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Proof step 4 from proof div-add-nosucc-neg (previous step)

i % j = 0, i < j, 0 < j, 1 < j, i < 0, j + i ≤ 0, i + 1 < j ⊦ j = 1, ~(~ i + (j - 1)) / j = i / j

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 5:
⊦ j ≠ 0 → ~ i / j = ~(i / j)
Proof step 6:
j ≠ 0, i % j = 0, i < j, 0 < j, 1 < j, i < 0, j + i ≤ 0, i + 1 < j ⊦ j = 1, ~((~ i + (j - 1)) / j) = i / j

Rule argument

spec: int-div inst: name: Div-left seq: ⊦ j ≠ 0 → ~ i / j = ~(i / j) subst: [j, i] → [j, ~ i + (j - 1)] flags: rewrite, current

Additional information

Used simplifier rules: 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%minus-sub: ⊦ ~(i - j) = j - i; from specification int-abs%minus-neg: j < 0 ⊦ i - (* j) = i + ~ j; from specification int-abs%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs Used terms: j, ~ i + (j - 1)