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Proof step 4 from proof div-elim-posneg (previous step)

i = i1 * j + j1 ∧ 0 ≤ j1 ∧ j1 < abs(j), 0 ≤ i, j < 0 ⊦ i0 = i1 ∧ j0 = j1 ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 + j < 0 ∧ i0 ≤ 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs%forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%lsone: ⊦ i < 1 ↔ i ≤ 0; from specification int-abs%ls-same-03: ⊦ i < 0 → (i < i * j ↔ j < 1); from specification int-mult%ls-01: ⊦ i + j < i + i0 ↔ j < i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%ls: ⊦ i + j < j ↔ i < 0; from specification int-basic2%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%neg-07: ⊦ i < ~ j ↔ i + j < 0; from specification int-abs%abs-08: ⊦ i < 0 → abs(i) = ~ i; from specification int-abs