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Proof step 3 from proof prod-div-less (previous step)

0 < (i * i1) / j, 1 < j, 0 < i, i < j, 0 < i1, i1 < j, 0 < j, i ≤ (i * i1) / j ⊦

User applied rule insert elim lemma

And obtained the following new goals

Proof step 4:
⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0)
Proof step 5:
i * i1 = i0 * j + j0, 0 < j, i1 < j, 0 < i1, i < j, 0 < i, 1 < j, 0 < i0, j0 < j, i ≤ i0, 0 ≤ j0, 0 ≤ i0, 0 ≤ i * i1 ⊦

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 * i1, j, i0, j0] flags: <rotated> <all paths>

Additional information

Used simplifier rules: ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ i ≤ j ↔ ¬ j < i; (ss)% ⊦ i < j → ¬ j < i; (s) Used terms: i * i1, j, i0, j0