Let x0 of type ι be given.

Assume H0: x0 ∈ int.

Let x1 of type ι be given.

Assume H1: x1 ∈ int.

Assume H2: not (and (x0 = 0) (x1 = 0)).

Let x2 of type ι be given.

Apply iffI with 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) leaving 2 subgoals.

Assume H3: gcd_reln x0 x1 x2.

Apply least_pos_int_lin_comb_ex with x0, x1, and (and (int_lin_comb x0 x1 x2) (SNoLt 0 x2)) (∀ x3 . int_lin_comb x0 x1 x3 ⟶ SNoLt 0 x3 ⟶ SNoLe x2 x3) leaving 4 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H1.

The subproof is completed by applying H2.

Let x3 of type ι be given.

Assume H4: and (and (int_lin_comb x0 x1 x3) (SNoLt 0 x3)) (∀ x4 . int_lin_comb x0 x1 x4 ⟶ SNoLt 0 x4 ⟶ SNoLe x3 x4).

Apply H4 with and (and (int_lin_comb x0 x1 x2) (SNoLt 0 x2)) (∀ x4 . int_lin_comb x0 x1 x4 ⟶ SNoLt 0 x4 ⟶ SNoLe x2 x4).

Assume H5: and (int_lin_comb x0 x1 x3) (SNoLt 0 x3).

Apply H5 with (∀ x4 . int_lin_comb x0 x1 x4 ⟶ SNoLt 0 x4 ⟶ SNoLe x3 x4) ⟶ and (and (int_lin_comb x0 x1 x2) (SNoLt 0 x2)) (∀ x4 . int_lin_comb x0 x1 x4 ⟶ SNoLt 0 x4 ⟶ SNoLe x2 x4).

Assume H6: int_lin_comb x0 x1 x3.

Assume H7: SNoLt 0 x3.

Assume H8: ∀ x4 . int_lin_comb x0 x1 x4 ⟶ SNoLt 0 x4 ⟶ SNoLe x3 x4.

Apply gcd_reln_uniq with x0, x1, x2, x3, λ x4 x5 . and (and (int_lin_comb x0 x1 x5) (SNoLt 0 x5)) (∀ x6 . int_lin_comb x0 x1 x6 ⟶ SNoLt 0 x6 ⟶ SNoLe x5 x6) leaving 3 subgoals.

The subproof is completed by applying H3.

Apply least_pos_int_lin_comb_gcd with x0, x1, x3 leaving 3 subgoals.

The subproof is completed by applying H6.

The subproof is completed by applying H7.

The subproof is completed by applying H8.

The subproof is completed by applying H4.

Assume H3: and (and (int_lin_comb x0 x1 x2) (SNoLt 0 x2)) (∀ x3 . int_lin_comb x0 x1 x3 ⟶ SNoLt 0 x3 ⟶ SNoLe x2 x3).

Apply H3 with gcd_reln x0 x1 x2.

Assume H4: and (int_lin_comb x0 x1 x2) (SNoLt 0 x2).

Apply H4 with (∀ x3 . int_lin_comb x0 x1 x3 ⟶ SNoLt 0 x3 ⟶ SNoLe x2 x3) ⟶ gcd_reln x0 x1 x2.

The subproof is completed by applying least_pos_int_lin_comb_gcd with x0, x1, x2.

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Proofgold Proof