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Proof step 77 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, i * (j0 * i6 + i1) + i0 * (j0 * i5 + i2) + (i2 * i8 + i1 * i3) * j0 = (((j0 * i8) * i2 + i0 * i2) + i1 * (j0 * i3 + i)) + (i0 * i5 + i * i6) * 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) + i0 * (j0 * i3 + i) ∧ j0 | ((j0 * i8) * i2 + i0 * i2) + i1 * (j0 * i3 + i) - i0 * (j0 * i5 + i2) + i * (j0 * i6 + i1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 78:
⊦ (i + i0) * j = i * j + i0 * j
Proof step 79:
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, i * (j0 * i6 + i1) + i0 * (j0 * i5 + i2) + (i2 * i8 + i1 * i3) * j0 = (((j0 * i8) * i2 + i0 * i2) + i1 * (j0 * i3 + i)) + (i0 * i5) * j0 + (i * i6) * 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) + i0 * (j0 * i3 + i) ∧ j0 | ((j0 * i8) * i2 + i0 * i2) + i1 * (j0 * i3 + i) - i0 * (j0 * i5 + i2) + i * (j0 * i6 + i1)

Rule argument

spec: int-mult inst: name: dist-lsplit seq: ⊦ (i + i0) * j = i * j + i0 * j subst: [j, i0, i] → [j0, i * i6, i0 * i5] flags: rewrite

Additional information

Used terms: j0, i * i6, i0 * i5