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Proof step 40 from proof less-mult-div-01 (previous step)

1 < j, 0 < j0, 0 < (j0 * i1) / j + i0 * i1, 0 < j, j0 < j, i1 < j, 0 < j1, 0 < i0 * j + j0, i0 * i1 + (j0 * i1) / j < i0 * i1 + j0, 0 ≤ i1, 0 ≤ i0, i0 * j + j0 ≤ (j0 * i1) / j + i0 * i1, 0 ≤ (j0 * i1) / j + i0 * i1 ⊦ i1 = 0, j0 = 0

User applied rule insert spec-lemma

And obtained the following new goals

Rule argument

spec: int-abs name: forward-lels seq: ⊦ i ≤ j ∧ j < j0 → i < j0 subst: [i, j, j0] → [i0 * j + j0, i0 * i1 + (j0 * i1) / j, i0 * i1 + j0]

Additional information

Used simplifier rules: prod-div-less-01: ⊦ 0 < i ∧ i < j ∧ 0 < i1 ∧ i1 < j → (i * i1) / j < i1;% ⊦ i ≠ j → (i ≤ j ↔ i < j); (s)% ⊦ i + j < i + i0 ↔ j < i0; (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%ls-forward: ⊦ i * j < i * j0 → i ≠ 0; (forward) from specification int-mult%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs% ⊦ 0 < i → (i * j < i * j0 ↔ j < j0); (s)% ⊦ i < j → ¬ j < i; (s) Used terms: i0 * j + j0, i0 * i1 + (j0 * i1) / j, i0 * i1 + j0