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Proof step 17 from proof auxexdivmod (previous step)

i < (i0 + i2 - 1) * j, i0 * j ≤ i, 0 ≤ i, 0 < j, 0 ≤ i0, i0 * j + i2 * j ≤ i, 0 < i0 + i2, 1 < i0 + i2, 1 < i2, 0 < i2, 0 ≤ i2, i0 * j + i2 * j < (i0 + i2 - 1) * j, ∀ i1, j0, i3. i1 * j0 ≤ i3 ∧ i3 < i1 * j0 + i2 * j0 ∧ 0 ≤ i1 ∧ 0 < j0 ∧ 0 ≤ i3 → (∃ i4. i4 * j0 ≤ i3 ∧ i3 < i4 * j0 + j0 ∧ 0 ≤ i4) ⊦ ∃ i1. i1 * j ≤ i ∧ i < i1 * j + j ∧ 0 ≤ i1

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-mult inst: name: dist-sub-split seq: ⊦ (i - j) * j0 = i * j0 - j * j0 subst: [j0, j, i] → [j, 1, i0 + i2] flags: rewrite, current

Additional information

Used simplifier rules: dist-lsplit: ⊦ (i + i0) * j = i * j + i0 * j;%associativity: ⊦ i * j * j0 = (i * j) * j0;%commutativity: ⊦ i * j = j * i;%one: ⊦ 1 * i = i;%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%sub-ls-01: ⊦ i < j - i0 ↔ i + i0 < j; from specification int-abs%ls-01: ⊦ i + j < i + i0 ↔ j < i0; from specification int-basic2%ls: ⊦ i + j < j ↔ i < 0; from specification int-basic2%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1 Used terms: j, 1, i0 + i2