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Proof step 108 from proof Lagrange-SumOf4Squares-primes (previous step)

i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = j0 * j, 1 < j, 0 < j, 0 < j0, 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), ∀ i3, i4, i5, i6, j1. 0 < j1 ∧ j1 < j0 → i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2 ≠ j1 * j ⊦ i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2 > 0, j = 2, j0 = 1, even(j0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 109:
i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = j0 * j, ~1 ≤ j0, ≡mod(0, i, j0), 1 < j, 0 < j, 0 < j0, prime(j), j0 < j, ≡mod(0, i2, j0), ≡mod(0, i1, j0), ≡mod(0, i0, j0), 0 ≤ j, ∀ i3, i4, i5, i6, j1. 0 < j1 ∧ j1 < j0 → i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2 ≠ j1 * j ⊦ j = 2, j0 = 1, even(j0)

Rule argument

Additional information

Used simplifier rules: forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%mone-06: ⊦ ~1 < i ↔ 0 ≤ i; from specification int-abs%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%numleledel-01: i0 ≠ 0, i0 ≤ i ⊦ (* i) ≤ j → (* i0) ≤ j; from specification int-abs%div-two-zero-01: ⊦ 0 ≤ i / 2 ↔ ~1 ≤ i; from specification int-div%lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista]%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%four-square-le-zero: ⊦ i ^2 + i0 ^2 + i1 ^2 + i2 ^2 ≤ 0 ↔ i = 0 ∧ i0 = 0 ∧ i1 = 0 ∧ i2 = 0; from specification int-square%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%ax-02: ⊦ i > j ↔ j < i; from specification int-abs