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Proof step 34 from proof le-abs-numerator (previous step)

i0 * j + i0 < 0, 0 ≤ j0, j0 + j < 0, j < 0, i0 < 0, i0 < i0 * j + j0, j < i0 * j + j0, (i0 * j + j0) + i0 < 0 ⊦ j = 0

User applied rule cut formula

And obtained the following new goals

Proof step 35:
i0 * j + i0 < 0, 0 ≤ j0, j0 + j < 0, j < 0, i0 < 0, i0 < i0 * j + j0, j < i0 * j + j0, (i0 * j + j0) + i0 < 0 ⊦ i0 * (j + 1) + j0 < 0, j = 0

Rule argument

fma: i0 * (j + 1) + j0 < 0

Additional information

Used simplifier rules: forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%ls: ⊦ i + j < j ↔ i < 0; from specification int-basic2%zero-ls-01: ⊦ i < 0 → (i * j < 0 ↔ 0 < j); from specification int-mult%numls-02: ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0; from specification int-abs%mone-06: ⊦ ~1 < i ↔ 0 ≤ i; from specification int-abs%lels-02: ⊦ i ≠ j → (i ≤ j ↔ i < j); from specification int-abs%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult Used formulas: i0 * (j + 1) + j0 < 0