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Proof step 83 from proof euler-four-square-identity2 (previous step)

⊦ (((i1 * j1 + i2 * j2) + i3 * j3) + i4 * j4)^2 + 2 * i1 * j3 * i2 * j4 + 2 * i3 * j2 * i1 * j4 + 2 * i3 * j1 * i4 * j2 + 2 * i2 * j1 * i3 * j4 + 2 * i2 * j3 * i4 * j1 + 2 * i4 * j3 * i1 * j2 = 2 * (i1 * j3 + i2 * j4) * (i3 * j1 + i4 * j2) + 2 * (i4 * j3 + i1 * j2) * (i2 * j1 + i3 * j4) + 2 * (i3 * j2 + i1 * j4) * (i2 * j3 + i4 * j1) + i4 ^2 * j4 ^2 + i3 ^2 * j3 ^2 + i2 ^2 * j2 ^2 + i1 ^2 * j1 ^2

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-basic2 name: add-a seq: ⊦ i + j + i0 = (i + j) + i0 subst: [i0, j, i] → [i3 * j3 + i4 * j4, i2 * j2, i1 * j1] flags: <all paths>

Additional information

Used simplifier rules: add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: i3 * j3 + i4 * j4, i2 * j2, i1 * j1