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Proof step 47 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 ⊦ j = 2, j0 = 1, i2 * j0 * i5 + i1 * (j0 * i6 + i1) + i0 * (j0 * i8 + i0) + i * (j0 * i3 + i) = i1 * i1 + i0 * i0 + i * i + j0 * (i2 * i5 + i1 * i6 + i0 * i8 + i * i3), j0 | i2 * 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 48:
⊦ (i + i0) * j = i * j + i0 * j
Proof step 49:
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 ⊦ j = 2, j0 = 1, i2 * j0 * i5 + ((j0 * i6) * i1 + i1 * i1) + i0 * (j0 * i8 + i0) + i * (j0 * i3 + i) = i1 * i1 + i0 * i0 + i * i + j0 * (i2 * i5 + i1 * i6 + i0 * i8 + i * i3), j0 | i2 * i2 + ((j0 * i6) * i1 + i1 * i1) + i0 * (j0 * i8 + i0) + i * (j0 * i3 + i)

Rule argument

spec: int-mult inst: name: dist-lsplit seq: ⊦ (i + i0) * j = i * j + i0 * j subst: [j, i0, i] → [i1, i1, j0 * i6] flags: rewrite

Additional information

Used simplifier rules: commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: i1, i1, j0 * i6