Proofgold Address

Definition and^and := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x1 ⟶ x2) ⟶ x2
Param omega^omega : ι
Param mul_nat^mul_nat : ι → ι → ι
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)
Param add_SNo^add_SNo : ι → ι → ι
Param nat_p^nat_p : ι → ο
Known 068c4.. : ∀ x0 . nat_p x0 ⟶ ∀ x1 x2 . divides_nat x0 x1 ⟶ divides_nat x0 x2 ⟶ divides_nat x0 (add_SNo x1 x2)
Known omega_nat_p^omega_nat_p : ∀ x0 . x0 ∈ omega ⟶ nat_p x0
Theorem 77870.. : ∀ x0 x1 x2 x3 . divides_nat x0 x1 ⟶ divides_nat x0 x2 ⟶ divides_nat x0 x3 ⟶ divides_nat x0 (add_SNo x1 (add_SNo x2 x3)) (proof)
Theorem a82e4.. : ∀ x0 x1 x2 x3 x4 . divides_nat x0 x1 ⟶ divides_nat x0 x2 ⟶ divides_nat x0 x3 ⟶ divides_nat x0 x4 ⟶ divides_nat x0 (add_SNo x1 (add_SNo x2 (add_SNo x3 x4))) (proof)
Param mul_SNo^mul_SNo : ι → ι → ι
Known and3I^and3I : ∀ x0 x1 x2 : ο . x0 ⟶ x1 ⟶ x2 ⟶ and (and x0 x1) x2
Known mul_SNo_In_omega^{mul_SNo_In_omega} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ mul_SNo x0 x1 ∈ omega
Known andI^andI : ∀ x0 x1 : ο . x0 ⟶ x1 ⟶ and 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
Param SNo^SNo : ι → ο
Known mul_SNo_com_4_inner_mid^{mul_SNo_com_4_inner_mid} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ mul_SNo (mul_SNo x0 x1) (mul_SNo x2 x3) = mul_SNo (mul_SNo x0 x2) (mul_SNo x1 x3)
Known nat_p_SNo^nat_p_SNo : ∀ x0 . nat_p x0 ⟶ SNo x0
Theorem a7e3e.. : ∀ x0 x1 x2 x3 . divides_nat x0 x2 ⟶ divides_nat x1 x3 ⟶ divides_nat (mul_SNo x0 x1) (mul_SNo x2 x3) (proof)
Param int^int : 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 ordsucc^ordsucc : ι → ι
Known nat_p_int^nat_p_int : ∀ x0 . nat_p x0 ⟶ x0 ∈ int
Known nat_1^nat_1 : nat_p 1
Known mul_SNo_oneR^mul_SNo_oneR : ∀ x0 . SNo x0 ⟶ mul_SNo x0 1 = x0
Known int_SNo^int_SNo : ∀ x0 . x0 ∈ int ⟶ SNo x0
Theorem divides_int_ref^{divides_int_ref} : ∀ x0 . x0 ∈ int ⟶ divides_int x0 x0 (proof)
Known nat_0^nat_0 : nat_p 0
Known mul_SNo_zeroR^{mul_SNo_zeroR} : ∀ x0 . SNo x0 ⟶ mul_SNo x0 0 = 0
Theorem divides_int_0^{divides_int_0} : ∀ x0 . x0 ∈ int ⟶ divides_int x0 0 (proof)
Known int_add_SNo^int_add_SNo : ∀ x0 . x0 ∈ int ⟶ ∀ x1 . x1 ∈ int ⟶ add_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)
Theorem 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) (proof)
Theorem 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)) (proof)
Theorem 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))) (proof)
Known int_mul_SNo^int_mul_SNo : ∀ x0 . x0 ∈ int ⟶ ∀ x1 . x1 ∈ int ⟶ mul_SNo x0 x1 ∈ int
Theorem 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) (proof)
Theorem divides_nat_divides_int^{divides_nat_divides_int} : ∀ x0 x1 . divides_nat x0 x1 ⟶ divides_int x0 x1 (proof)
Param SNoLe^SNoLe : ι → ι → ο
Definition or^or := λ x0 x1 : ο . ∀ x2 : ο . (x0 ⟶ x2) ⟶ (x1 ⟶ x2) ⟶ x2
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 84495.. : ∀ x0 . nat_p x0 ⟶ SNoLe 0 x0
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 minus_SNo_0^minus_SNo_0 : minus_SNo 0 = 0
Known nat_p_omega^nat_p_omega : ∀ x0 . nat_p x0 ⟶ x0 ∈ omega
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
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 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_1^SNo_1 : SNo 1
Known SNo_minus_SNo^{SNo_minus_SNo} : ∀ x0 . SNo x0 ⟶ SNo (minus_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)
Definition False^False := ∀ x0 : ο . x0
Known FalseE^FalseE : False ⟶ ∀ x0 : ο . x0
Definition not^not := λ x0 : ο . x0 ⟶ False
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 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 SNo_add_SNo^SNo_add_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (add_SNo x0 x1)
Known SNoLtLe_tra^SNoLtLe_tra : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLt x0 x1 ⟶ SNoLe x1 x2 ⟶ SNoLt x0 x2
Known add_SNo_0R^add_SNo_0R : ∀ x0 . SNo x0 ⟶ add_SNo x0 0 = x0
Known add_SNo_Le2^add_SNo_Le2 : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNoLe x1 x2 ⟶ SNoLe (add_SNo x0 x1) (add_SNo x0 x2)
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 SNo_mul_SNo^SNo_mul_SNo : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ SNo (mul_SNo x0 x1)
Known abdca.. : ∀ x0 . nat_p x0 ⟶ divides_nat x0 0
Known mul_SNo_zeroL^{mul_SNo_zeroL} : ∀ x0 . SNo x0 ⟶ mul_SNo 0 x0 = 0
Theorem divides_int_divides_nat^{divides_int_divides_nat} : ∀ x0 . x0 ∈ omega ⟶ ∀ x1 . x1 ∈ omega ⟶ divides_int x0 x1 ⟶ divides_nat x0 x1 (proof)
Known int_minus_SNo^{int_minus_SNo} : ∀ x0 . x0 ∈ int ⟶ minus_SNo x0 ∈ int
Theorem divides_int_minus_SNo^{divides_int_minus_SNo} : ∀ x0 x1 . divides_int x0 x1 ⟶ divides_int x0 (minus_SNo x1) (proof)
Known minus_SNo_invol^{minus_SNo_invol} : ∀ x0 . SNo x0 ⟶ minus_SNo (minus_SNo x0) = x0
Theorem divides_int_minus_SNo_conv : ∀ x0 x1 . SNo x1 ⟶ divides_int x0 (minus_SNo x1) ⟶ divides_int x0 x1 (proof)
Known add_SNo_com^add_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ add_SNo x0 x1 = add_SNo x1 x0
Theorem 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)) (proof)
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 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) (proof)
Known mul_SNo_com^mul_SNo_com : ∀ x0 x1 . SNo x0 ⟶ SNo x1 ⟶ mul_SNo x0 x1 = mul_SNo x1 x0
Theorem 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) (proof)
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 add_SNo_com_4_inner_mid^{add_SNo_com_4_inner_mid} : ∀ x0 x1 x2 x3 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ SNo x3 ⟶ add_SNo (add_SNo x0 x1) (add_SNo x2 x3) = add_SNo (add_SNo x0 x2) (add_SNo x1 x3)
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 add_SNo_minus_SNo_rinv^{add_SNo_minus_SNo_rinv} : ∀ x0 . SNo x0 ⟶ add_SNo x0 (minus_SNo x0) = 0
Known add_SNo_0L^add_SNo_0L : ∀ x0 . SNo x0 ⟶ add_SNo 0 x0 = x0
Theorem a3ea1.. : ∀ x0 x1 . x1 ∈ int ⟶ ∀ x2 . x2 ∈ int ⟶ divides_int x0 (add_SNo x1 (minus_SNo x2)) ⟶ divides_int x0 (add_SNo (mul_SNo x1 x1) (minus_SNo (mul_SNo x2 x2))) (proof)
Known add_SNo_minus_SNo_linv^{add_SNo_minus_SNo_linv} : ∀ x0 . SNo x0 ⟶ add_SNo (minus_SNo x0) x0 = 0
Theorem 385cb.. : ∀ x0 x1 . x1 ∈ int ⟶ ∀ x2 . x2 ∈ int ⟶ ∀ x3 . x3 ∈ int ⟶ divides_int x0 (add_SNo x1 (minus_SNo x2)) ⟶ divides_int x0 (add_SNo x2 x3) ⟶ divides_int x0 (add_SNo x1 x3) (proof)
Theorem b1639.. : ∀ x0 x1 . x1 ∈ int ⟶ ∀ x2 . x2 ∈ int ⟶ ∀ x3 . x3 ∈ int ⟶ divides_int x0 (add_SNo x2 (minus_SNo x3)) ⟶ divides_int x0 (add_SNo x1 x2) ⟶ divides_int x0 (add_SNo x1 x3) (proof)
Known add_SNo_com_3_0_1^{add_SNo_com_3_0_1} : ∀ x0 x1 x2 . SNo x0 ⟶ SNo x1 ⟶ SNo x2 ⟶ add_SNo x0 (add_SNo x1 x2) = add_SNo x1 (add_SNo x0 x2)
Theorem 6248f.. : ∀ x0 x1 . x1 ∈ int ⟶ ∀ x2 . x2 ∈ int ⟶ ∀ x3 . x3 ∈ int ⟶ ∀ x4 . x4 ∈ int ⟶ divides_int x0 (add_SNo x2 (minus_SNo x3)) ⟶ divides_int x0 (add_SNo x1 (add_SNo x3 x4)) ⟶ divides_int x0 (add_SNo x1 (add_SNo x2 x4)) (proof)
Theorem 5f8d5.. : ∀ x0 x1 . x1 ∈ int ⟶ ∀ x2 . x2 ∈ int ⟶ divides_int x0 (add_SNo x1 (minus_SNo x2)) ⟶ divides_int x0 x2 ⟶ divides_int x0 x1 (proof)