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Proof step 9 from proof four-square-divides-all (previous step)

j0 * i7 = i3 ^2 + i4 ^2 + (j0 * i8 + i0)^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, ≡mod(i4, i1, j0), j0 | abs(i3 - i2), i7 < j0, j0 < j, j0 * i7 < j0 ^2, ≡mod(i6, i, j0) ⊦ j = 2, j0 = 1, j0 | ((i2 * i3 + i1 * i4) + i0 * (j0 * i8 + i0)) + i * i6 ∧ j0 | i * (j0 * i8 + i0) + i2 * i4 - i1 * i3 + i0 * i6 ∧ j0 | i2 * (j0 * i8 + i0) + i1 * i6 - i0 * i3 + i * i4 ∧ j0 | i0 * i4 + i2 * i6 - i1 * (j0 * i8 + i0) + i * i3

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
j0 * i7 = i3 ^2 + i4 ^2 + (j0 * i8 + i0)^2 + i6 ^2, i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = j0 * j, i7 < j0, j0 < j, j0 * i7 < j0 ^2, ≡mod(i6, i, j0), 0 < j, i7 < j, 1 < j0, 0 < j0 ^2, 1 < j0 ^2, 0 < j0, 0 < i7, 1 < j, ≡mod(i4, i1, j0), j0 | i3 - i2 ⊦ j = 2, j0 = 1, j0 | ((i2 * i3 + i1 * i4) + i0 * (j0 * i8 + i0)) + i * i6 ∧ j0 | i * (j0 * i8 + i0) + i2 * i4 - i1 * i3 + i0 * i6 ∧ j0 | i2 * (j0 * i8 + i0) + i1 * i6 - i0 * i3 + i * i4 ∧ j0 | i0 * i4 + i2 * i6 - i1 * (j0 * i8 + i0) + i * i3

Rule argument

Additional information

Used simplifier rules: div-abs-02: ⊦ i | abs(j) ↔ i | j; from specification int-gcd