Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι be given.

Assume H0: int_lin_comb x0 x1 x2.

Assume H1: SNoLt 0 x2.

Assume H2: ∀ x3 . int_lin_comb x0 x1 x3 ⟶ SNoLt 0 x3 ⟶ SNoLe x2 x3.

Apply int_lin_comb_E with x0, x1, x2, gcd_reln x0 x1 x2 leaving 2 subgoals.

The subproof is completed by applying H0.

Assume H3: x0 ∈ int.

Assume H4: x1 ∈ int.

Assume H5: x2 ∈ int.

Let x3 of type ι be given.

Assume H6: x3 ∈ int.

Let x4 of type ι be given.

Assume H7: x4 ∈ int.

Assume H8: add_SNo (mul_SNo x3 x0) (mul_SNo x4 x1) = x2.

Apply and3I with divides_int x2 x0, divides_int x2 x1, ∀ x5 . divides_int x5 x0 ⟶ divides_int x5 x1 ⟶ SNoLe x5 x2 leaving 3 subgoals.

Apply least_pos_int_lin_comb_divides_int with x0, x1, x2 leaving 3 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H1.

The subproof is completed by applying H2.

Apply least_pos_int_lin_comb_divides_int with x1, x0, x2 leaving 3 subgoals.

Apply int_lin_comb_sym with x0, x1, x2.

The subproof is completed by applying H0.

The subproof is completed by applying H1.

Let x5 of type ι be given.

Assume H9: int_lin_comb x1 x0 x5.

Apply H2 with x5.

Apply int_lin_comb_sym with x1, x0, x5.

The subproof is completed by applying H9.

Let x5 of type ι be given.

Assume H9: divides_int x5 x0.

Assume H10: divides_int x5 x1.

Apply divides_int_pos_Le with x5, x2 leaving 2 subgoals.

Apply H8 with λ x6 x7 . divides_int x5 x6.

Apply divides_int_add_SNo with x5, mul_SNo x3 x0, mul_SNo x4 x1 leaving 2 subgoals.

Apply divides_int_mul_SNo_R with x5, x0, x3 leaving 2 subgoals.

The subproof is completed by applying H6.

The subproof is completed by applying H9.

Apply divides_int_mul_SNo_R with x5, x1, x4 leaving 2 subgoals.

The subproof is completed by applying H7.

The subproof is completed by applying H10.

The subproof is completed by applying H1.

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Proofgold Proof