Proofgold Address

Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Param int^int : ι
Param SNoS_^SNoS_ : ι → ι
Param omega^omega : ι
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param mul_SNo^mul_SNo : ι → ι → ι
Param eps_^eps_ : ι → ι
Definition diadic_rational_p^{diadic_rational_p} := λ x0 . ∀ x1 : ο . (∀ x2 . and (x2 ∈ omega) (∀ x3 : ο . (∀ x4 . and (x4 ∈ int) (x0 = mul_SNo (eps_ x2) x4) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1
Known diadic_rational_p_SNoS_omega^{diadic_rational_p_SNoS_omega} : ∀ x0 . diadic_rational_p x0 ⟶ x0 ∈ SNoS_ omega
Known int_diadic_rational_p^{int_diadic_rational_p} : ∀ x0 . x0 ∈ int ⟶ diadic_rational_p x0
Theorem Subq_int_SNoS_omega^{Subq_int_SNoS_omega} : int ⊆ SNoS_ omega (proof)
Param Sep^Sep : ι → (ι → ο) → ι
Param real^real : ι
Param setminus^setminus : ι → ι → ι
Param Sing^Sing : ι → ι
Param recip_SNo^recip_SNo : ι → ι
Definition div_SNo^div_SNo := λ x0 x1 . mul_SNo x0 (recip_SNo x1)
Definition rational^rational := {x0 ∈ real|∀ x1 : ο . (∀ x2 . and (x2 ∈ int) (∀ x3 : ο . (∀ x4 . and (x4 ∈ setminus omega (Sing 0)) (x0 = div_SNo x2 x4) ⟶ x3) ⟶ x3) ⟶ x1) ⟶ x1}
Known SNoS_omega_diadic_rational_p^{SNoS_omega_diadic_rational_p} : ∀ x0 . x0 ∈ SNoS_ omega ⟶ diadic_rational_p x0
Param nat_p^nat_p : ι → ο
Known SepI^SepI : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ x0 ⟶ x1 x2 ⟶ x2 ∈ Sep x0 x1
Known SNoS_omega_real^{SNoS_omega_real} : SNoS_ omega ⊆ real
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Param exp_SNo_nat^exp_SNo_nat : ι → ι → ι
Param ordsucc^ordsucc : ι → ι
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
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 nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known nat_exp_SNo_nat^{nat_exp_SNo_nat} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (exp_SNo_nat x0 x1)
Known nat_2^nat_2 : nat_p 2
Known neq_1_0^neq_1_0 : 1 = 0 ⟶ ∀ x0 : ο . x0
Known mul_SNo_eps_power_2^{mul_SNo_eps_power_2} : ∀ x0 . nat_p x0 ⟶ mul_SNo (eps_ x0) (exp_SNo_nat 2 x0) = 1
Param SNo^SNo : ι → ο
Known mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Known SNo_eps_^SNo_eps_ : ∀ x0 . x0 ∈ omega ⟶ SNo (eps_ x0)
Known SingE^SingE : ∀ x0 x1 . x1 ∈ Sing x0 ⟶ x1 = x0
Known recip_SNo_pow_2^{recip_SNo_pow_2} : ∀ x0 . nat_p x0 ⟶ recip_SNo (exp_SNo_nat 2 x0) = eps_ x0
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
Known int_SNo^int_SNo : ∀ x0 . x0 ∈ int ⟶ SNo x0
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Theorem Subq_SNoS_omega_rational^{Subq_SNoS_omega_rational} : SNoS_ omega ⊆ rational (proof)
Known Sep_Subq^Sep_Subq : ∀ x0 . ∀ x1 : ι → ο . Sep x0 x1 ⊆ x0
Theorem Subq_rational_real^{Subq_rational_real} : rational ⊆ real (proof)
Param minus_SNo^minus_SNo : ι → ι
Known SepE^SepE : ∀ x0 . ∀ x1 : ι → ο . ∀ x2 . x2 ∈ Sep x0 x1 ⟶ and (x2 ∈ x0) (x1 x2)
Known setminusE^setminusE : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ and (x2 ∈ x0) (nIn x2 x1)
Known real_minus_SNo^{real_minus_SNo} : ∀ x0 . x0 ∈ real ⟶ minus_SNo x0 ∈ real
Known int_minus_SNo^{int_minus_SNo} : ∀ x0 . x0 ∈ int ⟶ minus_SNo x0 ∈ int
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 real_SNo^real_SNo : ∀ x0 . x0 ∈ real ⟶ SNo x0
Known SNo_div_SNo^SNo_div_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (div_SNo x0 x1)
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 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
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Known omega_SNo^omega_SNo : ∀ x0 . x0 ∈ omega ⟶ SNo x0
Theorem rational_minus_SNo^{rational_minus_SNo} : ∀ x0 . x0 ∈ rational ⟶ minus_SNo x0 ∈ rational (proof)
Param add_SNo^add_SNo : ι → ι → ι
Definition nat_pair^nat_pair := λ x0 x1 . mul_SNo (exp_SNo_nat 2 x0) (add_SNo (mul_SNo 2 x1) 1)
Known mul_SNo_In_omega^{mul_SNo_In_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_SNo x0 x1 ∈ omega
Known add_SNo_In_omega^{add_SNo_In_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ add_SNo x0 x1 ∈ omega
Known nat_1^nat_1 : nat_p 1
Theorem nat_pair_In_omega^{nat_pair_In_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ nat_pair x0 x1 ∈ omega (proof)
Param even_nat^even_nat : ι → ο
Param mul_nat^mul_nat : ι → ι → ι
Known mul_nat_mul_SNo^{mul_nat_mul_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_nat x0 x1 = mul_SNo x0 x1
Known even_nat_double^{even_nat_double} : ∀ x0 . nat_p x0 ⟶ even_nat (mul_nat 2 x0)
Theorem even_nat_2x : ∀ x0 . x0 ∈ omega ⟶ even_nat (mul_SNo 2 x0) (proof)
Param odd_nat^odd_nat : ι → ο
Param ordinal^ordinal : ι → ο
Known ordinal_ordsucc_SNo_eq^{ordinal_ordsucc_SNo_eq} : ∀ x0 . ordinal x0 ⟶ ordsucc x0 = add_SNo 1 x0
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known even_nat_odd_nat_S^{even_nat_odd_nat_S} : ∀ x0 . even_nat x0 ⟶ odd_nat (ordsucc x0)
Theorem odd_nat_1p2x : ∀ x0 . x0 ∈ omega ⟶ odd_nat (add_SNo 1 (mul_SNo 2 x0)) (proof)
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Known SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Known SNo_2^SNo_2 : SNo 2
Known SNo_1^SNo_1 : SNo 1
Theorem odd_nat_2xp1 : ∀ x0 . x0 ∈ omega ⟶ odd_nat (add_SNo (mul_SNo 2 x0) 1) (proof)
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_inv_impred^{nat_inv_impred} : ∀ x0 : ι → ο . x0 0 ⟶ (∀ x1 . nat_p x1 ⟶ x0 (ordsucc x1)) ⟶ ∀ x1 . nat_p x1 ⟶ x0 x1
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known even_nat_not_odd_nat^{even_nat_not_odd_nat} : ∀ x0 . even_nat x0 ⟶ not (odd_nat x0)
Known exp_SNo_nat_S^{exp_SNo_nat_S} : ∀ x0 . SNo x0 ⟶ ∀ x1 . nat_p x1 ⟶ exp_SNo_nat x0 (ordsucc x1) = mul_SNo x0 (exp_SNo_nat 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 SNo_exp_SNo_nat^{SNo_exp_SNo_nat} : ∀ x0 . SNo x0 ⟶ ∀ x1 . nat_p x1 ⟶ SNo (exp_SNo_nat x0 x1)
Known SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Known nat_p_SNo^nat_p_SNo : ∀ x0 . nat_p x0 ⟶ SNo x0
Known exp_SNo_nat_0^{exp_SNo_nat_0} : ∀ x0 . SNo x0 ⟶ exp_SNo_nat x0 0 = 1
Known mul_SNo_oneL^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 x0 = x0
Param SNoLt^SNoLt : ι → ι → ο
Known ordinal_SNoLt_In^{ordinal_SNoLt_In} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ SNoLt x0 x1 ⟶ x0 ∈ x1
Known pos_mul_SNo_Lt'^{pos_mul_SNo_Lt} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt 0 x2 ⟶ SNoLt x0 x1 ⟶ SNoLt (mul_SNo x0 x2) (mul_SNo x1 x2)
Known nat_ordsucc^nat_ordsucc : ∀ x0 . nat_p x0 ⟶ nat_p (ordsucc x0)
Param SNoLe^SNoLe : ι → ι → ο
Known SNoLeLt_tra^SNoLeLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known SNo_0^SNo_0 : SNo 0
Known mul_SNo_nonneg_nonneg^{mul_SNo_nonneg_nonneg} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe 0 x0 ⟶ SNoLe 0 x1 ⟶ SNoLe 0 (mul_SNo x0 x1)
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Known SNoLt_0_2^SNoLt_0_2 : SNoLt 0 2
Known omega_nonneg^omega_nonneg : ∀ x0 . x0 ∈ omega ⟶ SNoLe 0 x0
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
Known add_SNo_Lt2^add_SNo_Lt2 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x1 x2 ⟶ SNoLt (add_SNo x0 x1) (add_SNo x0 x2)
Known SNoLt_0_1^SNoLt_0_1 : SNoLt 0 1
Known exp_SNo_nat_pos^{exp_SNo_nat_pos} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ ∀ x1 . nat_p x1 ⟶ SNoLt 0 (exp_SNo_nat x0 x1)
Known SNoLt_1_2^SNoLt_1_2 : SNoLt 1 2
Known neq_2_0^neq_2_0 : 2 = 0 ⟶ ∀ x0 : ο . x0
Theorem nat_pair_0^nat_pair_0 : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ ∀ x2 . x2 ∈ omega ⟶ ∀ x3 . x3 ∈ omega ⟶ nat_pair x0 x1 = nat_pair x2 x3 ⟶ x0 = x2 (proof)
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
Theorem nat_pair_1^nat_pair_1 : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ ∀ x2 . x2 ∈ omega ⟶ ∀ x3 . x3 ∈ omega ⟶ nat_pair x0 x1 = nat_pair x2 x3 ⟶ x1 = x3 (proof)
Param equip^equip : ι → ι → ο
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 2c48a..^{atleastp_antisym_equip} : ∀ x0 x1 . atleastp x0 x1 ⟶ atleastp x1 x0 ⟶ equip x0 x1
Known Subq_atleastp^{Subq_atleastp} : ∀ x0 x1 . x0 ⊆ x1 ⟶ atleastp x0 x1
Known Subq_tra^Subq_tra : ∀ x0 x1 x2 . x0 ⊆ x1 ⟶ x1 ⊆ x2 ⟶ x0 ⊆ x2
Known Subq_omega_int^{Subq_omega_int} : omega ⊆ int
Param If_i^If_i : ο → ι → ι → ι
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 x0)
Known minus_SNo_Lt_contra2^{minus_SNo_Lt_contra2} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt x0 (minus_SNo x1) ⟶ SNoLt x1 (minus_SNo x0)
Known minus_SNo_0^minus_SNo_0 : minus_SNo 0 = 0
Known nat_0^nat_0 : nat_p 0
Known minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
Known neq_0_1^neq_0_1 : 0 = 1 ⟶ ∀ x0 : ο . x0
Known If_i_0^If_i_0 : ∀ x0 : ο . ∀ x1 x2 . not x0 ⟶ If_i x0 x1 x2 = x2
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Known SNoLtLe_tra^SNoLtLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLt x0 x2
Known If_i_1^If_i_1 : ∀ x0 : ο . ∀ x1 x2 . x0 ⟶ If_i x0 x1 x2 = x1
Known Eps_i_ex^Eps_i_ex : ∀ x0 : ι → ο . (∀ x1 : ο . (∀ x2 . x0 x2 ⟶ x1) ⟶ x1) ⟶ x0 (prim0 x0)
Known int_3_cases^int_3_cases : ∀ x0 . x0 ∈ int ⟶ ∀ x1 : ο . (∀ x2 . x2 ∈ omega ⟶ x0 = minus_SNo (ordsucc x2) ⟶ x1) ⟶ (x0 = 0 ⟶ x1) ⟶ (∀ x2 . x2 ∈ omega ⟶ x0 = ordsucc x2 ⟶ x1) ⟶ x1
Known mul_SNo_zeroL^{mul_SNo_zeroL} : ∀ x0 . SNo x0 ⟶ mul_SNo 0 x0 = 0
Known SNo_recip_SNo^{SNo_recip_SNo} : ∀ x0 . SNo x0 ⟶ SNo (recip_SNo x0)
Known recip_SNo_of_pos_is_pos^{recip_SNo_of_pos_is_pos} : ∀ x0 . SNo x0 ⟶ SNoLt 0 x0 ⟶ SNoLt 0 (recip_SNo x0)
Known SNoLeE^SNoLeE : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ or (SNoLt x0 x1) (x0 = x1)
Known minus_SNo_Lt_contra1^{minus_SNo_Lt_contra1} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt (minus_SNo x0) x1 ⟶ SNoLt (minus_SNo x1) x0
Known omega_ordsucc^{omega_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ ordsucc x0 ∈ omega
Known ordinal_In_SNoLt^{ordinal_In_SNoLt} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ SNoLt x1 x0
Known nat_0_in_ordsucc^{nat_0_in_ordsucc} : ∀ x0 . nat_p x0 ⟶ 0 ∈ ordsucc x0
Theorem form100_3^form100_3 : equip omega rational (proof)