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Proof step 1 from proof mult-01 (conclusion)

⊦ 0 ≤ i0 ∧ i0 < abs(j) ∧ j < 0 ∧ 0 ≤ i → (i * abs(j) + i0) / j = ~ i

User applied rule insert lemma

And obtained the following new goals

Rule argument

spec: int-div inst: name: mult seq: ⊦ 0 ≤ i0 ∧ i0 < j ∧ 0 ≤ i → (i * j + i0) / j = i subst: [i0, j, i] → [i0, ~ j, i] flags: current, []

Additional information

Used simplifier rules: neg: j ≠ 0 ⊦ i / ~ j = ~(i / j);%abs-08: ⊦ i < 0 → abs(i) = ~ i; from specification int-abs%minus-left: ⊦ ~ i * j = ~(i * j); from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%neg-15: ⊦ i + ~ j = i - j; 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%negeq: ⊦ ~ i = j ↔ i + j = 0; from specification int-abs%neg-07: ⊦ i < ~ j ↔ i + j < 0; from specification int-abs%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%zero-left: ⊦ 0 * i = 0; from specification int-mult%sub-zero-01: ⊦ i - 0 = i; from specification int-abs%add-zero-rev: ⊦ 0 + i = i; from specification int-basic2%lenls: ⊦ i ≤ j → ¬ j < i; from specification int-abs Used terms: i0, ~ j, i