Proofgold Address

Param struct_b_b_e_e^{struct_b_b_e_e} : ι → ο
Param pack_b_b_e_e^pack_b_b_e_e : ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι → ι
Param ap^ap : ι → ι → ι
Definition decode_b^decode_b := λ x0 x1 . ap (ap x0 x1)
Param ordsucc^ordsucc : ι → ι
Known struct_b_b_e_e_eta : ∀ x0 . struct_b_b_e_e x0 ⟶ x0 = pack_b_b_e_e (ap x0 0) (decode_b (ap x0 1)) (decode_b (ap x0 2)) (ap x0 3) (ap x0 4)
Known pack_struct_b_b_e_e_E1 : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . struct_b_b_e_e (pack_b_b_e_e x0 x1 x2 x3 x4) ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x1 x5 x6 ∈ x0
Known pack_struct_b_b_e_e_E2 : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . struct_b_b_e_e (pack_b_b_e_e x0 x1 x2 x3 x4) ⟶ ∀ x5 . x5 ∈ x0 ⟶ ∀ x6 . x6 ∈ x0 ⟶ x2 x5 x6 ∈ x0
Known pack_struct_b_b_e_e_E3 : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . struct_b_b_e_e (pack_b_b_e_e x0 x1 x2 x3 x4) ⟶ x3 ∈ x0
Known pack_struct_b_b_e_e_E4 : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . struct_b_b_e_e (pack_b_b_e_e x0 x1 x2 x3 x4) ⟶ x4 ∈ x0
Definition field0^{RealsStruct_carrier} := λ x0 . ap x0 0
Definition field1b^{RealsStruct_plus} := λ x0 . decode_b (ap x0 1)
Definition field2b^{RealsStruct_mult} := λ x0 . decode_b (ap x0 2)
Definition field3^Field_zero := λ x0 . ap x0 3
Definition field4^{RealsStruct_zero} := λ x0 . ap x0 4
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param unpack_b_b_e_e_o^{unpack_b_b_e_e_o} : ι → (ι → (ι → ι → ι) → (ι → ι → ι) → ι → ι → ο) → ο
Param explicit_CRing_with_id^{explicit_CRing_with_id} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → ο
Definition CRing_with_id^{CRing_with_id} := λ x0 . and (struct_b_b_e_e x0) (unpack_b_b_e_e_o x0 (λ x1 . λ x2 x3 : ι → ι → ι . λ x4 x5 . explicit_CRing_with_id x1 x4 x5 x2 x3))
Theorem CRing_with_id_eta^{CRing_with_id_eta} : ∀ x0 . CRing_with_id x0 ⟶ x0 = pack_b_b_e_e (field0 x0) (field1b x0) (field2b x0) (field3 x0) (field4 x0) (proof)
Known CRing_with_id_unpack_eq^{CRing_with_id_unpack_eq} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . unpack_b_b_e_e_o (pack_b_b_e_e x0 x1 x2 x3 x4) (λ x6 . λ x7 x8 : ι → ι → ι . λ x9 x10 . explicit_CRing_with_id x6 x9 x10 x7 x8) = explicit_CRing_with_id x0 x3 x4 x1 x2
Theorem CRing_with_id_explicit_CRing_with_id^{CRing_with_id_explicit_CRing_with_id} : ∀ x0 . CRing_with_id x0 ⟶ explicit_CRing_with_id (field0 x0) (field3 x0) (field4 x0) (field1b x0) (field2b x0) (proof)
Theorem CRing_with_id_zero_In^{CRing_with_id_zero_In} : ∀ x0 . CRing_with_id x0 ⟶ field3 x0 ∈ field0 x0 (proof)
Theorem CRing_with_id_one_In^{CRing_with_id_one_In} : ∀ x0 . CRing_with_id x0 ⟶ field4 x0 ∈ field0 x0 (proof)
Theorem CRing_with_id_plus_clos^{CRing_with_id_plus_clos} : ∀ x0 . CRing_with_id x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ field1b x0 x1 x2 ∈ field0 x0 (proof)
Theorem CRing_with_id_mult_clos^{CRing_with_id_mult_clos} : ∀ x0 . CRing_with_id x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ field2b x0 x1 x2 ∈ field0 x0 (proof)
Known explicit_CRing_with_id_E^{explicit_CRing_with_id_E} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ο . (explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x6 x7 ∈ x0) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x3 x6 (x3 x7 x8) = x3 (x3 x6 x7) x8) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x6 x7 = x3 x7 x6) ⟶ x1 ∈ x0 ⟶ (∀ x6 . x6 ∈ x0 ⟶ x3 x1 x6 = x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 : ο . (∀ x8 . and (x8 ∈ x0) (x3 x6 x8 = x1) ⟶ x7) ⟶ x7) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x6 x7 ∈ x0) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x6 (x4 x7 x8) = x4 (x4 x6 x7) x8) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x6 x7 = x4 x7 x6) ⟶ x2 ∈ x0 ⟶ (x2 = x1 ⟶ ∀ x6 : ο . x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ x4 x2 x6 = x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x6 (x3 x7 x8) = x3 (x4 x6 x7) (x4 x6 x8)) ⟶ x5) ⟶ explicit_CRing_with_id x0 x1 x2 x3 x4 ⟶ x5
Theorem CRing_with_id_plus_assoc^{CRing_with_id_plus_assoc} : ∀ x0 . CRing_with_id x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ ∀ x3 . x3 ∈ field0 x0 ⟶ field1b x0 x1 (field1b x0 x2 x3) = field1b x0 (field1b x0 x1 x2) x3 (proof)
Theorem CRing_with_id_plus_com^{CRing_with_id_plus_com} : ∀ x0 . CRing_with_id x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ field1b x0 x1 x2 = field1b x0 x2 x1 (proof)
Theorem CRing_with_id_zero_L^{CRing_with_id_zero_L} : ∀ x0 . CRing_with_id x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ field1b x0 (field3 x0) x1 = x1 (proof)
Theorem CRing_with_id_plus_inv^{CRing_with_id_plus_inv} : ∀ x0 . CRing_with_id x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ field0 x0) (field1b x0 x1 x3 = field3 x0) ⟶ x2) ⟶ x2 (proof)
Theorem CRing_with_id_mult_assoc^{CRing_with_id_mult_assoc} : ∀ x0 . CRing_with_id x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ ∀ x3 . x3 ∈ field0 x0 ⟶ field2b x0 x1 (field2b x0 x2 x3) = field2b x0 (field2b x0 x1 x2) x3 (proof)
Theorem CRing_with_id_mult_com^{CRing_with_id_mult_com} : ∀ x0 . CRing_with_id x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ field2b x0 x1 x2 = field2b x0 x2 x1 (proof)
Theorem CRing_with_id_one_neq_zero^{CRing_with_id_one_neq_zero} : ∀ x0 . CRing_with_id x0 ⟶ field4 x0 = field3 x0 ⟶ ∀ x1 : ο . x1 (proof)
Theorem CRing_with_id_one_L^{CRing_with_id_one_L} : ∀ x0 . CRing_with_id x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ field2b x0 (field4 x0) x1 = x1 (proof)
Theorem CRing_with_id_distr_L^{CRing_with_id_distr_L} : ∀ x0 . CRing_with_id x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ ∀ x3 . x3 ∈ field0 x0 ⟶ field2b x0 x1 (field1b x0 x2 x3) = field1b x0 (field2b x0 x1 x2) (field2b x0 x1 x3) (proof)
Param nat_primrec^nat_primrec : ι → (ι → ι → ι) → ι → ι
Definition CRing_with_id_omega_exp^{CRing_with_id_omega_exp} := λ x0 x1 . nat_primrec (field4 x0) (λ x2 . field2b x0 x1)
Known nat_primrec_0^{nat_primrec_0} : ∀ x0 . ∀ x1 : ι → ι → ι . nat_primrec x0 x1 0 = x0
Theorem CRing_with_id_omega_exp_0^{CRing_with_id_omega_exp_0} : ∀ x0 . CRing_with_id x0 ⟶ ∀ x1 . CRing_with_id_omega_exp x0 x1 0 = field4 x0 (proof)
Param omega^omega : ι
Param nat_p^nat_p : ι → ο
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 omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Theorem CRing_with_id_omega_exp_S^{CRing_with_id_omega_exp_S} : ∀ x0 . CRing_with_id x0 ⟶ ∀ x1 x2 . x2 ∈ omega ⟶ CRing_with_id_omega_exp x0 x1 (ordsucc x2) = field2b x0 x1 (CRing_with_id_omega_exp x0 x1 x2) (proof)
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known nat_0^nat_0 : nat_p 0
Theorem CRing_with_id_omega_exp_1^{CRing_with_id_omega_exp_1} : ∀ x0 . CRing_with_id x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ CRing_with_id_omega_exp x0 x1 1 = 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
Theorem CRing_with_id_omega_exp_clos^{CRing_with_id_omega_exp_clos} : ∀ x0 . CRing_with_id x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ omega ⟶ CRing_with_id_omega_exp x0 x1 x2 ∈ field0 x0 (proof)
Definition CRing_with_id_eval_poly^{CRing_with_id_eval_poly} := λ x0 x1 x2 x3 . nat_primrec (field3 x0) (λ x4 . field1b x0 (field2b x0 (ap x2 x4) (CRing_with_id_omega_exp x0 x3 x4))) x1
Param Pi^Pi : ι → (ι → ι) → ι
Definition setexp^setexp := λ x0 x1 . Pi x1 (λ x2 . x0)
Known ap_Pi^ap_Pi : ∀ x0 . ∀ x1 : ι → ι . ∀ x2 x3 . x2 ∈ Pi x0 x1 ⟶ x3 ∈ x0 ⟶ ap x2 x3 ∈ x1 x3
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known ordsuccI1^ordsuccI1 : ∀ x0 . x0 ⊆ ordsucc x0
Theorem CRing_with_id_eval_poly_clos^{CRing_with_id_eval_poly_clos} : ∀ x0 . CRing_with_id x0 ⟶ ∀ x1 . x1 ∈ omega ⟶ ∀ x2 . x2 ∈ setexp (field0 x0) x1 ⟶ ∀ x3 . x3 ∈ field0 x0 ⟶ CRing_with_id_eval_poly x0 x1 x2 x3 ∈ field0 x0 (proof)
Definition field0^{RealsStruct_carrier} := λ x0 . ap x0 0
Definition field1b^{RealsStruct_plus} := λ x0 . decode_b (ap x0 1)
Definition field2b^{RealsStruct_mult} := λ x0 . decode_b (ap x0 2)
Definition field3^Field_zero := λ x0 . ap x0 3
Definition field4^{RealsStruct_zero} := λ x0 . ap x0 4
Param explicit_Field^{explicit_Field} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → ο
Definition Field^Field := λ x0 . and (struct_b_b_e_e x0) (unpack_b_b_e_e_o x0 (λ x1 . λ x2 x3 : ι → ι → ι . λ x4 x5 . explicit_Field x1 x4 x5 x2 x3))
Theorem Field_eta^Field_eta : ∀ x0 . Field x0 ⟶ x0 = pack_b_b_e_e (field0 x0) (field1b x0) (field2b x0) (field3 x0) (field4 x0) (proof)
Known Field_unpack_eq^{Field_unpack_eq} : ∀ x0 . ∀ x1 x2 : ι → ι → ι . ∀ x3 x4 . unpack_b_b_e_e_o (pack_b_b_e_e x0 x1 x2 x3 x4) (λ x6 . λ x7 x8 : ι → ι → ι . λ x9 x10 . explicit_Field x6 x9 x10 x7 x8) = explicit_Field x0 x3 x4 x1 x2
Theorem Field_explicit_Field^{Field_explicit_Field} : ∀ x0 . Field x0 ⟶ explicit_Field (field0 x0) (field3 x0) (field4 x0) (field1b x0) (field2b x0) (proof)
Theorem Field_zero_In^{Field_zero_In} : ∀ x0 . Field x0 ⟶ field3 x0 ∈ field0 x0 (proof)
Theorem Field_one_In^Field_one_In : ∀ x0 . Field x0 ⟶ field4 x0 ∈ field0 x0 (proof)
Theorem Field_plus_clos^{Field_plus_clos} : ∀ x0 . Field x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ field1b x0 x1 x2 ∈ field0 x0 (proof)
Theorem Field_mult_clos^{Field_mult_clos} : ∀ x0 . Field x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ field2b x0 x1 x2 ∈ field0 x0 (proof)
Known explicit_Field_E^{explicit_Field_E} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ο . (explicit_Field x0 x1 x2 x3 x4 ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x6 x7 ∈ x0) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x3 x6 (x3 x7 x8) = x3 (x3 x6 x7) x8) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x3 x6 x7 = x3 x7 x6) ⟶ x1 ∈ x0 ⟶ (∀ x6 . x6 ∈ x0 ⟶ x3 x1 x6 = x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 : ο . (∀ x8 . and (x8 ∈ x0) (x3 x6 x8 = x1) ⟶ x7) ⟶ x7) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x6 x7 ∈ x0) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x6 (x4 x7 x8) = x4 (x4 x6 x7) x8) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x4 x6 x7 = x4 x7 x6) ⟶ x2 ∈ x0 ⟶ (x2 = x1 ⟶ ∀ x6 : ο . x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ x4 x2 x6 = x6) ⟶ (∀ x6 . x6 ∈ x0 ⟶ (x6 = x1 ⟶ ∀ x7 : ο . x7) ⟶ ∀ x7 : ο . (∀ x8 . and (x8 ∈ x0) (x4 x6 x8 = x2) ⟶ x7) ⟶ x7) ⟶ (∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x4 x6 (x3 x7 x8) = x3 (x4 x6 x7) (x4 x6 x8)) ⟶ x5) ⟶ explicit_Field x0 x1 x2 x3 x4 ⟶ x5
Theorem Field_plus_assoc^{Field_plus_assoc} : ∀ x0 . Field x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ ∀ x3 . x3 ∈ field0 x0 ⟶ field1b x0 x1 (field1b x0 x2 x3) = field1b x0 (field1b x0 x1 x2) x3 (proof)
Theorem Field_plus_com^{Field_plus_com} : ∀ x0 . Field x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ field1b x0 x1 x2 = field1b x0 x2 x1 (proof)
Theorem Field_zero_L^Field_zero_L : ∀ x0 . Field x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ field1b x0 (field3 x0) x1 = x1 (proof)
Theorem Field_plus_inv^{Field_plus_inv} : ∀ x0 . Field x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ field0 x0) (field1b x0 x1 x3 = field3 x0) ⟶ x2) ⟶ x2 (proof)
Theorem Field_mult_assoc^{Field_mult_assoc} : ∀ x0 . Field x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ ∀ x3 . x3 ∈ field0 x0 ⟶ field2b x0 x1 (field2b x0 x2 x3) = field2b x0 (field2b x0 x1 x2) x3 (proof)
Theorem Field_mult_com^{Field_mult_com} : ∀ x0 . Field x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ field2b x0 x1 x2 = field2b x0 x2 x1 (proof)
Theorem Field_one_neq_zero^{Field_one_neq_zero} : ∀ x0 . Field x0 ⟶ field4 x0 = field3 x0 ⟶ ∀ x1 : ο . x1 (proof)
Theorem Field_one_L^Field_one_L : ∀ x0 . Field x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ field2b x0 (field4 x0) x1 = x1 (proof)
Theorem Field_mult_inv_L^{Field_mult_inv_L} : ∀ x0 . Field x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ (x1 = field3 x0 ⟶ ∀ x2 : ο . x2) ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ field0 x0) (field2b x0 x1 x3 = field4 x0) ⟶ x2) ⟶ x2 (proof)
Theorem Field_distr_L^{Field_distr_L} : ∀ x0 . Field x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ ∀ x3 . x3 ∈ field0 x0 ⟶ field2b x0 x1 (field1b x0 x2 x3) = field1b x0 (field2b x0 x1 x2) (field2b x0 x1 x3) (proof)
Param If_i^If_i : ο → ι → ι → ι
Param setminus^setminus : ι → ι → ι
Param Sing^Sing : ι → ι
Definition Field_div^Field_div := λ x0 x1 x2 . If_i (and (x1 ∈ field0 x0) (x2 ∈ setminus (field0 x0) (Sing (field3 x0)))) (prim0 (λ x3 . and (x3 ∈ field0 x0) (x1 = field2b x0 x2 x3))) 0
Definition False^False := ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
Definition nIn^nIn := λ x0 x1 . not (x0 ∈ x1)
Known setminusE^setminusE : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ and (x2 ∈ x0) (nIn x2 x1)
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 andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Theorem Field_div_prop^{Field_div_prop} : ∀ x0 . Field x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ setminus (field0 x0) (Sing (field3 x0)) ⟶ and (Field_div x0 x1 x2 ∈ field0 x0) (x1 = field2b x0 x2 (Field_div x0 x1 x2)) (proof)
Theorem Field_div_clos^{Field_div_clos} : ∀ x0 . Field x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ setminus (field0 x0) (Sing (field3 x0)) ⟶ Field_div x0 x1 x2 ∈ field0 x0 (proof)
Theorem Field_mult_div^{Field_mult_div} : ∀ x0 . Field x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ setminus (field0 x0) (Sing (field3 x0)) ⟶ x1 = field2b x0 x2 (Field_div x0 x1 x2) (proof)
Known If_i_0^If_i_0 : ∀ x0 : ο . ∀ x1 x2 . not x0 ⟶ If_i x0 x1 x2 = x2
Theorem Field_div_undef1^{Field_div_undef1} : ∀ x0 . Field x0 ⟶ ∀ x1 x2 . nIn x1 (field0 x0) ⟶ Field_div x0 x1 x2 = 0 (proof)
Theorem Field_div_undef2^{Field_div_undef2} : ∀ x0 . Field x0 ⟶ ∀ x1 x2 . nIn x2 (field0 x0) ⟶ Field_div x0 x1 x2 = 0 (proof)
Theorem Field_div_undef3^{Field_div_undef3} : ∀ x0 . Field x0 ⟶ ∀ x1 . Field_div x0 x1 (field3 x0) = 0 (proof)
Known Field_is_CRing_with_id^{Field_is_CRing_with_id} : ∀ x0 . Field x0 ⟶ CRing_with_id x0
Theorem Field_omega_exp_0^{Field_omega_exp_0} : ∀ x0 . Field x0 ⟶ ∀ x1 . CRing_with_id_omega_exp x0 x1 0 = field4 x0 (proof)
Theorem Field_omega_exp_S^{Field_omega_exp_S} : ∀ x0 . Field x0 ⟶ ∀ x1 x2 . x2 ∈ omega ⟶ CRing_with_id_omega_exp x0 x1 (ordsucc x2) = field2b x0 x1 (CRing_with_id_omega_exp x0 x1 x2) (proof)
Theorem Field_omega_exp_1^{Field_omega_exp_1} : ∀ x0 . Field x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ CRing_with_id_omega_exp x0 x1 1 = x1 (proof)
Theorem Field_omega_exp_clos^{Field_omega_exp_clos} : ∀ x0 . Field x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ omega ⟶ CRing_with_id_omega_exp x0 x1 x2 ∈ field0 x0 (proof)
Theorem Field_eval_poly_clos^{Field_eval_poly_clos} : ∀ x0 . Field x0 ⟶ ∀ x1 . x1 ∈ omega ⟶ ∀ x2 . x2 ∈ setexp (field0 x0) x1 ⟶ ∀ x3 . x3 ∈ field0 x0 ⟶ CRing_with_id_eval_poly x0 x1 x2 x3 ∈ field0 x0 (proof)
Param RealsStruct^RealsStruct : ι → ο
Param Field_of_RealsStruct^{Field_of_RealsStruct} : ι → ι
Known Field_Field_of_RealsStruct^{Field_Field_of_RealsStruct} : ∀ x0 . RealsStruct x0 ⟶ Field (Field_of_RealsStruct x0)
Theorem Field_of_RealsStruct_is_CRing_with_id^{Field_of_RealsStruct_is_CRing_with_id} : ∀ x0 . RealsStruct x0 ⟶ CRing_with_id (Field_of_RealsStruct x0) (proof)
Param RealsStruct_leq^{RealsStruct_leq} : ι → ι → ι → ο
Definition RealsStruct_lt^{RealsStruct_lt} := λ x0 x1 x2 . and (RealsStruct_leq x0 x1 x2) (x1 = x2 ⟶ ∀ x3 : ο . x3)
Theorem RealsStruct_lt_leq^{RealsStruct_lt_leq} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ RealsStruct_lt x0 x1 x2 ⟶ RealsStruct_leq x0 x1 x2 (proof)
Theorem RealsStruct_lt_irref^{RealsStruct_lt_irref} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ not (RealsStruct_lt x0 x1 x1) (proof)
Known RealsStruct_leq_antisym^{RealsStruct_leq_antisym} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ RealsStruct_leq x0 x1 x2 ⟶ RealsStruct_leq x0 x2 x1 ⟶ x1 = x2
Theorem RealsStruct_lt_leq_asym^{RealsStruct_lt_leq_asym} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ RealsStruct_lt x0 x1 x2 ⟶ not (RealsStruct_leq x0 x2 x1) (proof)
Theorem RealsStruct_leq_lt_asym^{RealsStruct_leq_lt_asym} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ RealsStruct_leq x0 x1 x2 ⟶ not (RealsStruct_lt x0 x2 x1) (proof)
Theorem RealsStruct_lt_asym^{RealsStruct_lt_asym} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ RealsStruct_lt x0 x1 x2 ⟶ not (RealsStruct_lt x0 x2 x1) (proof)
Known RealsStruct_leq_tra^{RealsStruct_leq_tra} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ ∀ x3 . x3 ∈ field0 x0 ⟶ RealsStruct_leq x0 x1 x2 ⟶ RealsStruct_leq x0 x2 x3 ⟶ RealsStruct_leq x0 x1 x3
Theorem RealsStruct_lt_leq_tra^{RealsStruct_lt_leq_tra} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ ∀ x3 . x3 ∈ field0 x0 ⟶ RealsStruct_lt x0 x1 x2 ⟶ RealsStruct_leq x0 x2 x3 ⟶ RealsStruct_lt x0 x1 x3 (proof)
Theorem RealsStruct_leq_lt_tra^{RealsStruct_leq_lt_tra} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ ∀ x3 . x3 ∈ field0 x0 ⟶ RealsStruct_leq x0 x1 x2 ⟶ RealsStruct_lt x0 x2 x3 ⟶ RealsStruct_lt x0 x1 x3 (proof)
Theorem RealsStruct_lt_tra^{RealsStruct_lt_tra} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ ∀ x3 . x3 ∈ field0 x0 ⟶ RealsStruct_lt x0 x1 x2 ⟶ RealsStruct_lt x0 x2 x3 ⟶ RealsStruct_lt x0 x1 x3 (proof)
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known RealsStruct_leq_linear^{RealsStruct_leq_linear} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ or (RealsStruct_leq x0 x1 x2) (RealsStruct_leq x0 x2 x1)
Known neq_i_sym^neq_i_sym : ∀ x0 x1 . (x0 = x1 ⟶ ∀ x2 : ο . x2) ⟶ x1 = x0 ⟶ ∀ x2 : ο . x2
Theorem RealsStruct_lt_trich_impred^{RealsStruct_lt_trich_impred} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ ∀ x3 : ο . (RealsStruct_lt x0 x1 x2 ⟶ x3) ⟶ (x1 = x2 ⟶ x3) ⟶ (RealsStruct_lt x0 x2 x1 ⟶ x3) ⟶ x3 (proof)
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Theorem RealsStruct_lt_trich^{RealsStruct_lt_trich} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ or (or (RealsStruct_lt x0 x1 x2) (x1 = x2)) (RealsStruct_lt x0 x2 x1) (proof)
Known RealsStruct_leq_refl^{RealsStruct_leq_refl} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ RealsStruct_leq x0 x1 x1
Theorem RealsStruct_leq_lt_linear^{RealsStruct_leq_lt_linear} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ or (RealsStruct_leq x0 x1 x2) (RealsStruct_lt x0 x2 x1) (proof)
Param Sep^Sep : ι → (ι → ο) → ι
Param natOfOrderedField_p^{natOfOrderedField_p} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ι → ο
Param RealsStruct_one^{RealsStruct_one} : ι → ι
Definition RealsStruct_N^{RealsStruct_N} := λ x0 . Sep (field0 x0) (natOfOrderedField_p (field0 x0) (field4 x0) (RealsStruct_one x0) (field1b x0) (field2b x0) (RealsStruct_leq x0))
Param explicit_Reals^{explicit_Reals} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ο
Param explicit_OrderedField^{explicit_OrderedField} : ι → ι → ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ο
Definition lt^lt := λ x0 x1 x2 . λ x3 x4 : ι → ι → ι . λ x5 : ι → ι → ο . λ x6 x7 . and (x5 x6 x7) (x6 = x7 ⟶ ∀ x8 : ο . x8)
Known explicit_Reals_E^{explicit_Reals_E} : ∀ x0 x1 x2 . ∀ x3 x4 : ι → ι → ι . ∀ x5 : ι → ι → ο . ∀ x6 : ο . (explicit_Reals x0 x1 x2 x3 x4 x5 ⟶ explicit_OrderedField x0 x1 x2 x3 x4 x5 ⟶ (∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ lt x0 x1 x2 x3 x4 x5 x1 x7 ⟶ x5 x1 x8 ⟶ ∀ x9 : ο . (∀ x10 . and (x10 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) (x5 x8 (x4 x10 x7)) ⟶ x9) ⟶ x9) ⟶ (∀ x7 . x7 ∈ setexp x0 (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ ∀ x8 . x8 ∈ setexp x0 (Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5)) ⟶ (∀ x9 . x9 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) ⟶ and (and (x5 (ap x7 x9) (ap x8 x9)) (x5 (ap x7 x9) (ap x7 (x3 x9 x2)))) (x5 (ap x8 (x3 x9 x2)) (ap x8 x9))) ⟶ ∀ x9 : ο . (∀ x10 . and (x10 ∈ x0) (∀ x11 . x11 ∈ Sep x0 (natOfOrderedField_p x0 x1 x2 x3 x4 x5) ⟶ and (x5 (ap x7 x11) x10) (x5 x10 (ap x8 x11))) ⟶ x9) ⟶ x9) ⟶ x6) ⟶ explicit_Reals x0 x1 x2 x3 x4 x5 ⟶ x6
Known RealsStruct_explicit_Reals^{RealsStruct_explicit_Reals} : ∀ x0 . RealsStruct x0 ⟶ explicit_Reals (field0 x0) (field4 x0) (RealsStruct_one x0) (field1b x0) (field2b x0) (RealsStruct_leq x0)
Theorem RealsStruct_Arch^{RealsStruct_Arch} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ RealsStruct_lt x0 (field4 x0) x1 ⟶ RealsStruct_leq x0 (field4 x0) x2 ⟶ ∀ x3 : ο . (∀ x4 . and (x4 ∈ RealsStruct_N x0) (RealsStruct_leq x0 x2 (field2b x0 x4 x1)) ⟶ x3) ⟶ x3 (proof)
Known Field_of_RealsStruct_0^{Field_of_RealsStruct_0} : ∀ x0 . ap (Field_of_RealsStruct x0) 0 = field0 x0
Known Field_of_RealsStruct_2f^{Field_of_RealsStruct_2f} : ∀ x0 . RealsStruct x0 ⟶ (λ x2 . ap (ap (ap (Field_of_RealsStruct x0) 2) x2)) = field2b x0
Known Field_of_RealsStruct_3^{Field_of_RealsStruct_3} : ∀ x0 . ap (Field_of_RealsStruct x0) 3 = field4 x0
Theorem 241a7.. : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ setminus (field0 x0) (Sing (field4 x0)) ⟶ and (Field_div (Field_of_RealsStruct x0) x1 x2 ∈ field0 x0) (x1 = field2b x0 x2 (Field_div (Field_of_RealsStruct x0) x1 x2)) (proof)
Theorem RealsStruct_div_clos^{RealsStruct_div_clos} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ setminus (field0 x0) (Sing (field4 x0)) ⟶ Field_div (Field_of_RealsStruct x0) x1 x2 ∈ field0 x0 (proof)
Theorem RealsStruct_mult_div^{RealsStruct_mult_div} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ setminus (field0 x0) (Sing (field4 x0)) ⟶ x1 = field2b x0 x2 (Field_div (Field_of_RealsStruct x0) x1 x2) (proof)
Theorem RealsStruct_div_undef1^{RealsStruct_div_undef1} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 x2 . nIn x1 (field0 x0) ⟶ Field_div (Field_of_RealsStruct x0) x1 x2 = 0 (proof)
Theorem RealsStruct_div_undef2^{RealsStruct_div_undef2} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 x2 . nIn x2 (field0 x0) ⟶ Field_div (Field_of_RealsStruct x0) x1 x2 = 0 (proof)
Theorem RealsStruct_div_undef3^{RealsStruct_div_undef3} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . Field_div (Field_of_RealsStruct x0) x1 (field4 x0) = 0 (proof)
Known Field_of_RealsStruct_4^{Field_of_RealsStruct_4} : ∀ x0 . ap (Field_of_RealsStruct x0) 4 = RealsStruct_one x0
Theorem RealsStruct_omega_exp_0^{RealsStruct_omega_exp_0} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . CRing_with_id_omega_exp (Field_of_RealsStruct x0) x1 0 = RealsStruct_one x0 (proof)
Theorem RealsStruct_omega_exp_S^{RealsStruct_omega_exp_S} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 x2 . x2 ∈ omega ⟶ CRing_with_id_omega_exp (Field_of_RealsStruct x0) x1 (ordsucc x2) = field2b x0 x1 (CRing_with_id_omega_exp (Field_of_RealsStruct x0) x1 x2) (proof)
Theorem RealsStruct_omega_exp_1^{RealsStruct_omega_exp_1} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ CRing_with_id_omega_exp (Field_of_RealsStruct x0) x1 1 = x1 (proof)
Theorem RealsStruct_omega_exp_clos^{RealsStruct_omega_exp_clos} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ omega ⟶ CRing_with_id_omega_exp (Field_of_RealsStruct x0) x1 x2 ∈ field0 x0 (proof)
Param RealsStruct_Npos^{RealsStruct_Npos} : ι → ι
Definition RealsStruct_divides^{RealsStruct_divides} := λ x0 x1 x2 . ∀ x3 : ο . (∀ x4 . and (x4 ∈ RealsStruct_Npos x0) (field2b x0 x1 x4 = x2) ⟶ x3) ⟶ x3
Definition RealsStruct_Primes^{RealsStruct_Primes} := λ x0 . {x1 ∈ RealsStruct_N x0|and (RealsStruct_lt x0 (RealsStruct_one x0) x1) (∀ x2 . x2 ∈ RealsStruct_Npos x0 ⟶ RealsStruct_divides x0 x2 x1 ⟶ or (x2 = RealsStruct_one x0) (x2 = x1))}
Definition RealsStruct_coprime^{RealsStruct_coprime} := λ x0 x1 x2 . ∀ x3 . x3 ∈ RealsStruct_Npos x0 ⟶ RealsStruct_divides x0 x3 x1 ⟶ RealsStruct_divides x0 x3 x2 ⟶ x3 = RealsStruct_one x0
Param Field_minus^Field_minus : ι → ι → ι
Definition RealsStruct_abs^{RealsStruct_abs} := λ x0 x1 . If_i (RealsStruct_leq x0 (field4 x0) x1) x1 (Field_minus (Field_of_RealsStruct x0) x1)
Known RealsStruct_minus_clos^{RealsStruct_minus_clos} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ Field_minus (Field_of_RealsStruct x0) x1 ∈ field0 x0
Theorem RealsStruct_abs_clos^{RealsStruct_abs_clos} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ RealsStruct_abs x0 x1 ∈ field0 x0 (proof)
Theorem RealsStruct_abs_nonneg_case^{RealsStruct_abs_nonneg_case} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ RealsStruct_leq x0 (field4 x0) x1 ⟶ RealsStruct_abs x0 x1 = x1 (proof)
Known RealsStruct_zero_In^{RealsStruct_zero_In} : ∀ x0 . RealsStruct x0 ⟶ field4 x0 ∈ field0 x0
Theorem RealsStruct_abs_neg_case^{RealsStruct_abs_neg_case} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ RealsStruct_lt x0 x1 (field4 x0) ⟶ RealsStruct_abs x0 x1 = Field_minus (Field_of_RealsStruct x0) x1 (proof)
Known RealsStruct_minus_zero^{RealsStruct_minus_zero} : ∀ x0 . RealsStruct x0 ⟶ Field_minus (Field_of_RealsStruct x0) (field4 x0) = field4 x0
Known RealsStruct_minus_leq^{RealsStruct_minus_leq} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ RealsStruct_leq x0 x1 x2 ⟶ RealsStruct_leq x0 (Field_minus (Field_of_RealsStruct x0) x2) (Field_minus (Field_of_RealsStruct x0) x1)
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Theorem RealsStruct_abs_nonneg^{RealsStruct_abs_nonneg} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ RealsStruct_leq x0 (field4 x0) (RealsStruct_abs x0 x1) (proof)
Known RealsStruct_minus_invol^{RealsStruct_minus_invol} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ Field_minus (Field_of_RealsStruct x0) (Field_minus (Field_of_RealsStruct x0) x1) = x1
Theorem RealsStruct_abs_zero_inv^{RealsStruct_abs_zero_inv} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ RealsStruct_abs x0 x1 = field4 x0 ⟶ x1 = field4 x0 (proof)
Known RealsStruct_plus_cancelR^{RealsStruct_plus_cancelR} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ ∀ x3 . x3 ∈ field0 x0 ⟶ field1b x0 x1 x3 = field1b x0 x2 x3 ⟶ x1 = x2
Known RealsStruct_minus_R^{RealsStruct_minus_R} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ field1b x0 x1 (Field_minus (Field_of_RealsStruct x0) x1) = field4 x0
Known RealsStruct_plus_clos^{RealsStruct_plus_clos} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ field1b x0 x1 x2 ∈ field0 x0
Theorem RealsStruct_dist_zero_eq^{RealsStruct_dist_zero_eq} : ∀ x0 . RealsStruct x0 ⟶ ∀ x1 . x1 ∈ field0 x0 ⟶ ∀ x2 . x2 ∈ field0 x0 ⟶ RealsStruct_abs x0 (field1b x0 x1 (Field_minus (Field_of_RealsStruct x0) x2)) = field4 x0 ⟶ x1 = x2 (proof)