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Proof step 44 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, 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 ⊦ ((i2 * (j0 * i5 + i2) + i1 * (j0 * i6 + i1)) + i0 * (j0 * i8 + i0)) + i * (j0 * i3 + i) = (i2 * i2 + i1 * i1 + i0 * i0 + i * i) + j0 * (i2 * i5 + i1 * i6 + i0 * i8 + i * i3), j = 2, j0 = 1, j0 | ((i2 * (j0 * i5 + i2) + i1 * (j0 * i6 + i1)) + i0 * (j0 * i8 + i0)) + i * (j0 * i3 + i)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 45:
⊦ i * (j + j0) = i * j + i * j0
Proof step 46:
j0 * i7 = (j0 * i5 + i2)^2 + (j0 * i6 + i1)^2 + (j0 * i8 + i0)^2 + (j0 * i3 + i)^2, 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 ⊦ (((i2 * j0 * i5 + i2 * i2) + i1 * (j0 * i6 + i1)) + i0 * (j0 * i8 + i0)) + i * (j0 * i3 + i) = (i2 * i2 + i1 * i1 + i0 * i0 + i * i) + j0 * (i2 * i5 + i1 * i6 + i0 * i8 + i * i3), j = 2, j0 = 1, j0 | (((i2 * j0 * i5 + i2 * i2) + i1 * (j0 * i6 + i1)) + i0 * (j0 * i8 + i0)) + i * (j0 * i3 + i)

Rule argument

spec: int-mult inst: name: dist-rsplit seq: ⊦ i * (j + j0) = i * j + i * j0 subst: [j0, j, i] → [i2, j0 * i5, i2] flags: rewrite

Additional information

Used terms: i2, j0 * i5, i2