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Proof step 5 from proof div-add-succ-neg (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ j ≠ 0 → ~ i / j = ~(i / j)
Proof step 7:
1 < j, 0 < j, i < 0 ⊦ i % j = 0, i / j = ~((~ i + (j - 1)) / j) + 1, j = 0

Rule argument

spec: <toplevel> name: Div-left seq: ⊦ j ≠ 0 → ~ i / j = ~(i / j) subst: [j, i] → [j, ~ i + (j - 1)] flags: <all paths>

Additional information

Used simplifier rules: i < 0 ⊦ (* i) + j = j - ~ i; (s)% ⊦ ~ 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 = i; (s) Used terms: j, ~ i + (j - 1)