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Proof step 20 from proof div-ls-mone (previous step)

j0 < 0, j0 < j, i0 ≤ 0, i0 < j, i0 * j + j0 < 0, i0 * j + j0 < j, (i0 * j + j0) + 1 < j, 0 < j + j0, i0 < j + j0, j0 + 1 < j, 1 < j, 0 < j ⊦ i0 < ~1 ↔ (i0 * j + j0) + 2 * j ≤ 0

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Used simplifier rules: lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs%mone-06: ⊦ ~1 < i ↔ 0 ≤ i; from specification int-abs%numls-03: ⊦ (* i2) * i + j < i + j0 ↔ (i2 - 1) * i + j < j0; from specification int-mult%one: ⊦ 1 * i = i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%ls: ⊦ i + j < j ↔ i < 0; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%lst-04: ⊦ i + j < i0 ∧ j0 < j → i + j0 < i0; from specification int-basic2%lf-01: ⊦ i < j ∧ j < i0 → i +1 < i0; (forward) from specification int-basic1%trans: ⊦ i < j ∧ j < j0 → i < j0; (forward) from specification int-basic1%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) 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%succ-01: ⊦ i +1 = i + 1; from specification int-basic2%ls-01: ⊦ i + j < i + i0 ↔ j < i0; from specification int-basic2%numls-03: ⊦ i + (* j) < (* j0) ↔ i < j0 - j; from specification int-abs%zero-ls: ⊦ 0 < i → (i * j < 0 ↔ j < 0); from specification int-mult%zero-gels-03: ⊦ i < 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%numle-03: ⊦ (* i2) * i + j ≤ i + j0 ↔ (i2 - 1) * i + j ≤ j0; from specification int-mult%lep-01: ⊦ i + j ≤ i ↔ j ≤ 0; from specification int-abs%mone-08: ⊦ i < 0 → (~1 ≤ i ↔ i = ~1); from specification int-abs%lf-02: ⊦ i < j ∧ j = i0 → i < i0; (forward) from specification int-basic1%minus-one-02: ⊦ ~1 * i = ~ i; from specification int-mult%neg-15: ⊦ i + ~ j = i - j; from specification int-abs%sub-ls: ⊦ i - j < i0 ↔ i < j + i0; from specification int-abs%add-zero-rev: ⊦ 0 + i = i; from specification int-basic2%neg-05: ⊦ ~ i < 0 ↔ 0 < i; from specification int-abs%neg-13: ⊦ ~ i < j ↔ 0 < j + i; from specification int-abs%minus-add: i < 0 ⊦ (* i) + j = j - ~ i; from specification int-abs%sub-ls-01: ⊦ i < j - i0 ↔ i + i0 < j; from specification int-abs%sub-add-01: ⊦ i + (j - i0) = i + j - i0; from specification int-abs%add-mult-two: ⊦ i + i = 2 * i; from specification int-mult%numsub: ⊦ i * (* j) + j0 - i = i * (j - 1) + j0; from specification int-mult%neg-03: ⊦ ~ i + i = 0; from specification int-abs