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Proof step 32 from proof div-div-pred-01 (previous step)

8 * i1 + j0 + j1 * 2 < 0, 8 * i1 + j0 + j1 * 2 < ~7, abs(j0) < 2, abs(j1) < 4, i1 * 4 + j1 < 0, j0 ≤ 0, j1 ≤ 0, i1 ≤ 0, i1 * 4 + j1 ≤ ~3 ⊦ i1 = (8 * i1 + j0 + j1 * 2) / 8, i1 = 0, i1 < 0 ∧ j0 + j1 * 2 < 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 33:
i1 < 0, 8 * i1 + j0 + j1 * 2 < 0, 8 * i1 + j0 + j1 * 2 < ~7, abs(j0) < 2, abs(j1) < 4, i1 * 4 + j1 < 0, i1 * 4 < j1 + j0, 0 ≤ j0 + j1 * 2, i1 * 4 + j1 ≤ ~3, j0 ≤ 0, j1 ≤ 0 ⊦ i1 = (8 * i1 + j0 + j1 * 2) / 8, i1 = 0

Rule argument

Additional information

Used simplifier rules: ⊦ 1 * i = i; (s)% ⊦ i + j < (* i1) * i + j0 ↔ j < (i1 - 1) * i + j0; (s)% ⊦ 0 < i → (i * j < 0 ↔ j < 0); (s)%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult% ⊦ i * 0 = 0; (s)% ⊦ (* i2) * i + j < (* i1) * i ↔ j < (i1 - i2) * i; (s)% ⊦ i + j < i + i0 ↔ j < i0; (s)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%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%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs% ⊦ i < j ↔ ¬ j ≤ i; (ss)% ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)% ⊦ i ≠ j → (i ≤ j ↔ i < j); (s)