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Proof step 12 from proof div-div-rec (previous step)

j + 1 = 2 ^ i * (i0 + 1) + j0, i0 - 1 = i1 * 2 + j1, i0 < i, j0 < i, abs(j0) < 2 ^ i, 0 < i, j < 0, 2 ^ i * i0 + j0 < 0, abs(j1) < 2, 0 < 2, i0 - 1 < 0, i0 ≤ 0, j0 ≤ 0, 0 ≤ i, j1 ≤ 0, i1 ≤ 0 ⊦ i1 = (j + 1 - 2 ^ i * 2) / (2 ^ i * 2)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
j + 1 = 2 ^ i * (2 * (i1 + 1) + j1) + j0, i1 * 2 + j1 < 0, abs(j1) < 2, 2 ^ i * ((i1 * 2 + j1) + 1) + j0 < 0, j < 0, 0 < i, abs(j0) < 2 ^ i, j0 < i, (i1 * 2 + j1) + 1 < i, i1 < i, j1 < i, i1 ≤ 0, j1 ≤ 0, 0 ≤ i, j0 ≤ 0 ⊦ i1 = (j + 1 - 2 ^ i * 2) / (2 ^ i * 2)

Rule argument

Additional information

Used simplifier rules: ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)%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% ⊦ i ≤ ~1 ↔ i < 0; (s)% ⊦ i + (* j) ≤ (* j0) ↔ i ≤ j0 - j; (s)%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult% ⊦ i * j + i = i * (j + 1); (s)%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%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)% ⊦ i < 1 ↔ i ≤ 0; (s)%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2% ⊦ i - j < i0 ↔ i < j + i0; (s)% ⊦ i - j = i0 ↔ i = i0 + j; (s)