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Proof step 12 from proof lagrange-primedivisors (previous step)

(i4 ^2 + i5 ^2 + i6 ^2 + i7 ^2) * (i0 ^2 + i1 ^2 + i2 ^2 + i3 ^2) = (((i4 * i0 + i5 * i1) + i6 * i2) + i7 * i3)^2 + ((i4 * i1 - i5 * i0 - i6 * i3) + i7 * i2)^2 + (i4 * i2 + i5 * i3 - i6 * i0 - i7 * i1)^2 + ((i4 * i3 - i5 * i2) + i6 * i1 - i7 * i0)^2, Π ints0 = i0 ^2 + i1 ^2 + i2 ^2 + i3 ^2, ∀ j0. j0 ∈ ints0 → (∃ i, i0, i1, i2. j0 = i ^2 + i0 ^2 + i1 ^2 + i2 ^2) ⊦ ∃ i0, i1, i2, i3. (i4 ^2 + i5 ^2 + i6 ^2 + i7 ^2) * Π ints0 = i0 ^2 + i1 ^2 + i2 ^2 + i3 ^2

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
Π ints0 = i0 ^2 + i1 ^2 + i2 ^2 + i3 ^2, (i4 ^2 + i5 ^2 + i6 ^2 + i7 ^2) * Π ints0 = (((i4 * i0 + i5 * i1) + i6 * i2) + i7 * i3)^2 + (i7 * i2 + i4 * i1 - i5 * i0 + i6 * i3)^2 + (i4 * i2 + i5 * i3 - i6 * i0 + i7 * i1)^2 + (i6 * i1 + i4 * i3 - i5 * i2 + i7 * i0)^2, ∀ j0. j0 ∈ ints0 → (∃ i, i0, i1, i2. j0 = i ^2 + i0 ^2 + i1 ^2 + i2 ^2) ⊦ ∃ i0, i1, i2, i3. (i4 ^2 + i5 ^2 + i6 ^2 + i7 ^2) * Π ints0 = i0 ^2 + i1 ^2 + i2 ^2 + i3 ^2

Rule argument

Additional information

Used simplifier rules: add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%sub-add-01: ⊦ i + (j - i0) = i + j - i0; from specification int-abs%sub-sub: ⊦ i - j - i0 = i - j + i0; from specification int-abs