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Proof step 39 from proof div-eq-zero-01 (previous step)

i0 * j + j < 0, 0 ≤ i0 * j + j0, 0 ≤ j0, j0 + j < 0, i0 < 0, j < 0, (i0 * j + j0) + j < 0 ⊦ i0 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-mult inst: name: succ-lsplit seq: ⊦ (i + 1) * j = i * j + j subst: [j, i] → [j, i0] flags: rewrite

Additional information

Used simplifier rules: zero-ls-01: ⊦ i < 0 → (i * j < 0 ↔ 0 < j); from specification int-mult%numls-02: ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0; from specification int-abs%mone-06: ⊦ ~1 < i ↔ 0 ≤ i; from specification int-abs%lels-03: ⊦ i < j → (¬ ¬ j ≤ i ↔ false); 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 Used terms: j, i0