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Proof step 4 from proof four-square-divides-all (previous step)

j0 * i7 = i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2, i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = j0 * j, 0 < j, i7 < j, 1 < j0, 0 < j0 ^2, 1 < j0 ^2, 0 < j0, 0 < i7, 1 < j, j0 | i5 - i0, ≡mod(i4, i1, j0), ≡mod(i3, i2, j0), i7 < j0, j0 < j, j0 * i7 < j0 ^2, ≡mod(i6, i, j0) ⊦ j = 2, j0 = 1, j0 | ((i2 * i3 + i1 * i4) + i0 * i5) + i * i6 ∧ j0 | i * i5 + i2 * i4 - i1 * i3 + i0 * i6 ∧ j0 | i2 * i5 + i1 * i6 - i0 * i3 + i * i4 ∧ j0 | i0 * i4 + i2 * i6 - i1 * i5 + i * i3

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 5:
⊦ i | j ↔ (∃ i0. i * i0 = j)
Proof step 6:
j0 * i7 = i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2, i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = j0 * j, 0 < j, i7 < j, 1 < j0, 0 < j0 ^2, 1 < j0 ^2, 0 < j0, 0 < i7, 1 < j, ∃ i1. j0 * i1 = i5 - i0, ≡mod(i4, i1, j0), ≡mod(i3, i2, j0), i7 < j0, j0 < j, j0 * i7 < j0 ^2, ≡mod(i6, i, j0) ⊦ j = 2, j0 = 1, j0 | ((i2 * i3 + i1 * i4) + i0 * i5) + i * i6 ∧ j0 | i * i5 + i2 * i4 - i1 * i3 + i0 * i6 ∧ j0 | i2 * i5 + i1 * i6 - i0 * i3 + i * i4 ∧ j0 | i0 * i4 + i2 * i6 - i1 * i5 + i * i3

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