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Proof step 61 from proof four-square-divides-all (previous step)

j0 * i7 = (j0 * i5 + i2)^2 + (j0 * i6 + i1)^2 + (j0 * i8 + i0)^2 + (j0 * i3 + i)^2, i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = j0 * j, i7 < j0, j0 < j, j0 * i7 < j0 ^2, 0 < j, i7 < j, 1 < j0, 0 < j0 ^2, 1 < j0 ^2, 0 < j0, 0 < i7, 1 < j ⊦ j = 2, j0 = 1, ((j0 * i3) * i0 + i * i0) + i1 * (j0 * i5 + i2) + (i * i8 + i2 * i6) * j0 = (i * (j0 * i8 + i0) + i2 * (j0 * i6 + i1)) + (i1 * i5 + i0 * i3) * j0, j0 | ((i2 * (j0 * i5 + i2) + i1 * (j0 * i6 + i1)) + i0 * (j0 * i8 + i0)) + i * (j0 * i3 + i) ∧ j0 | i * (j0 * i8 + i0) + i2 * (j0 * i6 + i1) - i1 * (j0 * i5 + i2) + (j0 * i3) * i0 + i * i0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 62:
⊦ (i + i0) * j = i * j + i0 * j
Proof step 63:
j0 * i7 = (j0 * i5 + i2)^2 + (j0 * i6 + i1)^2 + (j0 * i8 + i0)^2 + (j0 * i3 + i)^2, i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = j0 * j, i7 < j0, j0 < j, j0 * i7 < j0 ^2, 0 < j, i7 < j, 1 < j0, 0 < j0 ^2, 1 < j0 ^2, 0 < j0, 0 < i7, 1 < j ⊦ j = 2, j0 = 1, ((j0 * i3) * i0 + i * i0) + i1 * (j0 * i5 + i2) + (i * i8) * j0 + (i2 * i6) * j0 = (i * (j0 * i8 + i0) + i2 * (j0 * i6 + i1)) + (i1 * i5 + i0 * i3) * j0, j0 | ((i2 * (j0 * i5 + i2) + i1 * (j0 * i6 + i1)) + i0 * (j0 * i8 + i0)) + i * (j0 * i3 + i) ∧ j0 | i * (j0 * i8 + i0) + i2 * (j0 * i6 + i1) - i1 * (j0 * i5 + i2) + (j0 * i3) * i0 + i * i0

Rule argument

spec: int-mult inst: name: dist-lsplit seq: ⊦ (i + i0) * j = i * j + i0 * j subst: [j, i0, i] → [j0, i2 * i6, i * i8] flags: rewrite

Additional information

Used terms: j0, i2 * i6, i * i8