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Proof step 5 from proof div-mult-neg (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ j ≠ 0 → ~ i / j = ~(i / j)
Proof step 7:
0 < j, i < j, i0 < j, abs(i0) < j, i ≤ 0, i0 ≤ 0 ⊦ j = 0, ~((j * ~ i + ~ i0) / j) = i

Rule argument

spec: <toplevel> name: Div-left seq: ⊦ j ≠ 0 → ~ i / j = ~(i / j) subst: [j, i] → [j, j * ~ i + ~ i0] flags: <all paths>

Additional information

Used simplifier rules: ⊦ i * 0 = 0; (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ i + 0 = i; (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)% ⊦ i < 0 → abs(i) = ~ i; (s)% ⊦ ~ i < 0 ↔ 0 < i; (s)% ⊦ i < j → ¬ j < i; (s) Used terms: j, j * ~ i + ~ i0