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Proof step 3 from proof div-mult-shift-holds (previous step)

i % j * i0 < j, 0 < j, 0 ≤ i0, 0 ≤ i ⊦ (i * i0) / j = i / j * i0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 < j → (i * i0) / j = i / j * i0 + (i % j * i0) / j
Proof step 5:
i % j * i0 < j, 0 < j, 0 ≤ i, 0 ≤ i0 ⊦ i / j * i0 + (i % j * i0) / j = i / j * i0

Rule argument

spec: <toplevel> name: div-mult-rew seq: ⊦ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 < j → (i * i0) / j = i / j * i0 + (i % j * i0) / j subst: [j, i0, i] → [j, i0, i] flags: <all paths>

Additional information

Used simplifier rules: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s) Used terms: j, i0, i