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Proof step 12 from proof div-add1-posneg-neg (previous step)

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

User applied rule insert spec-lemma

And obtained the following new goals

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, ~ j1, i2, ~ j0]

Additional information

Used simplifier rules: ⊦ i + ~ j = i - j; (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ i - j = i0 ↔ i = i0 + j; (s)% ⊦ i + (j - i0) = i + j - i0; (s)% ⊦ i ≤ ~ j ↔ i + j ≤ 0; (s)% ⊦ i + 0 = i; (s)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)% ⊦ ~ i < j ↔ 0 < j + i; (s)%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ i < j ↔ ¬ j ≤ i; (ss)%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs% ⊦ i + j < j ↔ i < 0; (s)% ⊦ i < 0 → abs(i) = ~ i; (s)% ⊦ i ≤ j → ¬ j < i; (s) Used terms: i0 + i1, j, ~ j1, i2, ~ j0