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Proof step 20 from proof four-square-congmod (previous step)

i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = j0 * j, ≡mod(i4, i1, j0), 0 < i3 ^2 + i4 ^2 + (j0 * i7 + i0)^2 + (j0 * i5 + i)^2, 1 < j, 0 < j, 0 < j0, j0 | i3 - i2 ⊦ j0 = 1, j = 2, j0 | i3 ^2 + i4 ^2 + (j0 * i7 + i0)^2 + (j0 * i5 + i)^2

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 21:
⊦ i | j ↔ (∃ i0. i * i0 = j)
Proof step 22:
i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = j0 * j, ≡mod(i4, i1, j0), 0 < i3 ^2 + i4 ^2 + (j0 * i7 + i0)^2 + (j0 * i5 + i)^2, 1 < j, 0 < j, 0 < j0, ∃ i0. j0 * i0 = i3 - i2 ⊦ j0 = 1, j = 2, j0 | i3 ^2 + i4 ^2 + (j0 * i7 + i0)^2 + (j0 * i5 + i)^2

Rule argument

spec: int-gcd inst: name: divides seq: ⊦ i | j ↔ (∃ i0. i * i0 = j) subst: [j, i] → [i3 - i2, j0] flags: rewrite

Additional information

Used terms: i3 - i2, j0