Proofgold Address

Param nat_p^nat_p : ι → ο
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param add_nat^add_nat : ι → ι → ι
Param ordsucc^ordsucc : 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 add_nat_0R^add_nat_0R : ∀ x0 . add_nat x0 0 = x0
Known Subq_ref^Subq_ref : ∀ x0 . x0 ⊆ x0
Known add_nat_SR^add_nat_SR : ∀ x0 x1 . nat_p x1 ⟶ add_nat x0 (ordsucc x1) = ordsucc (add_nat x0 x1)
Known ordsuccI1^ordsuccI1 : ∀ x0 . x0 ⊆ ordsucc x0
Theorem 4d87b.. : ∀ x0 x1 . nat_p x1 ⟶ x0 ⊆ add_nat x0 x1 (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 nat_0^nat_0 : nat_p 0
Known mul_nat_0R^mul_nat_0R : ∀ x0 . mul_nat x0 0 = 0
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
Known nat_1^nat_1 : nat_p 1
Theorem f11bb.. : ∀ x0 . nat_p x0 ⟶ mul_nat 1 x0 = x0 (proof)
Param nat_primrec^nat_primrec : ι → (ι → ι → ι) → ι → ι
Definition exp_nat^exp_nat := λ x0 . nat_primrec 1 (λ x1 . mul_nat x0)
Known nat_primrec_0^{nat_primrec_0} : ∀ x0 . ∀ x1 : ι → ι → ι . nat_primrec x0 x1 0 = x0
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)
Theorem 4f402..^exp_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (exp_nat x0 x1) (proof)
Param setminus^setminus : ι → ι → ι
Param Sing^Sing : ι → ι
Known PowerI^PowerI : ∀ x0 x1 . x1 ⊆ x0 ⟶ x1 ∈ prim4 x0
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Known PowerE^PowerE : ∀ x0 x1 . x1 ∈ prim4 x0 ⟶ x1 ⊆ x0
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known setminusI^setminusI : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ nIn x2 x1 ⟶ x2 ∈ setminus x0 x1
Known SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Theorem 39f67.. : ∀ x0 x1 . setminus x1 (Sing x0) ∈ prim4 x0 ⟶ x1 ∈ prim4 (ordsucc x0) (proof)
Param binunion^binunion : ι → ι → ι
Known binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Theorem 35356.. : ∀ x0 x1 . x1 ∈ prim4 x0 ⟶ binunion x1 (Sing x0) ∈ prim4 (ordsucc x0) (proof)
Param Inj0^Inj0 : ι → ι
Param Inj1^Inj1 : ι → ι
Param setsum^setsum : ι → ι → ι
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Known setsum_Inj_inv^{setsum_Inj_inv} : ∀ x0 x1 x2 . x2 ∈ setsum x0 x1 ⟶ or (∀ x3 : ο . (∀ x4 . and (x4 ∈ x0) (x2 = Inj0 x4) ⟶ x3) ⟶ x3) (∀ x3 : ο . (∀ x4 . and (x4 ∈ x1) (x2 = Inj1 x4) ⟶ x3) ⟶ x3)
Theorem f4c7c.. : ∀ x0 x1 . ∀ x2 : ι → ο . (∀ x3 . x3 ∈ x0 ⟶ x2 (Inj0 x3)) ⟶ (∀ x3 . x3 ∈ x1 ⟶ x2 (Inj1 x3)) ⟶ ∀ x3 . x3 ∈ setsum x0 x1 ⟶ x2 x3 (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
Known Power_0_Sing_0^{Power_0_Sing_0} : prim4 0 = Sing 0
Known eq_1_Sing0^eq_1_Sing0 : 1 = Sing 0
Known equip_ref^equip_ref : ∀ x0 . equip x0 x0
Known add_nat_1_1_2^{add_nat_1_1_2} : add_nat 1 1 = 2
Known mul_add_nat_distrR^{mul_add_nat_distrR} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ ∀ x2 . nat_p x2 ⟶ mul_nat (add_nat x0 x1) x2 = add_nat (mul_nat x0 x2) (mul_nat x1 x2)
Known equip_sym^equip_sym : ∀ x0 x1 . equip x0 x1 ⟶ equip x1 x0
Known equip_tra^equip_tra : ∀ x0 x1 x2 . equip x0 x1 ⟶ equip x1 x2 ⟶ equip x0 x2
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 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
Param combine_funcs^{combine_funcs} : ι → ι → (ι → ι) → (ι → 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 combine_funcs_eq1^{combine_funcs_eq1} : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 . combine_funcs x0 x1 x2 x3 (Inj0 x4) = x2 x4
Known Power_Subq^Power_Subq : ∀ x0 x1 . x0 ⊆ x1 ⟶ prim4 x0 ⊆ prim4 x1
Known combine_funcs_eq2^{combine_funcs_eq2} : ∀ x0 x1 . ∀ x2 x3 : ι → ι . ∀ x4 . combine_funcs x0 x1 x2 x3 (Inj1 x4) = x3 x4
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Known binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known Inj1_setsum^Inj1_setsum : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ Inj1 x2 ∈ setsum x0 x1
Known eb0c4..^{binunion_remove1_eq} : ∀ x0 x1 . x1 ∈ x0 ⟶ x0 = binunion (setminus x0 (Sing x1)) (Sing 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 Inj0_setsum^Inj0_setsum : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ Inj0 x2 ∈ setsum x0 x1
Known nat_2^nat_2 : nat_p 2
Theorem 293d3.. : ∀ x0 . nat_p x0 ⟶ equip (prim4 x0) (exp_nat 2 x0) (proof)
Definition inj^inj := λ x0 x1 . λ x2 : ι → ι . and (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1) (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4)
Definition atleastp^atleastp := λ x0 x1 . ∀ x2 : ο . (∀ x3 : ι → ι . inj x0 x1 x3 ⟶ x2) ⟶ x2
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Theorem 71267.. : ∀ x0 . atleastp 0 x0 (proof)
Param If_i^If_i : ο → ι → ι → ι
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 nat_ordsucc_in_ordsucc^{nat_ordsucc_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ∈ ordsucc x0
Known nat_trans^nat_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ x1 ⊆ x0
Known ordsucc_inj^ordsucc_inj : ∀ x0 x1 . ordsucc x0 = ordsucc x1 ⟶ x0 = x1
Known nat_ordsucc_trans^{nat_ordsucc_trans} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ ordsucc x0 ⟶ x1 ⊆ x0
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 nat_0_in_ordsucc^{nat_0_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ 0 ∈ ordsucc x0
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 In_no2cycle^In_no2cycle : ∀ x0 x1 . x0 ∈ x1 ⟶ x1 ∈ x0 ⟶ False
Known nat_p_trans^nat_p_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ nat_p x1
Known nat_ordsucc^nat_ordsucc : ∀ x0 . nat_p x0 ⟶ nat_p (ordsucc x0)
Theorem 8ac0e.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ ordsucc x0 ⟶ equip x0 (setminus (ordsucc x0) (Sing x1)) (proof)
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known dneg^dneg : ∀ x0 : ο . not (not x0) ⟶ x0
Known setminusE1^setminusE1 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ x2 ∈ x0
Known setminusE2^setminusE2 : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ nIn x2 x1
Known atleastp_tra^atleastp_tra : ∀ x0 x1 x2 . atleastp x0 x1 ⟶ atleastp x1 x2 ⟶ atleastp x0 x2
Known equip_atleastp^{equip_atleastp} : ∀ x0 x1 . equip x0 x1 ⟶ atleastp x0 x1
Known add_nat_p^add_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (add_nat x0 x1)
Theorem 9da24.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ ∀ x2 x3 . atleastp (add_nat x0 x1) (binunion x2 x3) ⟶ or (atleastp x0 x2) (atleastp x1 x3) (proof)