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Proof step 11 from proof zero-04 (previous step)

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

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Rule argument

spec: int-div inst: name: div-elim2 seq: ⊦ 0 < j ∧ i < 0 → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j) subst: [i, j, i0, j0] → [j, i0, i, j1] flags: rewrite, current

Additional information

Used simplifier rules: dist-rmerge: ⊦ i * j + i * j0 = i * (j + j0); from specification int-mult%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%lf-01: ⊦ i < j ∧ j < i0 → i +1 < i0; (forward) from specification int-basic1%trans: ⊦ i < j ∧ j < j0 → i < j0; (forward) from specification int-basic1%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%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%succ-01: ⊦ i +1 = i + 1; from specification int-basic2%lst-03: ⊦ i + j < i0 ∧ ¬ j < j0 → i + j0 < i0; from specification int-basic2%ls: ⊦ i + j < j ↔ i < 0; from specification int-basic2%lep: ⊦ i ≤ i + j ↔ 0 ≤ j; from specification int-abs%modzero-03: ⊦ i < 0 ∧ j0 < j ∧ 0 < j → ¬ 0 ≤ i * j + j0; from specification int-mult Used terms: j, i0, i, j1