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Proof step 3 from proof mult (previous step)

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

User applied rule insert elim lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: div-elim seq: ⊦ 0 < j ∧ 0 < i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j) subst: [i, j, i0, j0] → [i * j + i0, j, i1, j0] flags: <rotated> <all paths>

Additional information

Used simplifier rules: zero: j ≠ 0 ⊦ 0 / j = 0;% ⊦ i < j ↔ ¬ j ≤ i; (ss)%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs% ⊦ 0 ≤ j1 ∧ j1 < j → (i0 * j + j1 < j ↔ i0 ≤ 0); (s)% ⊦ i ≤ j → (j ≤ i ↔ i = j); (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ i + j ≤ i ↔ j ≤ 0; (s)% ⊦ 0 < i → (i * j ≤ 0 ↔ j ≤ 0); (s)%lf-03: ⊦ i < j ∧ i = i0 → i0 < j; (forward) from specification int-basic1% ⊦ i * 0 = 0; (s)% ⊦ i + 0 = i; (s) Used terms: i * j + i0, j, i1, j0