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Proof step 9 from proof diveq-pos (previous step)

j0 < j, 0 < j, i0 * j + j0 < ((i0 * j + j0) / j + 1) * j, 0 ≤ j0, 0 ≤ i0, (i0 * j + j0) / j * j ≤ i0 * j + j0, i0 ≤ (i0 * j + j0) / j + 1, ~1 ≤ (i0 * j + j0) / j ⊦ (i0 * j + j0) / j = i0

User applied rule insert spec-lemma

And obtained the following new goals

Rule argument

spec: int-mult name: modlep seq: j * i ≤ j * i0 + j0, 0 ≤ j0, j0 < j ⊦ i ≤ i0 subst: [j, i, i0, j0] → [j, (i0 * j + j0) / j, i0, j0]

Additional information

Used simplifier rules: ⊦ i ≠ j → (i ≤ j ↔ i < j); (s)% ⊦ i < j → (j ≤ i + 1 ↔ j = i + 1); (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs% ⊦ 0 < i → (i * j < i * j0 ↔ j < j0); (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ ~1 < i ↔ 0 ≤ i; (s)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)% ⊦ j0 < j ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 ≤ j0 → (i * j + j0 < i0 * j ↔ i < i0); (s)% ⊦ ¬ i < i; (s) Used terms: j, (i0 * j + j0) / j, i0, j0