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Proof step 34 from proof four-square-divides-all (previous step)

j0 * i7 = (j0 * i5 + i2)^2 + (j0 * i6 + i1)^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 ⊦ j = 2, j0 = 1, j0 | ((i2 * (j0 * i5 + i2) + i1 * (j0 * i6 + i1)) + i0 * (j0 * i8 + i0)) + i * (j0 * i3 + i)

User applied rule cut formula

And obtained the following new goals

Proof step 35:
j0 * i7 = (j0 * i5 + i2)^2 + (j0 * i6 + i1)^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 ⊦ ((i2 * (j0 * i5 + i2) + i1 * (j0 * i6 + i1)) + i0 * (j0 * i8 + i0)) + i * (j0 * i3 + i) = (i2 ^2 + i1 ^2 + i0 ^2 + i ^2) + j0 * (i2 * i5 + i1 * i6 + i0 * i8 + i * i3), j = 2, j0 = 1, j0 | ((i2 * (j0 * i5 + i2) + i1 * (j0 * i6 + i1)) + i0 * (j0 * i8 + i0)) + i * (j0 * i3 + i)

Rule argument

fma: ( (i2 * (j0 * i5 + i2) + i1 * (j0 * i6 + i1)) + i0 * (j0 * i8 + i0)) + i * (j0 * i3 + i) = (i2 ^2 + i1 ^2 + i0 ^2 + i ^2) + j0 * (i2 * i5 + i1 * i6 + i0 * i8 + i * i3)

Additional information

Used simplifier rules: divides-mult-add: ⊦ i | i * j + j0 ↔ i | j0; from specification int-gcd%div-mult: ⊦ i | i * j; from specification int-gcd%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2 Used formulas: ((i2 * (j0 * i5 + i2) + i1 * (j0 * i6 + i1)) + i0 * (j0 * i8 + i0)) + i * (j0 * i3 + i) = (i2 ^2 + i1 ^2 + i0 ^2 + i ^2) + j0 * (i2 * i5 + i1 * i6 + i0 * i8 + i * i3)