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Proof step 22 from proof four-square-divides-all (previous step)

j0 * i7 = (j0 * i5 + i2)^2 + i4 ^2 + (j0 * i8 + i0)^2 + (j0 * i3 + i)^2, i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = j0 * j, 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, j0 | i4 - i1 ⊦ j = 2, j0 = 1, j0 | ((i2 * (j0 * i5 + i2) + i1 * i4) + i0 * (j0 * i8 + i0)) + i * (j0 * i3 + i) ∧ j0 | i * (j0 * i8 + i0) + i2 * i4 - i1 * (j0 * i5 + i2) + i0 * (j0 * i3 + i) ∧ j0 | i2 * (j0 * i8 + i0) + i1 * (j0 * i3 + i) - i0 * (j0 * i5 + i2) + i * i4 ∧ j0 | i0 * i4 + i2 * (j0 * i3 + i) - i1 * (j0 * i8 + i0) + i * (j0 * i5 + i2)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 23:
⊦ i | j ↔ (∃ i0. i * i0 = j)
Proof step 24:
j0 * i7 = (j0 * i5 + i2)^2 + i4 ^2 + (j0 * i8 + i0)^2 + (j0 * i3 + i)^2, i2 ^2 + i1 ^2 + i0 ^2 + i ^2 = j0 * j, 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, ∃ i0. j0 * i0 = i4 - i1 ⊦ j = 2, j0 = 1, j0 | ((i2 * (j0 * i5 + i2) + i1 * i4) + i0 * (j0 * i8 + i0)) + i * (j0 * i3 + i) ∧ j0 | i * (j0 * i8 + i0) + i2 * i4 - i1 * (j0 * i5 + i2) + i0 * (j0 * i3 + i) ∧ j0 | i2 * (j0 * i8 + i0) + i1 * (j0 * i3 + i) - i0 * (j0 * i5 + i2) + i * i4 ∧ j0 | i0 * i4 + i2 * (j0 * i3 + i) - i1 * (j0 * i8 + i0) + i * (j0 * i5 + i2)

Rule argument

spec: int-gcd inst: name: divides seq: ⊦ i | j ↔ (∃ i0. i * i0 = j) subst: [j, i] → [i4 - i1, j0] flags: rewrite

Additional information

Used terms: i4 - i1, j0