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Proof step 27 from proof mod-ls-same-01 (previous step)

j0 + j < 0, 0 ≤ j0, j < i0 * j + j0, j + 1 < i0 * j + j0, j < 0, 0 < i0 * j + j0, i0 < i0 * j + j0, i0 ≤ 0, j + 1 < i0 * j ⊦ i0 < 0 ↔ ¬ (i0 * j + j0) + j < 0, ~1 = j, j = 1, i0 * j + j0 = 0

User applied rule cut formula

And closed the goal!

Rule argument

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

Additional information

Used simplifier rules: lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%ls-01: ⊦ i + j < i + i0 ↔ j < i0; from specification int-basic2%ls-same-03: ⊦ i < 0 → (i < i * j ↔ j < 1); from specification int-mult%numls-03: ⊦ i + (* j) < (* j0) ↔ i < j0 - j; from specification int-abs%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%zero-gels-03: ⊦ i < 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%lf-01: ⊦ i < j ∧ j < i0 → i +1 < i0; (forward) from specification int-basic1%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%trans: ⊦ i < j ∧ j < j0 → i < j0; (forward) from specification int-basic1%succ-01: ⊦ i +1 = i + 1; from specification int-basic2%ls: ⊦ i + j < j ↔ i < 0; from specification int-basic2%ls-add-03: ⊦ i * j + i0 < i * (j + j1) + j0 ↔ i0 < i * j1 + j0; from specification int-mult%one: ⊦ 1 * i = i; from specification int-mult%irref: ⊦ ¬ i < i; from specification int-basic1%ls-add-02: ⊦ i * (j + j1) + i0 < i * j + j0 ↔ i * j1 + i0 < j0; from specification int-mult%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-03: ⊦ i < j → (¬ ¬ j ≤ i ↔ false); from specification int-abs%anti: ⊦ i ≤ j → (j ≤ i ↔ i = j); from specification int-abs%lf-02: ⊦ i < j ∧ j = i0 → i < i0; (forward) from specification int-basic1%ls-02: ⊦ i < i + j ↔ 0 < j; from specification int-basic2%zero-gr-01: ⊦ i < 0 → (0 < i * j ↔ j < 0); from specification int-mult Used formulas: (i0 + 1) * j + j0 < 0