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Proof step 15 from proof mod-add1-posneg-pos (previous step)

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

User applied rule insert spec-lemma

And obtained the following new goals

Proof step 16:
i * j + j0 = i0 * j + j1, 0 ≤ j0, j0 < j, 0 ≤ j1, j1 < j ⊦ i = i0 ∧ j0 = j1
Proof step 17:
j * (i1 + i0) + j0 = i2 * j + j1, 0 < j, 1 < j, i1 < j1, 0 < j1, j1 < j, 0 < i0, i1 < i0, i1 < i0 * j, i1 < j, i1 < j + j0, 0 < j + j0, i1 * j + j0 < 0, j0 < 0, j0 + j < j1, j0 < j, 1 < j1, 0 ≤ i2, 0 ≤ j1, i1 ≤ 0, 0 ≤ i0, j0 ≤ j1 ⊦ j0 = 0, j1 = j0 + j, (i0 + i1) * j + 0 = (i2 + 1) * j + (j1 - j0 - j) ∧ 0 ≤ 0 ∧ 0 ≤ j1 - j0 - j ∧ j1 - j0 - j < j

Rule argument

spec: int-mult 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, 0, i2 + 1, j1 - j0 - j]

Additional information

Used simplifier rules: ⊦ i - j - i0 = i - j + i0; (s)% ⊦ i - j = i0 ↔ i = i0 + j; (s)% ⊦ i + 0 = i; (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: i0 + i1, j, 0, i2 + 1, j1 - j0 - j