Proofgold Signed Transaction

Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
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)
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Theorem 71c93..^injI : ∀ x0 x1 . ∀ x2 : ι → ι . (∀ x3 . x3 ∈ x0 ⟶ x2 x3 ∈ x1) ⟶ (∀ x3 . x3 ∈ x0 ⟶ ∀ x4 . x4 ∈ x0 ⟶ x2 x3 = x2 x4 ⟶ x3 = x4) ⟶ inj x0 x1 x2 (proof)
Param nat_p^nat_p : ι → ο
Param add_nat^add_nat : ι → ι → ι
Known add_nat_com^add_nat_com : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ add_nat x0 x1 = add_nat x1 x0
Known nat_p_trans^nat_p_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ nat_p x1
Known 85b3a..^add_nat_In_R : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ∀ x2 . nat_p x2 ⟶ add_nat x1 x2 ∈ add_nat x0 x2
Theorem eaa26..^add_nat_In_L : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ ∀ x2 . x2 ∈ x1 ⟶ add_nat x0 x2 ∈ add_nat x0 x1 (proof)
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param ordinal^ordinal : ι → ο
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
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 add_nat_p^add_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (add_nat x0 x1)
Known set_ext^set_ext : ∀ x0 x1 . x0 ⊆ x1 ⟶ x1 ⊆ x0 ⟶ x0 = x1
Known Subq_ref^Subq_ref : ∀ x0 . x0 ⊆ x0
Theorem d59a0..^{add_nat_Subq_R} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ x0 ⊆ x1 ⟶ ∀ x2 . nat_p x2 ⟶ add_nat x0 x2 ⊆ add_nat x1 x2 (proof)
Theorem 586cf..^{add_nat_Subq_L} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ ∀ x2 . nat_p x2 ⟶ x1 ⊆ x2 ⟶ add_nat x0 x1 ⊆ add_nat x0 x2 (proof)
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 add_nat_SR^add_nat_SR : ∀ x0 x1 . nat_p x1 ⟶ add_nat x0 (ordsucc x1) = ordsucc (add_nat x0 x1)
Known ordsucc_inj^ordsucc_inj : ∀ x0 x1 . ordsucc x0 = ordsucc x1 ⟶ x0 = x1
Theorem 1c3ee..^{add_nat_cancel_R} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ ∀ x2 . nat_p x2 ⟶ add_nat x0 x2 = add_nat x1 x2 ⟶ x0 = x1 (proof)
Param mul_nat^mul_nat : ι → ι → ι
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
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 EmptyE^EmptyE : ∀ x0 . nIn x0 0
Known In_no2cycle^In_no2cycle : ∀ x0 x1 . x0 ∈ x1 ⟶ x1 ∈ x0 ⟶ False
Known In_0_1^In_0_1 : 0 ∈ 1
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
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_ordsucc^nat_ordsucc : ∀ x0 . nat_p x0 ⟶ nat_p (ordsucc x0)
Known mul_nat_p^mul_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (mul_nat 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 nat_0_in_ordsucc^{nat_0_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ 0 ∈ ordsucc x0
Theorem 5d96a..^{mul_nat_0m_1n_In} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ 0 ∈ x0 ⟶ 1 ∈ x1 ⟶ x0 ∈ mul_nat x0 x1 (proof)
Known Subq_tra^Subq_tra : ∀ x0 x1 x2 . x0 ⊆ x1 ⟶ x1 ⊆ x2 ⟶ x0 ⊆ x2
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 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
Theorem ad387.. : ∀ x0 x1 . nat_p x0 ⟶ nat_p x1 ⟶ x0 ⊆ x1 ⟶ mul_nat x0 x0 ⊆ mul_nat x1 x1 (proof)
Param exp_nat^exp_nat : ι → ι → ι
Known caaf4..^exp_nat_S : ∀ x0 x1 . nat_p x1 ⟶ exp_nat x0 (ordsucc x1) = mul_nat x0 (exp_nat x0 x1)
Known nat_1^nat_1 : nat_p 1
Known 3e9f7..^exp_nat_1 : ∀ x0 . exp_nat x0 1 = x0
Theorem 81ebf.. : ∀ x0 . exp_nat x0 2 = mul_nat x0 x0 (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
Known equip_0_Empty^{equip_0_Empty} : ∀ x0 . equip x0 0 ⟶ x0 = 0
Known 856b4..^exp_nat_0 : ∀ x0 . exp_nat x0 0 = 1
Param Sing^Sing : ι → ι
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 mul_nat_com^mul_nat_com : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ mul_nat x0 x1 = mul_nat x1 x0
Known nat_2^nat_2 : nat_p 2
Known add_nat_1_1_2^{add_nat_1_1_2} : add_nat 1 1 = 2
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)
Known mul_nat_1R^mul_nat_1R : ∀ x0 . mul_nat x0 1 = x0
Known equip_sym^equip_sym : ∀ x0 x1 . equip x0 x1 ⟶ equip x1 x0
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Param setminus^setminus : ι → ι → ι
Known 9c223..^{equip_ordsucc_remove1} : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ equip x0 (ordsucc x1) ⟶ equip (setminus x0 (Sing x2)) x1
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 xm^xm : ∀ x0 : ο . or x0 (not x0)
Known If_i_1^If_i_1 : ∀ x0 : ο . ∀ x1 x2 . x0 ⟶ If_i x0 x1 x2 = x1
Known PowerI^PowerI : ∀ x0 x1 . x1 ⊆ x0 ⟶ x1 ∈ prim4 x0
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 PowerE^PowerE : ∀ x0 x1 . x1 ∈ prim4 x0 ⟶ x1 ⊆ x0
Known If_i_0^If_i_0 : ∀ x0 : ο . ∀ x1 x2 . not x0 ⟶ If_i x0 x1 x2 = x2
Known add_nat_Subq_L^{add_nat_Subq_R} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ x0 ⊆ add_nat x0 x1
Known SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = 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 SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Known nat_Subq_add_ex^{nat_Subq_add_ex} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ x0 ⊆ x1 ⟶ ∀ x2 : ο . (∀ x3 . and (nat_p x3) (x1 = add_nat x3 x0) ⟶ x2) ⟶ x2
Param binunion^binunion : ι → ι → ι
Known binunion_Subq_min^{binunion_Subq_min} : ∀ x0 x1 x2 . x0 ⊆ x2 ⟶ x1 ⊆ x2 ⟶ binunion x0 x1 ⊆ x2
Known setminus_Subq^{setminus_Subq} : ∀ x0 x1 . setminus x0 x1 ⊆ x0
Known binunionI2^binunionI2 : ∀ x0 x1 x2 . x2 ∈ x1 ⟶ x2 ∈ binunion x0 x1
Known binunionE^binunionE : ∀ x0 x1 x2 . x2 ∈ binunion x0 x1 ⟶ or (x2 ∈ x0) (x2 ∈ x1)
Known binunionI1^binunionI1 : ∀ x0 x1 x2 . x2 ∈ x0 ⟶ x2 ∈ binunion x0 x1
Known 4f402..^exp_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (exp_nat x0 x1)
Theorem b4d9c..^{equip_finite_Power} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . equip x1 x0 ⟶ equip (prim4 x1) (exp_nat 2 x0) (proof)
Param infinite^infinite : ι → ο
Param omega^omega : ι
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
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 omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Known omega_ordsucc^{omega_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ ordsucc x0 ∈ omega
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known ordinal_ordsucc_In_Subq^{ordinal_ordsucc_In_Subq} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ⊆ x0
Theorem c1cc5..^{infiniteRamsey_lem} : ∀ x0 . ∀ x1 x2 x3 : ι → ι . infinite x0 ⟶ (∀ x4 . x4 ⊆ x0 ⟶ infinite x4 ⟶ and (x1 x4 ⊆ x4) (infinite (x1 x4))) ⟶ (∀ x4 . x4 ⊆ x0 ⟶ infinite x4 ⟶ and (x2 x4 ∈ x4) (nIn (x2 x4) (x1 x4))) ⟶ x3 0 = x1 x0 ⟶ (∀ x4 . nat_p x4 ⟶ x3 (ordsucc x4) = x1 (x3 x4)) ⟶ and (and (∀ x4 . nat_p x4 ⟶ and (x3 x4 ⊆ x0) (infinite (x3 x4))) (∀ x4 . x4 ∈ omega ⟶ ∀ x5 . x5 ∈ omega ⟶ x4 ⊆ x5 ⟶ x3 x5 ⊆ x3 x4)) (∀ x4 . x4 ∈ omega ⟶ ∀ x5 . x5 ∈ omega ⟶ x2 (x3 x4) = x2 (x3 x5) ⟶ x4 = x5) (proof)
Param SNo^SNo : ι → ο
Param int^int : ι
Param mul_SNo^mul_SNo : ι → ι → ι
Definition divides_int^divides_int := λ x0 x1 . and (and (x0 ∈ int) (x1 ∈ int)) (∀ x2 : ο . (∀ x3 . and (x3 ∈ int) (mul_SNo x0 x3 = x1) ⟶ x2) ⟶ x2)
Param add_SNo^add_SNo : ι → ι → ι
Param minus_SNo^minus_SNo : ι → ι
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Known minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
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 divides_int_minus_SNo^{divides_int_minus_SNo} : ∀ x0 x1 . divides_int x0 x1 ⟶ divides_int x0 (minus_SNo x1)
Theorem 03b8a.. : ∀ x0 x1 x2 . SNo x1 ⟶ SNo x2 ⟶ divides_int x0 (add_SNo x1 (minus_SNo x2)) ⟶ divides_int x0 (add_SNo x2 (minus_SNo x1)) (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)
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 SNoLt^SNoLt : ι → ι → ο
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Param SNoLe^SNoLe : ι → ι → ο
Known SNoLtLe_tra^SNoLtLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLt x0 x2
Known SNo_1^SNo_1 : SNo 1
Known nat_p_SNo^nat_p_SNo : ∀ x0 . nat_p x0 ⟶ SNo x0
Known ordinal_In_SNoLt^{ordinal_In_SNoLt} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ SNoLt x1 x0
Known divides_int_pos_Le^{divides_int_pos_Le} : ∀ x0 x1 . divides_int x0 x1 ⟶ SNoLt 0 x1 ⟶ SNoLe x0 x1
Known SNoLt_0_1^SNoLt_0_1 : SNoLt 0 1
Theorem 10be6..^{prime_not_divides_int_1} : ∀ x0 . prime_nat x0 ⟶ not (divides_int x0 1) (proof)
Param nat_primrec^nat_primrec : ι → (ι → ι → ι) → ι → ι
Definition 05ecb..^Pi_SNo := λ x0 : ι → ι . nat_primrec 1 (λ x1 x2 . mul_SNo x2 (x0 x1))
Known nat_primrec_0^{nat_primrec_0} : ∀ x0 . ∀ x1 : ι → ι → ι . nat_primrec x0 x1 0 = x0
Theorem b6e18..^Pi_SNo_0 : ∀ x0 : ι → ι . 05ecb.. 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 33475..^Pi_SNo_S : ∀ x0 : ι → ι . ∀ x1 . nat_p x1 ⟶ 05ecb.. x0 (ordsucc x1) = mul_SNo (05ecb.. x0 x1) (x0 x1) (proof)
Known mul_SNo_In_omega^{mul_SNo_In_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_SNo x0 x1 ∈ omega
Known ordsuccI1^ordsuccI1 : ∀ x0 . x0 ⊆ ordsucc x0
Theorem 63bfe..^{Pi_SNo_In_omega} : ∀ x0 : ι → ι . ∀ x1 . nat_p x1 ⟶ (∀ x2 . x2 ∈ x1 ⟶ x0 x2 ∈ omega) ⟶ 05ecb.. x0 x1 ∈ omega (proof)
Known Subq_omega_int^{Subq_omega_int} : omega ⊆ int
Known int_mul_SNo^int_mul_SNo : ∀ x0 . x0 ∈ int ⟶ ∀ x1 . x1 ∈ int ⟶ mul_SNo x0 x1 ∈ int
Theorem 22c84..^{Pi_SNo_In_int} : ∀ x0 : ι → ι . ∀ x1 . nat_p x1 ⟶ (∀ x2 . x2 ∈ x1 ⟶ x0 x2 ∈ int) ⟶ 05ecb.. x0 x1 ∈ int (proof)
Known divides_int_divides_nat^{divides_int_divides_nat} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ divides_int x0 x1 ⟶ divides_nat x0 x1
Theorem d07df..^{divides_int_prime_nat_eq} : ∀ x0 x1 . prime_nat x0 ⟶ prime_nat x1 ⟶ divides_int x0 x1 ⟶ x0 = x1 (proof)
Known Euclid_lemma^Euclid_lemma : ∀ x0 . prime_nat x0 ⟶ ∀ x1 . x1 ∈ int ⟶ ∀ x2 . x2 ∈ int ⟶ divides_int x0 (mul_SNo x1 x2) ⟶ or (divides_int x0 x1) (divides_int x0 x2)
Theorem 2326d..^{Euclid_lemma_Pi_SNo} : ∀ x0 : ι → ι . ∀ x1 . prime_nat x1 ⟶ ∀ x2 . nat_p x2 ⟶ (∀ x3 . x3 ∈ x2 ⟶ x0 x3 ∈ int) ⟶ divides_int x1 (05ecb.. x0 x2) ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ x2) (divides_int x1 (x0 x4)) ⟶ x3) ⟶ x3 (proof)
Known mul_nat_mul_SNo^{mul_nat_mul_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_nat x0 x1 = mul_SNo x0 x1
Theorem a297e..^{divides_nat_mul_SNo_R} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ divides_nat x0 (mul_SNo x0 x1) (proof)
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
Known omega_SNo^omega_SNo : ∀ x0 . x0 ∈ omega ⟶ SNo x0
Theorem a0cc9..^{divides_nat_mul_SNo_L} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ divides_nat x1 (mul_SNo x0 x1) (proof)
Known ordsuccE^ordsuccE : ∀ x0 x1 . x1 ∈ ordsucc x0 ⟶ or (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 3899c..^{Pi_SNo_divides} : ∀ x0 : ι → ι . ∀ x1 . nat_p x1 ⟶ (∀ x2 . x2 ∈ x1 ⟶ x0 x2 ∈ omega) ⟶ ∀ x2 . x2 ∈ x1 ⟶ divides_nat (x0 x2) (05ecb.. x0 x1) (proof)
Theorem 9a5d6..^Pi_SNo_eq : ∀ x0 x1 : ι → ι . ∀ x2 . nat_p x2 ⟶ (∀ x3 . x3 ∈ x2 ⟶ x0 x3 = x1 x3) ⟶ 05ecb.. x0 x2 = 05ecb.. x1 x2 (proof)
Definition 168aa..^{nonincrfinseq} := λ x0 : ι → ο . λ x1 . λ x2 : ι → ι . ∀ x3 . x3 ∈ x1 ⟶ and (x0 (x2 x3)) (∀ x4 . x4 ∈ x3 ⟶ SNoLe (x2 x3) (x2 x4))
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 nat_0^nat_0 : nat_p 0
Known Empty_eq^Empty_eq : ∀ x0 . (∀ x1 . nIn x1 x0) ⟶ x0 = 0
Known divides_nat_divides_int^{divides_nat_divides_int} : ∀ x0 x1 . divides_nat x0 x1 ⟶ divides_int x0 x1
Known least_ordinal_ex^{least_ordinal_ex} : ∀ x0 : ι → ο . (∀ x1 : ο . (∀ x2 . and (ordinal x2) (x0 x2) ⟶ x1) ⟶ x1) ⟶ ∀ x1 : ο . (∀ x2 . and (and (ordinal x2) (x0 x2)) (∀ x3 . x3 ∈ x2 ⟶ not (x0 x3)) ⟶ x1) ⟶ x1
Known 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
Known cases_1^cases_1 : ∀ x0 . x0 ∈ 1 ⟶ ∀ x1 : ι → ο . x1 0 ⟶ x1 x0
Known ordinal_SNoLt_In^{ordinal_SNoLt_In} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ SNoLt x0 x1 ⟶ x0 ∈ x1
Known mul_SNo_oneR^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = x0
Known pos_mul_SNo_Lt^{pos_mul_SNo_Lt} : ∀ x0 x1 x2 . SNo x0 ⟶ SNoLt 0 x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x1 x2 ⟶ SNoLt (mul_SNo x0 x1) (mul_SNo x0 x2)
Known ordinal_Empty^{ordinal_Empty} : ordinal 0
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe 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 mul_SNo_nonzero_cancel^{mul_SNo_nonzero_cancel_L} : ∀ x0 x1 x2 . SNo x0 ⟶ (x0 = 0 ⟶ ∀ x3 : ο . x3) ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 x1 = mul_SNo x0 x2 ⟶ x1 = x2
Known SNoLt_tra^SNoLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known SNo_0^SNo_0 : SNo 0
Known SNoLt_trichotomy_or_impred^{SNoLt_trichotomy_or_impred} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ ∀ x2 : ο . (SNoLt x0 x1 ⟶ x2) ⟶ (x0 = x1 ⟶ x2) ⟶ (SNoLt x1 x0 ⟶ x2) ⟶ x2
Known SNoLeE^SNoLeE : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ or (SNoLt x0 x1) (x0 = x1)
Known omega_nonneg^omega_nonneg : ∀ x0 . x0 ∈ omega ⟶ SNoLe 0 x0
Known mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Theorem a96cd..^{prime_factorization_ex_uniq} : ∀ x0 . nat_p x0 ⟶ 0 ∈ x0 ⟶ ∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (∀ x3 : ο . (∀ x4 : ι → ι . and (and (168aa.. prime_nat x2 x4) (05ecb.. x4 x2 = x0)) (∀ x5 . x5 ∈ omega ⟶ ∀ x6 : ι → ι . 168aa.. prime_nat x5 x6 ⟶ 05ecb.. x6 x5 = x0 ⟶ and (x5 = x2) (∀ x7 . x7 ∈ x2 ⟶ x6 x7 = x4 x7)) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1 (proof)
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Known nat_ordsucc_in_ordsucc^{nat_ordsucc_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ∈ ordsucc x0
Theorem 16680..^{nat_le2_cases} : ∀ x0 . nat_p x0 ⟶ x0 ⊆ 2 ⟶ or (or (x0 = 0) (x0 = 1)) (x0 = 2) (proof)
Known 2bbf0..^{mul_nat_0_or_Subq} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ or (x1 = 0) (x0 ⊆ mul_nat x0 x1)
Known neq_2_0^neq_2_0 : 2 = 0 ⟶ ∀ x0 : ο . x0
Known mul_nat_0R^mul_nat_0R : ∀ x0 . mul_nat x0 0 = 0
Known mul_nat_0L^mul_nat_0L : ∀ x0 . nat_p x0 ⟶ mul_nat 0 x0 = 0
Theorem 4e2fe..^{prime_nat_2_lem} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ mul_nat x0 x1 = 2 ⟶ or (x0 = 1) (x0 = 2) (proof)
Known In_1_2^In_1_2 : 1 ∈ 2
Theorem 9dc41..^prime_nat_2 : prime_nat 2 (proof)
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 int_SNo^int_SNo : ∀ x0 . x0 ∈ int ⟶ SNo x0
Known mul_SNo_minus_distrR^{mul_minus_SNo_distrR} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 (minus_SNo x1) = minus_SNo (mul_SNo x0 x1)
Known add_SNo_cancel_L^{add_SNo_cancel_L} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 x1 = add_SNo x0 x2 ⟶ x1 = x2
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Known add_SNo_rotate_3_1^{add_SNo_rotate_3_1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 (add_SNo x1 x2) = add_SNo x2 (add_SNo x0 x1)
Known add_SNo_minus_L2^{add_SNo_minus_L2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo (minus_SNo x0) (add_SNo x0 x1) = x1
Known divides_int_mul_SNo_L^{divides_int_mul_SNo_L} : ∀ x0 x1 x2 . x2 ∈ int ⟶ divides_int x0 x1 ⟶ divides_int x0 (mul_SNo x1 x2)
Known int_add_SNo^int_add_SNo : ∀ x0 . x0 ∈ int ⟶ ∀ x1 . x1 ∈ int ⟶ add_SNo x0 x1 ∈ int
Known int_minus_SNo^{int_minus_SNo} : ∀ x0 . x0 ∈ int ⟶ minus_SNo x0 ∈ int
Known divides_int_ref^{divides_int_ref} : ∀ x0 . x0 ∈ int ⟶ divides_int x0 x0
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Theorem af26f..^{not_eq_2m_2n1} : ∀ x0 . x0 ∈ int ⟶ ∀ x1 . x1 ∈ int ⟶ mul_SNo 2 x0 = add_SNo (mul_SNo 2 x1) 1 ⟶ ∀ x2 : ο . x2 (proof)
Param div_SNo^div_SNo : ι → ι → ι
Known div_SNo_0_denum^{div_SNo_0_denum} : ∀ x0 . SNo x0 ⟶ div_SNo x0 0 = 0
Known SNo_div_SNo^SNo_div_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (div_SNo x0 x1)
Known mul_div_SNo_invR^{mul_div_SNo_invR} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ (x1 = 0 ⟶ ∀ x2 : ο . x2) ⟶ mul_SNo x1 (div_SNo x0 x1) = x0
Theorem a3726..^{divides_int_div_SNo_int} : ∀ x0 x1 . divides_int x0 x1 ⟶ div_SNo x1 x0 ∈ int (proof)
Known div_SNo_pos_LtL^{div_SNo_pos_LtL} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt 0 x1 ⟶ SNoLt x0 (mul_SNo x2 x1) ⟶ SNoLt (div_SNo x0 x1) x2
Known SNoLt_0_2^SNoLt_0_2 : SNoLt 0 2
Known add_SNo_Lt1_cancel^{add_SNo_Lt1_cancel} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x2 x1) ⟶ SNoLt x0 x2
Known add_SNo_minus_SNo_rinv^{add_SNo_minus_SNo_rinv} : ∀ x0 . SNo x0 ⟶ add_SNo x0 (minus_SNo x0) = 0
Known add_SNo_1_1_2^{add_SNo_1_1_2} : add_SNo 1 1 = 2
Known add_SNo_minus_R2^{add_SNo_minus_R2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo (add_SNo x0 x1) (minus_SNo x1) = x0
Known aa7e8..^{nonneg_int_nat_p} : ∀ x0 . x0 ∈ int ⟶ SNoLe 0 x0 ⟶ nat_p x0
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_div_SNo_L^{mul_div_SNo_L} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x2 (div_SNo x0 x1) = div_SNo (mul_SNo x2 x0) x1
Known div_mul_SNo_invL^{div_mul_SNo_invL} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ (x1 = 0 ⟶ ∀ x2 : ο . x2) ⟶ div_SNo (mul_SNo x0 x1) x1 = x0
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Known div_SNo_pos_pos^{div_SNo_pos_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt 0 x0 ⟶ SNoLt 0 x1 ⟶ SNoLt 0 (div_SNo x0 x1)
Known mul_div_SNo_both^{mul_div_SNo_both} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ mul_SNo (div_SNo x0 x1) (div_SNo x2 x3) = div_SNo (mul_SNo x0 x2) (mul_SNo x1 x3)
Known mul_SNo_pos_pos^{mul_SNo_pos_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt 0 x0 ⟶ SNoLt 0 x1 ⟶ SNoLt 0 (mul_SNo x0 x1)
Theorem 9bd18..^{form100_1_lem1} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ mul_SNo x0 x0 = mul_SNo 2 (mul_SNo x1 x1) ⟶ x1 = 0 (proof)
Theorem 1129e..^{form100_1_lem2} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ setminus omega 1 ⟶ mul_SNo x0 x0 = mul_SNo 2 (mul_SNo x1 x1) ⟶ ∀ x2 : ο . x2 (proof)