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Proof step 93 from proof Lagrange-SumOf4Squares-primes (previous step)

j0 * j * j0 * i7 = (((i2 * i3 + i1 * i4) + i0 * i5) + i * i6)^2 + ((i2 * i4 - i1 * i3 - i0 * i6) + i * i5)^2 + (i2 * i5 + i1 * i6 - i0 * i3 - i * i4)^2 + ((i2 * i6 - i1 * i5) + i0 * i4 - i * i3)^2, i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = j0 * j, j0 * i7 = i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2, i7 < j, 0 < j0, 1 < j0, 0 < i7, j0 * i7 < j0 ^2, 0 < j, prime(j), j0 < j, abs(i3) ≤ j0 / 2, ≡mod(i3, i2, j0), abs(i4) ≤ j0 / 2, ≡mod(i4, i1, j0), abs(i5) ≤ j0 / 2, ≡mod(i5, i0, j0), abs(i6) ≤ j0 / 2, ≡mod(i6, i, j0), 1 < j0 ^2, 0 < j0 ^2, i7 < j0, 1 < j, ∀ i3, i4, i5, i6, j1. 0 < j1 ∧ j1 < j0 → i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2 ≠ j1 * j ⊦ j0 | ((i2 * i3 + i1 * i4) + i0 * i5) + i * i6 ∧ j0 | (i2 * i4 - i1 * i3 - i0 * i6) + i * i5 ∧ j0 | i2 * i5 + i1 * i6 - i0 * i3 - i * i4 ∧ j0 | (i2 * i6 - i1 * i5) + i0 * i4 - i * i3, j = 2, j0 = 1, even(j0)

User applied rule insert lemma

And obtained the following new goals

Proof step 94:
i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = j0 * j, j0 * i7 = i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2, ≡mod(i3, i2, j0), ≡mod(i4, i1, j0), ≡mod(i5, i0, j0), ≡mod(i6, i, j0), i7 < j, 0 < j0, 1 < j0, 0 < i7, j0 * i7 < j0 ^2, 0 < j, j0 < j, 1 < j0 ^2, 0 < j0 ^2, i7 < j0, 1 < j, j ≠ 2, j0 ≠ 1 ⊦ j0 | ((i2 * i3 + i1 * i4) + i0 * i5) + i * i6 ∧ j0 | i * i5 + i2 * i4 - i1 * i3 + i0 * i6 ∧ j0 | i2 * i5 + i1 * i6 - i0 * i3 + i * i4 ∧ j0 | i0 * i4 + i2 * i6 - i1 * i5 + i * i3

Rule argument

spec: inst: name: four-square-divides-all seq: i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = j0 * j, j0 * i7 = i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2, ≡mod(i3, i2, j0), ≡mod(i4, i1, j0), ≡mod(i5, i0, j0), ≡mod(i6, i, j0), i7 < j, 0 < j0, 1 < j0, 0 < i7, j0 * i7 < j0 ^2, 0 < j, j0 < j, 1 < j0 ^2, 0 < j0 ^2, i7 < j0, 1 < j, j ≠ 2, j0 ≠ 1 ⊦ j0 | ((i2 * i3 + i1 * i4) + i0 * i5) + i * i6 ∧ j0 | i * i5 + i2 * i4 - i1 * i3 + i0 * i6 ∧ j0 | i2 * i5 + i1 * i6 - i0 * i3 + i * i4 ∧ j0 | i0 * i4 + i2 * i6 - i1 * i5 + i * i3 subst: [i2, i1, i0, i, j0, j, i7, i3, i4, i5, i6] → [i2, i1, i0, i, j0, j, i7, i3, i4, i5, i6] flags: current, []

Additional information

Used simplifier rules: sub-sub: ⊦ i - j - i0 = i - j + i0; from specification int-abs%sub-add-01: ⊦ i + (j - i0) = i + j - i0; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: i2, i1, i0, i, j0, j, i7, i3, i4, i5, i6