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Proof step 9 from proof div-le-modnotzero (previous step)

0 < j0, 0 < j, 1 < j, j0 < j, i1 * j + j0 ≤ i0 * j, 0 ≤ i1, 0 ≤ i1 * j + j0, i0 ≤ i1, 0 ≤ i0 ⊦ j0 = 0

User applied rule insert spec-lemma

And obtained the following new goals

Rule argument

spec: int-mult name: modlep-02 seq: j * i + j0 ≤ j * i0, abs(j0) < j ⊦ i ≤ i0 subst: [j, i, j0, i0] → [j, i1, j0, i0]

Additional information

Used simplifier rules: ⊦ i ≤ j → (j ≤ i ↔ i = j); (s)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)% ⊦ i ≤ i; (s)% ⊦ i + j ≤ i ↔ j ≤ 0; (s)% ⊦ i ≠ j → (i ≤ j ↔ i < j); (s)% ⊦ i < j → ¬ j < i; (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ 0 < j → abs(j) = j; (s) Used terms: j, i1, j0, i0