Proofgold Address

Param add_nat^add_nat : ι → ι → ι
Param ordsucc^ordsucc : ι → ι
Definition u1 := 1
Definition u2 := ordsucc u1
Definition u3 := ordsucc u2
Definition u4 := ordsucc u3
Definition u5 := ordsucc u4
Definition u6 := ordsucc u5
Definition u7 := ordsucc u6
Definition u8 := ordsucc u7
Definition u9 := ordsucc u8
Param nat_p^nat_p : ι → ο
Known add_nat_SR^add_nat_SR : ∀ x0 x1 . nat_p x1 ⟶ add_nat x0 (ordsucc x1) = ordsucc (add_nat x0 x1)
Known nat_3^nat_3 : nat_p 3
Known e705e.. : add_nat u5 u3 = u8
Theorem 0aa1b.. : add_nat u5 u4 = u9 (proof)
Definition u10 := ordsucc u9
Known nat_4^nat_4 : nat_p 4
Theorem 9e8b0.. : add_nat u5 u5 = u10 (proof)
Definition u11 := ordsucc u10
Known nat_5^nat_5 : nat_p 5
Theorem fe08b.. : add_nat u5 u6 = u11 (proof)
Definition u12 := ordsucc u11
Known nat_6^nat_6 : nat_p 6
Theorem ecf06.. : add_nat u5 u7 = u12 (proof)
Definition u13 := ordsucc u12
Known nat_7^nat_7 : nat_p 7
Theorem b1ec3.. : add_nat u5 u8 = u13 (proof)
Definition u14 := ordsucc u13
Known nat_8^nat_8 : nat_p 8
Theorem ec3c1.. : add_nat u5 u9 = u14 (proof)
Definition u15 := ordsucc u14
Known nat_9^nat_9 : nat_p 9
Theorem 0f811.. : add_nat u5 u10 = u15 (proof)
Definition u16 := ordsucc u15
Known nat_10^nat_10 : nat_p 10
Theorem 4cfca.. : add_nat u5 u11 = u16 (proof)
Definition u17 := ordsucc u16
Known nat_11^nat_11 : nat_p 11
Theorem df221.. : add_nat u5 u12 = u17 (proof)
Definition u18 := ordsucc u17
Known nat_12^nat_12 : nat_p 12
Theorem 18365.. : add_nat u5 u13 = u18 (proof)
Definition u19 := ordsucc u18
Known nat_13^nat_13 : nat_p 13
Theorem 0564f.. : add_nat u5 u14 = u19 (proof)
Definition u20 := ordsucc u19
Known nat_14^nat_14 : nat_p 14
Theorem b3144.. : add_nat u5 u15 = u20 (proof)
Known nat_2^nat_2 : nat_p 2
Known nat_1^nat_1 : nat_p 1
Known nat_0^nat_0 : nat_p 0
Known add_nat_0R^add_nat_0R : ∀ x0 . add_nat x0 0 = x0
Theorem 2f205.. : add_nat u13 u6 = u19 (proof)
Param mul_nat^mul_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 mul_nat_com^mul_nat_com : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ mul_nat x0 x1 = mul_nat x1 x0
Known 6cce6.. : ∀ x0 . nat_p x0 ⟶ mul_nat u2 x0 = add_nat x0 x0
Theorem c6b87.. : mul_nat u5 u4 = u20 (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 and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param Sing^Sing : ι → ι
Known bijE^bijE : ∀ x0 x1 . ∀ x2 : ι → ι . bij x0 x1 x2 ⟶ ∀ x3 : ο . ((∀ x4 . x4 ∈ x0 ⟶ x2 x4 ∈ x1) ⟶ (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ x2 x4 = x2 x5 ⟶ x4 = x5) ⟶ (∀ x4 . x4 ∈ x1 ⟶ ∀ x5 : ο . (∀ x6 . and (x6 ∈ x0) (x2 x6 = x4) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known In_0_1^In_0_1 : 0 ∈ 1
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Known cases_1^cases_1 : ∀ x0 . x0 ∈ 1 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 x0
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Known SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Theorem f5939.. : ∀ x0 . equip u1 x0 ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ x0) (x0 = Sing x2) ⟶ x1) ⟶ x1 (proof)
Param UPair^UPair : ι → ι → ι
Param atleastp^atleastp : ι → ι → ο
Known e8716.. : ∀ x0 . atleastp u2 x0 ⟶ ∀ x1 : ο . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x1) ⟶ x1
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known 4fb58..^{Pigeonhole_not_atleastp_ordsucc} : ∀ x0 . nat_p x0 ⟶ not (atleastp (ordsucc x0) x0)
Known atleastp_tra^atleastp_tra : ∀ x0 x1 x2 . atleastp x0 x1 ⟶ atleastp x1 x2 ⟶ atleastp x0 x2
Known 5d098.. : ∀ x0 x1 . x1 ∈ x0 ⟶ ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ (x1 = x2 ⟶ ∀ x4 : ο . x4) ⟶ (x1 = x3 ⟶ ∀ x4 : ο . x4) ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ atleastp u3 x0
Known UPairI1^UPairI1 : ∀ x0 x1 . x0 ∈ UPair x0 x1
Known UPairI2^UPairI2 : ∀ x0 x1 . x1 ∈ UPair x0 x1
Known equip_atleastp^{equip_atleastp} : ∀ x0 x1 . equip x0 x1 ⟶ atleastp x0 x1
Known equip_sym^equip_sym : ∀ x0 x1 . equip x0 x1 ⟶ equip x1 x0
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known UPairE^UPairE : ∀ x0 x1 x2 . x0 ∈ UPair x1 x2 ⟶ or (x0 = x1) (x0 = x2)
Theorem feddd.. : ∀ x0 . equip u2 x0 ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ x0) (∀ x3 : ο . (∀ x4 . and (x4 ∈ x0) (and (x2 = x4 ⟶ ∀ x5 : ο . x5) (x0 = UPair x2 x4)) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1 (proof)
Param omega^omega : ι
Definition finite^finite := λ x0 . ∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (equip x0 x2) ⟶ x1) ⟶ x1
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known equip_ref^equip_ref : ∀ x0 . equip x0 x0
Theorem 9b83b..^nat_finite : ∀ x0 . nat_p x0 ⟶ finite x0 (proof)
Theorem 6cd03.. : ∀ x0 x1 . x1 ∈ x0 ⟶ Sing x1 ⊆ x0 (proof)
Param binunion^binunion : ι → ι → ι
Param setminus^setminus : ι → ι → ι
Known e6195.. : ∀ x0 x1 . x1 ⊆ x0 ⟶ x0 = binunion (setminus x0 x1) x1
Theorem eb0c4..^{binunion_remove1_eq} : ∀ x0 x1 . x1 ∈ x0 ⟶ x0 = binunion (setminus x0 (Sing x1)) (Sing x1) (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 nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known nat_trans^nat_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Known nat_ordsucc^nat_ordsucc : ∀ x0 . nat_p x0 ⟶ nat_p (ordsucc x0)
Known nat_ordsucc_in_ordsucc^{nat_ordsucc_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ∈ ordsucc x0
Known Subq_ref^Subq_ref : ∀ x0 . x0 ⊆ x0
Theorem 3d32d.. : ∀ x0 x1 . nat_p x0 ⟶ nat_p x1 ⟶ x0 ⊆ x1 ⟶ ordsucc x0 ⊆ ordsucc 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 and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known Empty_Subq_eq^{Empty_Subq_eq} : ∀ x0 . x0 ⊆ 0 ⟶ x0 = 0
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known e8bc0..^{equip_adjoin_ordsucc} : ∀ x0 x1 x2 . nIn x2 x1 ⟶ equip x0 x1 ⟶ equip (ordsucc x0) (binunion x1 (Sing x2))
Known setminus_nIn_I2^{setminus_nIn_I2} : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ nIn x2 (setminus x0 x1)
Known setminusE^setminusE : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ and (x2 ∈ x0) (nIn x2 x1)
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (x1 ∈ x0) (x1 = x0)
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known Subq_tra^Subq_tra : ∀ x0 x1 x2 . x0 ⊆ x1 ⟶ x1 ⊆ x2 ⟶ x0 ⊆ x2
Known ordsuccI1^ordsuccI1 : ∀ x0 . x0 ⊆ ordsucc x0
Theorem b70a6.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ⊆ x0 ⟶ ∀ x2 : ο . (∀ x3 . and (and (nat_p x3) (x3 ⊆ x0)) (equip x1 x3) ⟶ x2) ⟶ x2 (proof)
Known 2d48b.. : ∀ x0 x1 . atleastp x0 x1 ⟶ ∀ x2 : ο . (∀ x3 . x3 ⊆ x1 ⟶ equip x0 x3 ⟶ x2) ⟶ x2
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Known equip_tra^equip_tra : ∀ x0 x1 x2 . equip x0 x1 ⟶ equip x1 x2 ⟶ equip x0 x2
Known Subq_atleastp^{Subq_atleastp} : ∀ x0 x1 . x0 ⊆ x1 ⟶ atleastp x0 x1
Theorem db10e.. : ∀ x0 x1 . x1 ⊆ x0 ⟶ finite x0 ⟶ finite x1 (proof)
Definition u21 := ordsucc u20
Definition u22 := ordsucc u21
Definition u23 := ordsucc u22
Definition u24 := ordsucc u23
Theorem c7bb7.. : add_nat u20 u4 = u24 (proof)
Theorem 8f72e.. : add_nat u12 u8 = u20 (proof)
Theorem 663d6.. : add_nat u12 u12 = u24 (proof)
Param lam^Sigma : ι → (ι → ι) → ι
Definition setprod^setprod := λ x0 x1 . lam x0 (λ x2 . x1)
Param Pi^Pi : ι → (ι → ι) → ι
Definition setexp^setexp := λ x0 x1 . Pi x1 (λ x2 . x0)
Param ap^ap : ι → ι → ι
Param If_i^If_i : ο → ι → ι → ι
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 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 ap1_Sigma^ap1_Sigma : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ ap x2 1 ∈ x1 (ap x2 0)
Known ap0_Sigma^ap0_Sigma : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ lam x0 x1 ⟶ ap x2 0 ∈ x0
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 tuple_2_1_eq^tuple_2_1_eq : ∀ x0 x1 . ap (lam 2 (λ x3 . If_i (x3 = 0) x0 x1)) 1 = x1
Known tuple_2_0_eq^tuple_2_0_eq : ∀ x0 x1 . ap (lam 2 (λ x3 . If_i (x3 = 0) x0 x1)) 0 = x0
Known tuple_2_Sigma^{tuple_2_Sigma} : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x1 x2 ⟶ lam 2 (λ x4 . If_i (x4 = 0) x2 x3) ∈ lam x0 x1
Known Eps_i_ex^Eps_i_ex : ∀ x0 : ι → ο . (∀ x1 : ο . (∀ x2 . x0 x2 ⟶ x1) ⟶ x1) ⟶ x0 (prim0 x0)
Known lam_Pi^lam_Pi : ∀ x0 . ∀ x1 x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1 x3) ⟶ lam x0 x2 ∈ Pi x0 x1
Known beta^beta : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ x0 ⟶ ap (lam x0 x1) x2 = x1 x2
Theorem 8f044.. : ∀ x0 x1 . ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ equip (x2 x3) x1) ⟶ equip (lam x0 x2) (setprod x0 x1) (proof)
Param famunion^famunion : ι → (ι → ι) → ι
Known famunionI^famunionI : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ x0 ⟶ x3 ∈ x1 x2 ⟶ x3 ∈ famunion x0 x1
Known famunionE_impred^{famunionE_impred} : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 . x2 ∈ famunion x0 x1 ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ x0 ⟶ x2 ∈ x1 x4 ⟶ x3) ⟶ x3
Theorem 411b1.. : ∀ x0 . ∀ x1 : ι → ι . (∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x1 x2 ⟶ x4 ∈ x1 x3 ⟶ x2 = x3) ⟶ equip (lam x0 x1) (famunion x0 x1) (proof)
Theorem 4f936.. : ∀ x0 x1 . ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ equip (x2 x3) x1) ⟶ (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x2 x3 ⟶ x5 ∈ x2 x4 ⟶ x3 = x4) ⟶ equip (famunion x0 x2) (setprod x0 x1) (proof)
Theorem 34b98.. : ∀ x0 x1 x2 x3 . equip x0 x1 ⟶ equip x2 x3 ⟶ equip (setprod x0 x2) (setprod x1 x3) (proof)
Theorem 5983e.. : ∀ x0 x1 x2 . ∀ x3 : ι → ι . equip x0 x1 ⟶ (∀ x4 . x4 ∈ x0 ⟶ equip (x3 x4) x2) ⟶ (∀ x4 . x4 ∈ x0 ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x3 x4 ⟶ x6 ∈ x3 x5 ⟶ x4 = x5) ⟶ equip (famunion x0 x3) (setprod x1 x2) (proof)
Known Subq_Empty^Subq_Empty : ∀ x0 . 0 ⊆ x0
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Theorem 386bf.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ atleastp x1 x0 ⟶ x1 ⊆ x0 (proof)
Theorem e6d41.. : ∀ x0 x1 . nat_p x0 ⟶ nat_p x1 ⟶ equip x0 x1 ⟶ x0 = x1 (proof)
Param Sep^Sep : ι → (ι → ο) → ι
Known add_nat_p^add_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (add_nat x0 x1)
Param setsum^setsum : ι → ι → ι
Known c88e0.. : ∀ x0 x1 x2 x3 . nat_p x0 ⟶ nat_p x1 ⟶ equip x0 x2 ⟶ equip x1 x3 ⟶ equip (add_nat x0 x1) (setsum x2 x3)
Known d778e.. : ∀ x0 x1 x2 x3 . equip x0 x2 ⟶ equip x1 x3 ⟶ (∀ x4 . x4 ∈ x0 ⟶ nIn x4 x1) ⟶ equip (binunion x0 x1) (setsum x2 x3)
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Known SepE2^SepE2 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x1 x2
Known binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Known SepE1^SepE1 : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ x2 ∈ x0
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
Theorem 65ef6.. : ∀ x0 x1 . nat_p x1 ⟶ ∀ x2 : ι → ι . ∀ x3 x4 x5 x6 . nat_p x3 ⟶ nat_p x4 ⟶ (x3 = x4 ⟶ ∀ x7 : ο . x7) ⟶ nat_p x5 ⟶ nat_p x6 ⟶ (∀ x7 . x7 ∈ x1 ⟶ or (equip {x8 ∈ x0|x2 x8 = x7} x3) (equip {x8 ∈ x0|x2 x8 = x7} x4)) ⟶ equip {x7 ∈ x1|equip {x8 ∈ x0|x2 x8 = x7} x3} x5 ⟶ equip {x7 ∈ x1|equip {x8 ∈ x0|x2 x8 = x7} x4} x6 ⟶ add_nat x5 x6 = x1 (proof)
Known mul_nat_p^mul_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (mul_nat x0 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)
Theorem a1aba.. : ∀ x0 x1 . nat_p x0 ⟶ ∀ x2 : ι → ι . ∀ x3 x4 x5 x6 . nat_p x3 ⟶ nat_p x4 ⟶ (x3 = x4 ⟶ ∀ x7 : ο . x7) ⟶ nat_p x5 ⟶ nat_p x6 ⟶ (∀ x7 . x7 ∈ x0 ⟶ x2 x7 ∈ x1) ⟶ (∀ x7 . x7 ∈ x1 ⟶ or (equip {x8 ∈ x0|x2 x8 = x7} x3) (equip {x8 ∈ x0|x2 x8 = x7} x4)) ⟶ equip {x7 ∈ x1|equip {x8 ∈ x0|x2 x8 = x7} x3} x5 ⟶ equip {x7 ∈ x1|equip {x8 ∈ x0|x2 x8 = x7} x4} x6 ⟶ add_nat (mul_nat x5 x3) (mul_nat x6 x4) = x0 (proof)
Param SNo^SNo : ι → ο
Param mul_SNo^mul_SNo : ι → ι → ι
Param add_SNo^add_SNo : ι → ι → ι
Known add_SNo_1_1_2^{add_SNo_1_1_2} : add_SNo 1 1 = 2
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)
Known SNo_1^SNo_1 : SNo 1
Known mul_SNo_oneL^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 x0 = x0
Theorem d67ed.. : ∀ x0 . SNo x0 ⟶ mul_SNo 2 x0 = add_SNo x0 x0 (proof)
Param minus_SNo^minus_SNo : ι → ι
Known mul_SNo_distrL^{mul_SNo_distrL} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 (add_SNo x1 x2) = add_SNo (mul_SNo x0 x1) (mul_SNo x0 x2)
Known SNo_2^SNo_2 : SNo 2
Known minus_add_SNo_distr^{minus_add_SNo_distr} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ minus_SNo (add_SNo x0 x1) = add_SNo (minus_SNo x0) (minus_SNo x1)
Known add_SNo_com_4_inner_mid^{add_SNo_com_4_inner_mid} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ add_SNo (add_SNo x0 x1) (add_SNo x2 x3) = add_SNo (add_SNo x0 x2) (add_SNo x1 x3)
Known add_SNo_minus_SNo_rinv^{add_SNo_minus_SNo_rinv} : ∀ x0 . SNo x0 ⟶ add_SNo x0 (minus_SNo x0) = 0
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Known add_SNo_cancel_R^{add_SNo_cancel_R} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 x1 = add_SNo x2 x1 ⟶ x0 = x2
Known add_SNo_minus_R2'^{add_SNo_minus_R2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo (add_SNo x0 (minus_SNo x1)) x1 = x0
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Known add_SNo_assoc^{add_SNo_assoc} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 (add_SNo x1 x2) = add_SNo (add_SNo x0 x1) x2
Known add_SNo_minus_SNo_linv^{add_SNo_minus_SNo_linv} : ∀ x0 . SNo x0 ⟶ add_SNo (minus_SNo x0) x0 = 0
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Theorem 4b1ff.. : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ add_SNo x0 x1 = x2 ⟶ add_SNo x0 (mul_SNo 2 x1) = x3 ⟶ and (x1 = add_SNo x3 (minus_SNo x2)) (x0 = add_SNo (mul_SNo 2 x2) (minus_SNo x3)) (proof)
Known nat_p_SNo^nat_p_SNo : ∀ x0 . nat_p x0 ⟶ SNo x0
Known 2669c.. : nat_p u20
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Known add_nat_add_SNo^{add_nat_add_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ add_nat x0 x1 = add_SNo x0 x1
Theorem ff095.. : add_SNo u20 (minus_SNo u12) = u8 (proof)
Theorem d70cc.. : mul_SNo u2 u12 = u24 (proof)
Known 73189.. : nat_p u24
Theorem 7c521.. : add_SNo u24 (minus_SNo u20) = u4 (proof)
Theorem f7ea8.. : ∀ x0 x1 . nat_p x0 ⟶ nat_p x1 ⟶ add_SNo x0 x1 = u12 ⟶ add_SNo x0 (mul_SNo 2 x1) = u20 ⟶ and (x0 = u4) (x1 = u8) (proof)
Known mul_SNo_oneR^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = x0
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
Known mul_nat_mul_SNo^{mul_nat_mul_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_nat x0 x1 = mul_SNo x0 x1
Known neq_1_2^neq_1_2 : 1 = 2 ⟶ ∀ x0 : ο . x0
Theorem eeb21.. : ∀ x0 : ι → ι . ∀ x1 x2 . nat_p x1 ⟶ nat_p x2 ⟶ (∀ x3 . x3 ∈ u20 ⟶ x0 x3 ∈ u12) ⟶ (∀ x3 . x3 ∈ u12 ⟶ or (equip {x4 ∈ u20|x0 x4 = x3} u1) (equip {x4 ∈ u20|x0 x4 = x3} u2)) ⟶ equip {x3 ∈ u12|equip {x4 ∈ u20|x0 x4 = x3} u1} x1 ⟶ equip {x3 ∈ u12|equip {x4 ∈ u20|x0 x4 = x3} u2} x2 ⟶ and (x1 = u4) (x2 = u8) (proof)
Definition DirGraphOutNeighbors := λ x0 . λ x1 : ι → ι → ο . λ x2 . {x3 ∈ x0|and (x2 = x3 ⟶ ∀ x4 : ο . x4) (x1 x2 x3)}
Known 9c223..^{equip_ordsucc_remove1} : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ equip x0 (ordsucc x1) ⟶ equip (setminus x0 (Sing x2)) x1
Known cfabd.. : ∀ x0 . ∀ x1 : ι → ι → ο . (∀ x2 x3 . x1 x2 x3 ⟶ x1 x3 x2) ⟶ ∀ x2 . x2 ∈ x0 ⟶ ∀ x3 . x3 ∈ DirGraphOutNeighbors x0 x1 x2 ⟶ x2 ∈ DirGraphOutNeighbors x0 x1 x3
Known b4538.. : ∀ x0 : ι → ι → ο . (∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1) ⟶ (∀ x1 . x1 ⊆ u18 ⟶ atleastp u3 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x0 x2 x3)) ⟶ (∀ x1 . x1 ⊆ u18 ⟶ atleastp u6 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ not (x0 x2 x3))) ⟶ ∀ x1 . x1 ∈ u18 ⟶ equip (DirGraphOutNeighbors u18 x0 x1) u5
Theorem 141e8.. : ∀ x0 : ι → ι → ο . (∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1) ⟶ (∀ x1 . x1 ⊆ u18 ⟶ atleastp u3 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x0 x2 x3)) ⟶ (∀ x1 . x1 ⊆ u18 ⟶ atleastp u6 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ not (x0 x2 x3))) ⟶ ∀ x1 . x1 ∈ u18 ⟶ ∀ x2 . x2 ∈ DirGraphOutNeighbors u18 x0 x1 ⟶ equip (setminus (DirGraphOutNeighbors u18 x0 x2) (Sing x1)) u4 (proof)
Theorem 36d6a.. : ∀ x0 : ι → ι → ο . (∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1) ⟶ (∀ x1 . x1 ⊆ u18 ⟶ atleastp u3 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x0 x2 x3)) ⟶ (∀ x1 . x1 ⊆ u18 ⟶ atleastp u6 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ not (x0 x2 x3))) ⟶ ∀ x1 . x1 ∈ u18 ⟶ equip (lam (DirGraphOutNeighbors u18 x0 x1) (λ x2 . setminus (DirGraphOutNeighbors u18 x0 x2) (Sing x1))) u20 (proof)
Theorem 4948f.. : ∀ x0 : ι → ι → ο . (∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1) ⟶ (∀ x1 . x1 ⊆ u18 ⟶ atleastp u3 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x0 x2 x3)) ⟶ (∀ x1 . x1 ⊆ u18 ⟶ atleastp u6 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ not (x0 x2 x3))) ⟶ ∀ x1 . x1 ∈ u18 ⟶ equip (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1)) u6 (proof)
Known 2c48a..^{atleastp_antisym_equip} : ∀ x0 x1 . atleastp x0 x1 ⟶ atleastp x1 x0 ⟶ equip x0 x1
Known 48e0f.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . or (atleastp x1 x0) (atleastp (ordsucc x0) x1)
Known 86c65.. : nat_p u18
Known 1fe14.. : ∀ x0 x1 x2 x3 . atleastp x0 x2 ⟶ atleastp x1 x3 ⟶ (∀ x4 . x4 ∈ x2 ⟶ nIn x4 x3) ⟶ atleastp (setsum x0 x1) (binunion x2 x3)
Known setminusE2^setminusE2 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ nIn x2 x1
Known nat_17^nat_17 : nat_p 17
Known c558f.. : ∀ x0 x1 x2 x3 . atleastp x0 x2 ⟶ atleastp x1 x3 ⟶ atleastp (binunion x0 x1) (setsum x2 x3)
Known 91a85.. : add_nat 11 6 = 17
Known binunion_Subq_min^{binunion_Subq_min} : ∀ x0 x1 x2 . x0 ⊆ x2 ⟶ x1 ⊆ x2 ⟶ binunion x0 x1 ⊆ x2
Known Sep_Subq^Sep_Subq : ∀ x0 . ∀ x1 : ι → ο . Sep x0 x1 ⊆ x0
Theorem 05730.. : ∀ x0 : ι → ι → ο . (∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1) ⟶ (∀ x1 . x1 ⊆ u18 ⟶ atleastp u3 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x0 x2 x3)) ⟶ (∀ x1 . x1 ⊆ u18 ⟶ atleastp u6 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ not (x0 x2 x3))) ⟶ ∀ x1 . x1 ∈ u18 ⟶ equip (setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1))) u12 (proof)
Param binintersect^binintersect : ι → ι → ι
Known 86f86.. : ∀ x0 : ι → ι → ο . (∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1) ⟶ (∀ x1 . x1 ⊆ u18 ⟶ atleastp u3 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x0 x2 x3)) ⟶ (∀ x1 . x1 ⊆ u18 ⟶ atleastp u6 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ not (x0 x2 x3))) ⟶ ∀ x1 . x1 ∈ u18 ⟶ ∀ x2 . x2 ∈ u18 ⟶ (x1 = x2 ⟶ ∀ x3 : ο . x3) ⟶ not (x0 x1 x2) ⟶ or (equip (binintersect (DirGraphOutNeighbors u18 x0 x1) (DirGraphOutNeighbors u18 x0 x2)) u1) (equip (binintersect (DirGraphOutNeighbors u18 x0 x1) (DirGraphOutNeighbors u18 x0 x2)) u2)
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known binintersectE^{binintersectE} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ and (x2 ∈ x0) (x2 ∈ x1)
Known 5169f..^equip_Sing_1 : ∀ x0 . equip (Sing x0) u1
Known binintersectI^{binintersectI} : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ x1 ⟶ x2 ∈ binintersect x0 x1
Known setminusE1^setminusE1 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ x2 ∈ x0
Known setminusI^setminusI : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ nIn x2 x1 ⟶ x2 ∈ setminus x0 x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Known ced33.. : ∀ x0 x1 . (x0 = x1 ⟶ ∀ x2 : ο . x2) ⟶ equip (UPair x0 x1) u2
Known binintersectE2^{binintersectE2} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ x2 ∈ x1
Known binintersectE1^{binintersectE1} : ∀ x0 x1 x2 . x2 ∈ binintersect x0 x1 ⟶ x2 ∈ x0
Param SetAdjoin^SetAdjoin : ι → ι → ι
Known aa241.. : ∀ x0 x1 x2 . ∀ x3 : ι → ο . x3 x0 ⟶ x3 x1 ⟶ x3 x2 ⟶ ∀ x4 . x4 ∈ SetAdjoin (UPair x0 x1) x2 ⟶ x3 x4
Known a515c.. : ∀ x0 x1 x2 . (x0 = x1 ⟶ ∀ x3 : ο . x3) ⟶ (x0 = x2 ⟶ ∀ x3 : ο . x3) ⟶ (x1 = x2 ⟶ ∀ x3 : ο . x3) ⟶ equip (SetAdjoin (UPair x0 x1) x2) u3
Known setminus_Subq^{setminus_Subq} : ∀ x0 x1 . setminus x0 x1 ⊆ x0
Theorem 51de2.. : ∀ x0 : ι → ι → ο . (∀ x1 x2 . x0 x1 x2 ⟶ x0 x2 x1) ⟶ (∀ x1 . x1 ⊆ u18 ⟶ atleastp u3 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ x0 x2 x3)) ⟶ (∀ x1 . x1 ⊆ u18 ⟶ atleastp u6 x1 ⟶ not (∀ x2 . x2 ∈ x1 ⟶ ∀ x3 . x3 ∈ x1 ⟶ (x2 = x3 ⟶ ∀ x4 : ο . x4) ⟶ not (x0 x2 x3))) ⟶ ∀ x1 . x1 ∈ u18 ⟶ and (equip {x2 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1))|equip (binintersect (DirGraphOutNeighbors u18 x0 x2) (DirGraphOutNeighbors u18 x0 x1)) u1} u4) (equip {x2 ∈ setminus u18 (binunion (DirGraphOutNeighbors u18 x0 x1) (Sing x1))|equip (binintersect (DirGraphOutNeighbors u18 x0 x2) (DirGraphOutNeighbors u18 x0 x1)) u2} u8) (proof)