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Proof step 11 from proof div-add1-posneg-neg (previous step)

i0 * j + i1 * j + j0 = i2 * j + j1, i2 * j + j1 < j, i2 * j + j1 < 0, i2 < j, i2 ≤ 0, j1 < j, j1 ≤ 0, 0 < j, abs(j0) < j, 1 + i2 * j + j1 < j, (i1 * j + j0) + 1 < j, j0 ≤ 0, j0 < j, i1 ≤ 0, i1 < j, i1 * j + j0 < 0, i1 * j + j0 < j, 0 ≤ i0, abs(j1) < j ⊦ i2 = i0 + i1

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 12:
i * j + j0 = i0 * j + j1, 0 ≤ j0, j0 < j, 0 ≤ j1, j1 < j ⊦ i = i0 ∧ j0 = j1
Proof step 13:
i0 * j + i1 * j + j0 = i2 * j + j1, i2 * j + j1 < j, i2 * j + j1 < 0, i2 < j, i2 ≤ 0, j1 < j, j1 ≤ 0, 0 < j, abs(j0) < j, 1 + i2 * j + j1 < j, (i1 * j + j0) + 1 < j, j0 ≤ 0, j0 < j, i1 ≤ 0, i1 < j, i1 * j + j0 < 0, i1 * j + j0 < j, 0 ≤ i0, abs(j1) < j ⊦ (i0 + i1) * j + ~ j1 = i2 * j + ~ j0 ∧ 0 ≤ ~ j1 ∧ ~ j1 < j ∧ 0 ≤ ~ j0 ∧ ~ j0 < j, i2 = i0 + i1

Rule argument

spec: int-mult inst: name: modeqp seq: i * j + j0 = i0 * j + j1, 0 ≤ j0, j0 < j, 0 ≤ j1, j1 < j ⊦ i = i0 ∧ j0 = j1 subst: [i, j, j0, i0, j1] → [i0 + i1, j, ~ j1, i2, ~ j0] flags: []

Additional information

Used terms: i0 + i1, j, ~ j1, i2, ~ j0