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Proof step 8 from proof div-div-rec (previous step)

j + 1 = 2 ^ i * (i0 + 1) + j0, 0 ≤ i, i0 < i, i0 ≤ 0, j0 < i, j0 ≤ 0, 2 ^ i * (i0 + 1) + j0 < 2 ^ i, abs(j0) < 2 ^ i, 0 < i, j < 0 ⊦ (i0 - 1) / 2 = (2 ^ i * (i0 + 1) + j0 - 2 ^ i * 2) / (2 ^ i * 2)

User applied rule cut formula

And obtained the following new goals

Proof step 9:
2 ^ i * (i0 + 1) + j0 - 2 ^ i * 2 = 2 ^ i * (i0 - 1) + j0, j + 1 = 2 ^ i * (i0 + 1) + j0, 0 ≤ i, i0 < i, i0 ≤ 0, j0 < i, j0 ≤ 0, 2 ^ i * (i0 + 1) + j0 < 2 ^ i, abs(j0) < 2 ^ i, 0 < i, j < 0 ⊦ (i0 - 1) / 2 = (2 ^ i * (i0 + 1) + j0 - 2 ^ i * 2) / (2 ^ i * 2)

Rule argument

fma: 2 ^ i * (i0 + 1) + j0 - 2 ^ i * 2 = 2 ^ i * (i0 - 1) + j0

Additional information

Used simplifier rules: sub-eq: ⊦ i - j = i0 ↔ i = i0 + j; from specification int-abs%dist-rmerge: ⊦ i * j + i * j0 = i * (j + j0); from specification int-mult%sub-add-01: ⊦ i + (j - i0) = i + j - i0; from specification int-abs%numsub: ⊦ (* i) + i0 - (* j) = (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%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%add-in: ⊦ i + j = i + i0 ↔ j = i0; from specification int-basic2 Used formulas: 2 ^ i * (i0 + 1) + j0 - 2 ^ i * 2 = 2 ^ i * (i0 - 1) + j0