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Proof step 21 from proof mod-add1-negpos-neg (previous step)

j * (i2 + i1) + j0 = i0 * j + j1, j + j1 < 0, j1 + j < j, 2 * j + j1 ≤ j0, j1 + j ≤ j0, abs(j1) < j, 0 ≤ i2, 0 < j, 1 < j, 0 < j0, 0 ≤ i2 * j + j0, j0 < j, i1 < 0, j1 ≤ 0, j1 < j, j1 < j0, i0 ≤ 0, i0 < j, i0 < j0, i0 * j + j1 < 0, j1 ≤ j0, j1 + j < j0 ⊦ j0 = j1 + j, j0 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-abs inst: name: Abs-neg seq: ⊦ j < 0 → abs(j) = 0 - j subst: [j] → [j1] flags: rewrite

Additional information

Used simplifier rules: neg-06: ⊦ 0 - i = ~ i; from specification int-abs%neg-13: ⊦ ~ i < j ↔ 0 < j + i; from specification int-abs%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1%Abs-pos: ⊦ ¬ j < 0 → abs(j) = j; from specification int-abs%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%anti: ⊦ i ≤ j → (j ≤ i ↔ i = j); from specification int-abs%add-zero: ⊦ i + 0 = i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: j1