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Proof step 8 from proof div-add-succ (previous step)

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

User applied rule insert elim lemma

And obtained the following new goals

Proof step 9:
⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0)
Proof step 10:
i = i0 * j + j0, 0 < i, 1 < j, 0 < j, j0 < j, 0 ≤ j, 0 ≤ j0, 0 ≤ i0, 0 ≤ i ⊦ (i - 1) / j = i0, j0 = 0

Rule argument

spec: <toplevel> name: div-elim-pospos seq: ⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0) subst: [i, j, i0, j0] → [i, j, i0, j0] flags: <rotated> <all paths>

Additional information

Used simplifier rules: ⊦ i ≤ j ↔ ¬ j < i; (ss)% ⊦ i < j → ¬ j < i; (s) Used terms: i, j, i0, j0