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Proof step 10 from proof Lagrange-SumOf4Squares-primes (previous step)

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

User applied rule insert lemma

And obtained the following new goals

Proof step 11:
j = i ^2 + i0 ^2 + i1 ^2 + i2 ^2, even(j) ⊦ ∃ i3, i4, i5, i6. even(i3 + i4) ∧ even(i5 + i6) ∧ i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2 = j
Proof step 12:
∃ i3, i4, i5, i6. even(i3 + i4) ∧ even(i5 + i6) ∧ i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2 = j0 * j, even(j0), i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = j0 * j, j0 < j, prime(j), 0 < j0, 0 < j, 1 < j, ∀ i3, i4, i5, i6, j1. 0 < j1 ∧ j1 < j0 → i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2 ≠ j1 * j ⊦ j0 = 1, j = 2

Rule argument

spec: inst: name: even-sum seq: j = i ^2 + i0 ^2 + i1 ^2 + i2 ^2, even(j) ⊦ ∃ i3, i4, i5, i6. even(i3 + i4) ∧ even(i5 + i6) ∧ i3 ^2 + i4 ^2 + i5 ^2 + i6 ^2 = j subst: [j, i, i0, i1, i2] → [j0 * j, i2, i1, i0, i] flags: current, []

Additional information

Used simplifier rules: even-mult: ⊦ even(i * j) ↔ even(i) ∨ even(j); from specification int-min%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used terms: j0 * j, i2, i1, i0, i