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Proof step 8 from proof four-square-congmod (previous step)

i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = j0 * j, ≡mod(i6, i, j0), 1 < j, 0 < j, 0 < j0, 0 < i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2, ≡mod(i3, i2, j0), ≡mod(i4, i1, j0), j0 | i5 - i0 ⊦ j0 = 1, j = 2, j0 | i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ i | j ↔ (∃ i0. i * i0 = j)
Proof step 10:
i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = j0 * j, ≡mod(i6, i, j0), 1 < j, 0 < j, 0 < j0, 0 < i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2, ≡mod(i3, i2, j0), ≡mod(i4, i1, j0), ∃ i1. j0 * i1 = i5 - i0 ⊦ j0 = 1, j = 2, j0 | i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2

Rule argument

spec: int-gcd inst: name: divides seq: ⊦ i | j ↔ (∃ i0. i * i0 = j) subst: [j, i] → [i5 - i0, j0] flags: rewrite

Additional information

Used terms: i5 - i0, j0