Proofgold Signed Transaction

Param setminus^setminus : ι → ι → ι
Param omega^omega : ι
Param Sing^Sing : ι → ι
Param nat_p^nat_p : ι → ο
Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param add_nat^add_nat : ι → ι → ι
Param mul_nat^mul_nat : ι → 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 setminusE^setminusE : ∀ x0 x1 x2 . x2 ∈ setminus x0 x1 ⟶ and (x2 ∈ x0) (nIn x2 x1)
Param ordinal^ordinal : ι → ο
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 andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and x0 x1
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
Known nat_0^nat_0 : nat_p 0
Known mul_nat_0L^mul_nat_0L : ∀ x0 . nat_p x0 ⟶ mul_nat 0 x0 = 0
Known add_nat_0R^add_nat_0R : ∀ x0 . add_nat x0 0 = x0
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 ordinal_ordsucc^{ordinal_ordsucc} : ∀ x0 . ordinal x0 ⟶ ordinal (ordsucc x0)
Known nat_p_ordinal^{nat_p_ordinal} : ∀ x0 . nat_p x0 ⟶ ordinal x0
Known add_nat_SR^add_nat_SR : ∀ x0 x1 . nat_p x1 ⟶ add_nat x0 (ordsucc x1) = ordsucc (add_nat x0 x1)
Known omega_ordsucc^{omega_ordsucc} : ∀ x0 . x0 ∈ omega ⟶ ordsucc x0 ∈ omega
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 omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Known In_irref^In_irref : ∀ x0 . nIn x0 x0
Definition Subq^Subq := λ x0 x1 . ∀ x2 . x2 ∈ x0 ⟶ x2 ∈ x1
Known ordinal_ordsucc_In_Subq^{ordinal_ordsucc_In_Subq} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ ordsucc x1 ⊆ x0
Known nat_p_trans^nat_p_trans : ∀ x0 . nat_p x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ nat_p x1
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 ordinal_Empty^{ordinal_Empty} : ordinal 0
Known Empty_Subq_eq^{Empty_Subq_eq} : ∀ x0 . x0 ⊆ 0 ⟶ x0 = 0
Known SingI^SingI : ∀ x0 . x0 ∈ Sing x0
Theorem quotient_remainder_nat^{quotient_remainder_nat} : ∀ x0 . x0 ∈ setminus omega (Sing 0) ⟶ ∀ x1 . nat_p x1 ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ omega) (∀ x4 : ο . (∀ x5 . and (x5 ∈ x0) (x1 = add_nat (mul_nat x3 x0) x5) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2 (proof)
Param SNo^SNo : ι → ο
Param SNoLe^SNoLe : ι → ι → ο
Param mul_SNo^mul_SNo : ι → ι → ι
Param SNoLt^SNoLt : ι → ι → ο
Known SNoLeE^SNoLeE : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ or (SNoLt x0 x1) (x0 = x1)
Known SNo_0^SNo_0 : SNo 0
Known mul_SNo_nonpos_pos^{mul_SNo_nonpos_pos} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 0 ⟶ SNoLt 0 x1 ⟶ SNoLe (mul_SNo x0 x1) 0
Known mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Known SNoLe_ref^SNoLe_ref : ∀ x0 . SNoLe x0 x0
Theorem mul_SNo_nonpos_nonneg^{mul_SNo_nonpos_nonneg} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 0 ⟶ SNoLe 0 x1 ⟶ SNoLe (mul_SNo x0 x1) 0 (proof)
Known EmptyE^EmptyE : ∀ x0 . nIn x0 0
Known ordsuccI2^ordsuccI2 : ∀ x0 . x0 ∈ ordsucc x0
Theorem ordinal_0_In_ordsucc^{ordinal_0_In_ordsucc} : ∀ x0 . ordinal x0 ⟶ 0 ∈ ordsucc x0 (proof)
Known ordinal_In_SNoLt^{ordinal_In_SNoLt} : ∀ x0 . ordinal x0 ⟶ ∀ x1 . x1 ∈ x0 ⟶ SNoLt x1 x0
Theorem ordinal_ordsucc_pos^{ordinal_ordsucc_pos} : ∀ x0 . ordinal x0 ⟶ SNoLt 0 (ordsucc x0) (proof)
Param int^int : ι
Param add_SNo^add_SNo : ι → ι → ι
Param minus_SNo^minus_SNo : ι → ι
Known int_SNo_cases^{int_SNo_cases} : ∀ x0 : ι → ο . (∀ x1 . x1 ∈ omega ⟶ x0 x1) ⟶ (∀ x1 . x1 ∈ omega ⟶ x0 (minus_SNo x1)) ⟶ ∀ x1 . x1 ∈ int ⟶ x0 x1
Known Subq_omega_int^{Subq_omega_int} : omega ⊆ int
Known add_nat_add_SNo^{add_nat_add_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ add_nat x0 x1 = add_SNo x0 x1
Known mul_nat_p^mul_nat_p : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ nat_p (mul_nat x0 x1)
Known mul_nat_mul_SNo^{mul_nat_mul_SNo} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_nat x0 x1 = mul_SNo x0 x1
Known omega_nonneg^omega_nonneg : ∀ x0 . x0 ∈ omega ⟶ SNoLe 0 x0
Known int_add_SNo^int_add_SNo : ∀ x0 . x0 ∈ int ⟶ ∀ x1 . x1 ∈ int ⟶ add_SNo x0 x1 ∈ int
Known int_minus_SNo_omega^{int_minus_SNo_omega} : ∀ x0 . x0 ∈ omega ⟶ minus_SNo x0 ∈ int
Known nat_1^nat_1 : nat_p 1
Known ordinal_SNoLt_In^{ordinal_SNoLt_In} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ SNoLt x0 x1 ⟶ x0 ∈ x1
Known aa7e8..^{nonneg_int_nat_p} : ∀ x0 . x0 ∈ int ⟶ SNoLe 0 x0 ⟶ nat_p x0
Known int_minus_SNo^{int_minus_SNo} : ∀ x0 . x0 ∈ int ⟶ minus_SNo x0 ∈ int
Known add_SNo_minus_Le2b^{add_SNo_minus_Le2b} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe (add_SNo x2 x1) x0 ⟶ SNoLe x2 (add_SNo x0 (minus_SNo x1))
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Known SNoLtLe^SNoLtLe : ∀ x0 x1 . SNoLt x0 x1 ⟶ SNoLe x0 x1
Known add_SNo_minus_Lt1b^{add_SNo_minus_Lt1b} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 (add_SNo x2 x1) ⟶ SNoLt (add_SNo x0 (minus_SNo x1)) x2
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 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_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_SNo x0)
Known SNo_1^SNo_1 : SNo 1
Known mul_SNo_minus_distrL^{mul_SNo_minus_distrL} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo (minus_SNo x0) x1 = minus_SNo (mul_SNo x0 x1)
Known mul_SNo_oneL^mul_SNo_oneL : ∀ x0 . SNo x0 ⟶ mul_SNo 1 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 SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (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 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 SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Known nat_p_SNo^nat_p_SNo : ∀ x0 . nat_p x0 ⟶ SNo x0
Theorem quotient_remainder_int^{quotient_remainder_int} : ∀ x0 . x0 ∈ setminus omega (Sing 0) ⟶ ∀ x1 . x1 ∈ int ⟶ ∀ x2 : ο . (∀ x3 . and (x3 ∈ int) (∀ x4 : ο . (∀ x5 . and (x5 ∈ x0) (x1 = add_SNo (mul_SNo x3 x0) x5) ⟶ x4) ⟶ x4) ⟶ x2) ⟶ x2 (proof)
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)
Known divides_int_ref^{divides_int_ref} : ∀ x0 . x0 ∈ int ⟶ divides_int x0 x0
Known divides_int_0^{divides_int_0} : ∀ x0 . x0 ∈ int ⟶ divides_int x0 0
Known divides_int_add_SNo^{divides_int_add_SNo} : ∀ x0 x1 x2 . divides_int x0 x1 ⟶ divides_int x0 x2 ⟶ divides_int x0 (add_SNo x1 x2)
Known divides_int_add_SNo_3 : ∀ x0 x1 x2 x3 . divides_int x0 x1 ⟶ divides_int x0 x2 ⟶ divides_int x0 x3 ⟶ divides_int x0 (add_SNo x1 (add_SNo x2 x3))
Known divides_int_add_SNo_4 : ∀ x0 x1 x2 x3 x4 . divides_int x0 x1 ⟶ divides_int x0 x2 ⟶ divides_int x0 x3 ⟶ divides_int x0 x4 ⟶ divides_int x0 (add_SNo x1 (add_SNo x2 (add_SNo x3 x4)))
Known divides_int_mul_SNo^{divides_int_mul_SNo} : ∀ x0 x1 x2 x3 . divides_int x0 x2 ⟶ divides_int x1 x3 ⟶ divides_int (mul_SNo x0 x1) (mul_SNo x2 x3)
Param divides_nat^divides_nat : ι → ι → ο
Known divides_nat_divides_int^{divides_nat_divides_int} : ∀ x0 x1 . divides_nat x0 x1 ⟶ divides_int x0 x1
Known divides_int_divides_nat^{divides_int_divides_nat} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ divides_int x0 x1 ⟶ divides_nat x0 x1
Known divides_int_minus_SNo^{divides_int_minus_SNo} : ∀ x0 x1 . divides_int x0 x1 ⟶ divides_int x0 (minus_SNo x1)
Known divides_int_minus_SNo_conv : ∀ x0 x1 . SNo x1 ⟶ divides_int x0 (minus_SNo x1) ⟶ divides_int x0 x1
Known divides_int_diff_SNo_rev : ∀ x0 x1 . x1 ∈ int ⟶ ∀ x2 . x2 ∈ int ⟶ divides_int x0 (add_SNo x1 (minus_SNo x2)) ⟶ divides_int x0 (add_SNo x2 (minus_SNo 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 divides_int_mul_SNo_R^{divides_int_mul_SNo_R} : ∀ x0 x1 x2 . x2 ∈ int ⟶ divides_int x0 x1 ⟶ divides_int x0 (mul_SNo x2 x1)
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known int_SNo^int_SNo : ∀ x0 . x0 ∈ int ⟶ SNo x0
Theorem divides_int_1^{divides_int_1} : ∀ x0 . x0 ∈ int ⟶ divides_int 1 x0 (proof)
Known mul_SNo_oneR^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = x0
Known nonneg_mul_SNo_Le^{nonneg_mul_SNo_Le} : ∀ x0 x1 x2 . SNo x0 ⟶ SNoLe 0 x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x1 x2 ⟶ SNoLe (mul_SNo x0 x1) (mul_SNo x0 x2)
Known SNoLtLe_or^SNoLtLe_or : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ or (SNoLt x0 x1) (SNoLe x1 x0)
Known SNoLt_irref^SNoLt_irref : ∀ x0 . not (SNoLt x0 x0)
Known SNoLeLt_tra^SNoLeLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 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 ordinal_Subq_SNoLe^{ordinal_Subq_SNoLe} : ∀ x0 x1 . ordinal x0 ⟶ ordinal x1 ⟶ x0 ⊆ x1 ⟶ SNoLe x0 x1
Known omega_SNo^omega_SNo : ∀ x0 . x0 ∈ omega ⟶ SNo x0
Known SNoLe_tra^SNoLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLe x0 x2
Known minus_SNo_0^minus_SNo_0 : minus_SNo 0 = 0
Known minus_SNo_Le_contra^{minus_SNo_Le_contra} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ SNoLe (minus_SNo x1) (minus_SNo x0)
Theorem divides_int_pos_Le^{divides_int_pos_Le} : ∀ x0 x1 . divides_int x0 x1 ⟶ SNoLt 0 x1 ⟶ SNoLe x0 x1 (proof)
Definition gcd_reln^gcd_reln := λ x0 x1 x2 . and (and (divides_int x2 x0) (divides_int x2 x1)) (∀ x3 . divides_int x3 x0 ⟶ divides_int x3 x1 ⟶ SNoLe x3 x2)
Known SNoLe_antisym^{SNoLe_antisym} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLe x0 x1 ⟶ SNoLe x1 x0 ⟶ x0 = x1
Theorem gcd_reln_uniq^{gcd_reln_uniq} : ∀ x0 x1 x2 x3 . gcd_reln x0 x1 x2 ⟶ gcd_reln x0 x1 x3 ⟶ x2 = x3 (proof)
Definition int_lin_comb^int_lin_comb := λ x0 x1 x2 . and (and (and (x0 ∈ int) (x1 ∈ int)) (x2 ∈ int)) (∀ x3 : ο . (∀ x4 . and (x4 ∈ int) (∀ x5 : ο . (∀ x6 . and (x6 ∈ int) (add_SNo (mul_SNo x4 x0) (mul_SNo x6 x1) = x2) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3)
Known and4I^and4I : ∀ x0 x1 x2 x3 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ and (and (and x0 x1) x2) x3
Theorem int_lin_comb_I^{int_lin_comb_I} : ∀ x0 . x0 ∈ int ⟶ ∀ x1 . x1 ∈ int ⟶ ∀ x2 . x2 ∈ int ⟶ (∀ x3 : ο . (∀ x4 . and (x4 ∈ int) (∀ x5 : ο . (∀ x6 . and (x6 ∈ int) (add_SNo (mul_SNo x4 x0) (mul_SNo x6 x1) = x2) ⟶ x5) ⟶ x5) ⟶ x3) ⟶ x3) ⟶ int_lin_comb x0 x1 x2 (proof)
Theorem int_lin_comb_E^{int_lin_comb_E} : ∀ x0 x1 x2 . int_lin_comb x0 x1 x2 ⟶ ∀ x3 : ο . (x0 ∈ int ⟶ x1 ∈ int ⟶ x2 ∈ int ⟶ ∀ x4 . x4 ∈ int ⟶ ∀ x5 . x5 ∈ int ⟶ add_SNo (mul_SNo x4 x0) (mul_SNo x5 x1) = x2 ⟶ x3) ⟶ x3 (proof)
Theorem int_lin_comb_E1^{int_lin_comb_E1} : ∀ x0 x1 x2 . int_lin_comb x0 x1 x2 ⟶ x0 ∈ int (proof)
Theorem int_lin_comb_E2^{int_lin_comb_E2} : ∀ x0 x1 x2 . int_lin_comb x0 x1 x2 ⟶ x1 ∈ int (proof)
Theorem int_lin_comb_E3^{int_lin_comb_E3} : ∀ x0 x1 x2 . int_lin_comb x0 x1 x2 ⟶ x2 ∈ int (proof)
Theorem int_lin_comb_E4^{int_lin_comb_E4} : ∀ x0 x1 x2 . int_lin_comb x0 x1 x2 ⟶ ∀ x3 : ο . (∀ x4 . x4 ∈ int ⟶ ∀ x5 . x5 ∈ int ⟶ add_SNo (mul_SNo x4 x0) (mul_SNo x5 x1) = x2 ⟶ x3) ⟶ x3 (proof)
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 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 minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
Known nat_ordsucc^nat_ordsucc : ∀ x0 . nat_p x0 ⟶ nat_p (ordsucc x0)
Known ordinal_SNo^ordinal_SNo : ∀ x0 . ordinal x0 ⟶ SNo x0
Theorem least_pos_int_lin_comb_ex^{least_pos_int_lin_comb_ex} : ∀ x0 . x0 ∈ int ⟶ ∀ x1 . x1 ∈ int ⟶ not (and (x0 = 0) (x1 = 0)) ⟶ ∀ x2 : ο . (∀ x3 . and (and (int_lin_comb x0 x1 x3) (SNoLt 0 x3)) (∀ x4 . int_lin_comb x0 x1 x4 ⟶ SNoLt 0 x4 ⟶ SNoLe x3 x4) ⟶ x2) ⟶ x2 (proof)
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Theorem int_lin_comb_sym^{int_lin_comb_sym} : ∀ x0 x1 x2 . int_lin_comb x0 x1 x2 ⟶ int_lin_comb x1 x0 x2 (proof)
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
Known int_mul_SNo^int_mul_SNo : ∀ x0 . x0 ∈ int ⟶ ∀ x1 . x1 ∈ int ⟶ mul_SNo x0 x1 ∈ int
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 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
Theorem least_pos_int_lin_comb_divides_int^{least_pos_int_lin_comb_divides_int} : ∀ x0 x1 x2 . int_lin_comb x0 x1 x2 ⟶ SNoLt 0 x2 ⟶ (∀ x3 . int_lin_comb x0 x1 x3 ⟶ SNoLt 0 x3 ⟶ SNoLe x2 x3) ⟶ divides_int x2 x0 (proof)
Theorem least_pos_int_lin_comb_gcd^{least_pos_int_lin_comb_gcd} : ∀ x0 x1 x2 . int_lin_comb x0 x1 x2 ⟶ SNoLt 0 x2 ⟶ (∀ x3 . int_lin_comb x0 x1 x3 ⟶ SNoLt 0 x3 ⟶ SNoLe x2 x3) ⟶ gcd_reln x0 x1 x2 (proof)
Definition iff^iff := λ x0 x1 : ο . and (x0 ⟶ x1) (x1 ⟶ x0)
Known iffI^iffI : ∀ x0 x1 : ο . (x0 ⟶ x1) ⟶ (x1 ⟶ x0) ⟶ iff x0 x1
Theorem BezoutThm^BezoutThm : ∀ x0 . x0 ∈ int ⟶ ∀ x1 . x1 ∈ int ⟶ not (and (x0 = 0) (x1 = 0)) ⟶ ∀ x2 . iff (gcd_reln x0 x1 x2) (and (and (int_lin_comb x0 x1 x2) (SNoLt 0 x2)) (∀ x3 . int_lin_comb x0 x1 x3 ⟶ SNoLt 0 x3 ⟶ SNoLe x2 x3)) (proof)
Known SNoLt_tra^SNoLt_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLt x1 x2 ⟶ SNoLt x0 x2
Known mul_SNo_pos_neg^{mul_SNo_pos_neg} : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNoLt 0 x0 ⟶ SNoLt x1 0 ⟶ SNoLt (mul_SNo x0 x1) 0
Known 2bbf0..^{mul_nat_0_or_Subq} : ∀ x0 . nat_p x0 ⟶ ∀ x1 . nat_p x1 ⟶ or (x1 = 0) (x0 ⊆ mul_nat x0 x1)
Theorem gcd_id^gcd_id : ∀ x0 . x0 ∈ setminus omega (Sing 0) ⟶ gcd_reln x0 x0 x0 (proof)
Theorem gcd_0^gcd_0 : ∀ x0 . x0 ∈ setminus omega (Sing 0) ⟶ gcd_reln 0 x0 x0 (proof)
Theorem gcd_sym^gcd_sym : ∀ x0 x1 x2 . gcd_reln x0 x1 x2 ⟶ gcd_reln x1 x0 x2 (proof)
Theorem gcd_minus^gcd_minus : ∀ x0 x1 x2 . gcd_reln x0 x1 x2 ⟶ gcd_reln x0 (minus_SNo x1) x2 (proof)
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
Theorem euclidean_algorithm_prop_1^{euclidean_algorithm_prop_1} : ∀ x0 x1 x2 . x1 ∈ int ⟶ gcd_reln x0 (add_SNo x1 (minus_SNo x0)) x2 ⟶ gcd_reln x0 x1 x2 (proof)
Known and7I^and7I : ∀ x0 x1 x2 x3 x4 x5 x6 : ο . x0 ⟶ x1 ⟶ x2 ⟶ x3 ⟶ x4 ⟶ x5 ⟶ x6 ⟶ and (and (and (and (and (and x0 x1) x2) x3) x4) x5) x6
Theorem euclidean_algorithm^{euclidean_algorithm} : and (and (and (and (and (and (∀ x0 . x0 ∈ setminus omega (Sing 0) ⟶ gcd_reln x0 x0 x0) (∀ x0 . x0 ∈ setminus omega (Sing 0) ⟶ gcd_reln 0 x0 x0)) (∀ x0 . x0 ∈ setminus omega (Sing 0) ⟶ gcd_reln x0 0 x0)) (∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ SNoLt x0 x1 ⟶ ∀ x2 . gcd_reln x0 (add_SNo x1 (minus_SNo x0)) x2 ⟶ gcd_reln x0 x1 x2)) (∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ SNoLt x1 x0 ⟶ ∀ x2 . gcd_reln x1 x0 x2 ⟶ gcd_reln x0 x1 x2)) (∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ int ⟶ SNoLt x1 0 ⟶ ∀ x2 . gcd_reln x0 (minus_SNo x1) x2 ⟶ gcd_reln x0 x1 x2)) (∀ x0 . x0 ∈ int ⟶ ∀ x1 . x1 ∈ int ⟶ SNoLt x0 0 ⟶ ∀ x2 . gcd_reln (minus_SNo x0) x1 x2 ⟶ gcd_reln x0 x1 x2) (proof)
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))
Known xm^xm : ∀ x0 : ο . or x0 (not x0)
Known orIL^orIL : ∀ x0 x1 : ο . x0 ⟶ or x0 x1
Known orIR^orIR : ∀ x0 x1 : ο . x1 ⟶ or x0 x1
Known In_no2cycle^In_no2cycle : ∀ x0 x1 . x0 ∈ x1 ⟶ x1 ∈ x0 ⟶ False
Known In_0_1^In_0_1 : 0 ∈ 1
Known SNoLt_0_1^SNoLt_0_1 : SNoLt 0 1
Known mul_SNo_rotate_3_1^{mul_SNo_rotate_3_1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ mul_SNo x0 (mul_SNo x1 x2) = mul_SNo x2 (mul_SNo x0 x1)
Theorem 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) (proof)
Definition coprime_int := λ x0 x1 . gcd_reln x0 x1 1