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Proof step 46 from proof euler-four-square-identity2 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 47:
⊦ i + j = i + i0 ↔ j = i0
Proof step 48:
⊦ 2 * i3 * j2 * i1 * j4 + i1 ^2 * j4 ^2 + (i3 * j1 + i4 * j2)^2 + (i2 * j1 + i3 * j4)^2 + (i2 * j3 + i4 * j1)^2 + i4 ^2 * j3 ^2 + 2 * i4 * j3 * i1 * j2 + (((i1 * j1 + i2 * j2) + i3 * j3) + i4 * j4)^2 + (i1 * j3 + i2 * j4)^2 = 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 + i4 ^2 * j3 ^2 + i4 ^2 * j2 ^2 + i4 ^2 * j1 ^2 + i3 ^2 * j4 ^2 + i3 ^2 * j3 ^2 + i3 ^2 * j1 ^2 + i2 ^2 * j4 ^2 + i2 ^2 * j3 ^2 + i2 ^2 * j2 ^2 + i2 ^2 * j1 ^2 + i1 ^2 * j4 ^2 + i1 ^2 * j3 ^2 + i1 ^2 * j1 ^2

Rule argument

spec: int-basic2 name: add-in seq: ⊦ i + j = i + i0 ↔ j = i0 subst: [i0, j, i] → [ 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 + i4 ^2 * j3 ^2 + i4 ^2 * j2 ^2 + i4 ^2 * j1 ^2 + i3 ^2 * j4 ^2 + i3 ^2 * j3 ^2 + i3 ^2 * j1 ^2 + i2 ^2 * j4 ^2 + i2 ^2 * j3 ^2 + i2 ^2 * j2 ^2 + i2 ^2 * j1 ^2 + i1 ^2 * j4 ^2 + i1 ^2 * j3 ^2 + i1 ^2 * j1 ^2, 2 * i3 * j2 * i1 * j4 + i1 ^2 * j4 ^2 + (i3 * j1 + i4 * j2)^2 + (i2 * j1 + i3 * j4)^2 + (i2 * j3 + i4 * j1)^2 + i4 ^2 * j3 ^2 + 2 * i4 * j3 * i1 * j2 + (((i1 * j1 + i2 * j2) + i3 * j3) + i4 * j4)^2 + (i1 * j3 + i2 * j4)^2, i3 ^2 * j2 ^2] 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: 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 + i4 ^2 * j3 ^2 + i4 ^2 * j2 ^2 + i4 ^2 * j1 ^2 + i3 ^2 * j4 ^2 + i3 ^2 * j3 ^2 + i3 ^2 * j1 ^2 + i2 ^2 * j4 ^2 + i2 ^2 * j3 ^2 + i2 ^2 * j2 ^2 + i2 ^2 * j1 ^2 + i1 ^2 * j4 ^2 + i1 ^2 * j3 ^2 + i1 ^2 * j1 ^2, 2 * i3 * j2 * i1 * j4 + i1 ^2 * j4 ^2 + (i3 * j1 + i4 * j2)^2 + (i2 * j1 + i3 * j4)^2 + (i2 * j3 + i4 * j1)^2 + i4 ^2 * j3 ^2 + 2 * i4 * j3 * i1 * j2 + (((i1 * j1 + i2 * j2) + i3 * j3) + i4 * j4)^2 + (i1 * j3 + i2 * j4)^2, i3 ^2 * j2 ^2