . x3 ∈ x1 ⟶ and (inv x0 x2 x3 ∈ x0) (x2 (inv x0 x2 x3) = x3)
Known ap1_Sigma^ap1_Sigma : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ ap x2 1 ∈ x1 (ap x2 0)
Known If_i_1^If_i_1 : ∀ x0 : ο . ∀ x1 x2 . x0 ⟶ If_i x0 x1 x2 = x1
Known If_i_0^If_i_0 : ∀ x0 : ο . ∀ x1 x2 . not x0 ⟶ If_i x0 x1 x2 = x2
Known tuple_Sigma_eta^{tuple_Sigma_eta} : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ lam 2 (λ x4 . If_i (x4 = 0) (ap x2 0) (ap x2 1)) = x2
Known ap0_Sigma^ap0_Sigma : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ ap x2 0 ∈ x0
Theorem 17a87.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ equip (exp_nat x0 x1) (setexp x0 x1) (proof)
Param even_nat^even_nat : ι → ο
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 even_nat_0^even_nat_0 : even_nat 0
Definition odd_nat^odd_nat := λ x0 . and (x0 ∈ omega) (∀ x1 . x1 ∈ omega ⟶ x0 = mul_nat 2 x1 ⟶ ∀ x2 : ο . x2)
Known odd_nat_even_nat_S^{odd_nat_even_nat_S} : ∀ x0 . odd_nat x0 ⟶ even_nat (ordsucc x0)
Known even_nat_odd_nat_S^{even_nat_odd_nat_S} : ∀ x0 . even_nat x0 ⟶ odd_nat (ordsucc x0)
Param setminus^setminus : ι → ι → ι
Param UPair^UPair : ι → ι → ι
Known setminusI^setminusI : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ nIn x2 x1 ⟶ x2 ∈ setminus x0 x1
Known UPairE^UPairE : ∀ x0 x1 x2 . x0 ∈ UPair x1 x2 ⟶ or (x0 = x1) (x0 = x2)
Known In_no2cycle^In_no2cycle : ∀ x0 x1 . x0 ∈ x1 ⟶ x1 ∈ x0 ⟶ False
Known nat_trans^nat_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Known setminusE^setminusE : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ and (x2 ∈ x0) (nIn x2 x1)
Param ordinal^ordinal : ι → ο
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 nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known nat_p_trans^nat_p_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ nat_p x1
Known UPairI1^UPairI1 : ∀ x0 x1 . x0 ∈ UPair x0 x1
Known UPairI2^UPairI2 : ∀ x0 x1 . x1 ∈ UPair x0 x1
Known ordinal_ordsucc_In_Subq^{ordinal_ordsucc_In_Subq} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ⊆ x0
Known nat_ordsucc_in_ordsucc^{nat_ordsucc_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ∈ ordsucc x0
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Theorem 63511.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . equip x0 x1 ⟶ ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x1 ⟶ x2 x3 ∈ x1) ⟶ (∀ x3 . x3 ∈ x1 ⟶ x2 (x2 x3) = x3) ⟶ (∀ x3 . x3 ∈ x1 ⟶ x2 x3 = x3 ⟶ ∀ x4 : ο . x4) ⟶ even_nat x0 (proof)
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Known even_nat_not_odd_nat^{even_nat_not_odd_nat} : ∀ x0 . even_nat x0 ⟶ not (odd_nat x0)
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Theorem d3cb5.. : ∀ x0 x1 . odd_nat x1 ⟶ equip x0 x1 ⟶ ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x0) ⟶ (∀ x3 . x3 ∈ x0 ⟶ x2 (x2 x3) = x3) ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ x0) (x2 x4 = x4) ⟶ x3) ⟶ x3 (proof)
Param Sing^Sing : ι → ι
Known SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Known setminusE1^setminusE1 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ x2 ∈ x0
Theorem 74e07.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . equip x0 x1 ⟶ ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x1 ⟶ x2 x3 ∈ x1) ⟶ (∀ x3 . x3 ∈ x1 ⟶ x2 (x2 x3) = x3) ⟶ (∀ x3 : ο . (∀ x4 . and (x4 ∈ x1) (and (x2 x4 = x4) (∀ x5 . x5 ∈ x1 ⟶ x2 x5 = x5 ⟶ x4 = x5)) ⟶ x3) ⟶ x3) ⟶ odd_nat x0 (proof)
Param SNo^SNo : ι → ο
Param mul_SNo^mul_SNo : ι → ι → ι
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Known mul_SNo_assoc^{mul_SNo_assoc} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 (mul_SNo x1 x2) = mul_SNo (mul_SNo x0 x1) x2
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
Theorem 24b4c.. : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ mul_SNo x0 (mul_SNo x1 (mul_SNo x2 x3)) = mul_SNo x1 (mul_SNo x2 (mul_SNo x0 x3)) (proof)
Known mul_SNo_oneR^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = x0
Known SNo_1^SNo_1 : SNo 1
Theorem e032f.. : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 (mul_SNo x1 x2) = mul_SNo x1 (mul_SNo x2 x0) (proof)
Param add_SNo^add_SNo : ι → ι → ι
Known 547c4.. : ∀ x0 : ι → ο . ∀ x1 x2 : ι → ι → ι . (∀ x3 x4 . x0 x3 ⟶ x0 x4 ⟶ x0 (x1 x3 x4)) ⟶ (∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 (x1 x3 x4) x5 = x1 (x2 x3 x5) (x2 x4 x5)) ⟶ ∀ x3 x4 x5 x6 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x0 x6 ⟶ x2 (x1 x3 (x1 x4 x5)) x6 = x1 (x2 x3 x6) (x1 (x2 x4 x6) (x2 x5 x6))
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Known mul_SNo_distrR^{mul_SNo_distrR} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo (add_SNo x0 x1) x2 = add_SNo (mul_SNo x0 x2) (mul_SNo x1 x2)
Theorem 7bd74.. : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ mul_SNo (add_SNo x0 (add_SNo x1 x2)) x3 = add_SNo (mul_SNo x0 x3) (add_SNo (mul_SNo x1 x3) (mul_SNo x2 x3)) (proof)
Param SNoLt^SNoLt : ι → ι → ο
Known ordinal_In_SNoLt^{ordinal_In_SNoLt} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ SNoLt x1 x0
Known In_1_4^In_1_4 : 1 ∈ 4
Theorem c188a.. : SNoLt 1 4 (proof)