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Proof step 3 from proof four-square-divides-all (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 4:
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

Rule argument

Additional information

Used simplifier rules: lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista]%transitivity: ⊦ i < i0 ∧ i0 < i1 → i < i1; (forward) from specification oelem[intlista]%forward-pos: ⊦ i < j ∧ 0 < i → 0 < j ∧ 1 < j; (forward) from specification int-basic1%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%div-abs-02: ⊦ i | abs(j) ↔ i | j; from specification int-gcd