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Proof step 17 from proof div-ls-mone (previous step)

~1 = (~ i + ~1) + i, ~ i + ~1 < i, (~ i + ~1) + 1 < (~ i + ~1) * j + 2 * j, 0 < (~ i + ~1) * j + 2 * j, ~ i + ~1 < 0, 1 + (~ i + ~1) * j < j, ~ i + ~1 < j, 0 < j, (~ i + ~1) + 1 < j, ~ i + ~1 < (~ i + ~1) * j + 2 * j, 0 < i, (~ i + ~1) + 1 < i ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-mult inst: name: dist-lsplit seq: ⊦ (i + i0) * j = i * j + i0 * j subst: [j, i0, i] → [j, ~1, ~ i] flags: rewrite

Additional information

Used simplifier rules: neg-03: ⊦ ~ i + i = 0; 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%neg-15: ⊦ i + ~ j = i - j; from specification int-abs%sub-ls: ⊦ i - j < i0 ↔ i < j + i0; from specification int-abs%add-mult-two: ⊦ i + i = 2 * i; from specification int-mult%mone-06: ⊦ ~1 < i ↔ 0 ≤ i; from specification int-abs%zero-ge: ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%add-zero-rev: ⊦ 0 + i = i; from specification int-basic2%minus-left: ⊦ ~ i * j = ~(i * j); from specification int-mult%minus-one-02: ⊦ ~1 * i = ~ i; from specification int-mult%neg-11: ⊦ ~ i + ~ j = ~(i + j); from specification int-abs%neg-13: ⊦ ~ i < j ↔ 0 < j + i; from specification int-abs%sub-add-01: ⊦ i + (j - i0) = i + j - i0; from specification int-abs%two-sub-01: ⊦ 2 * i0 + j - i0 + j0 = i0 + j - j0; from specification int-mult%sub-ls-01: ⊦ i < j - i0 ↔ i + i0 < j; from specification int-abs%numls-08: ⊦ i + j < (* i1) * i ↔ j < (i1 - 1) * i; from specification int-mult%one: ⊦ 1 * i = i; from specification int-mult%ls-same: ⊦ 0 < i → (i * j < i ↔ j < 1); from specification int-mult%bet-01: ⊦ 0 ≤ i → (i < 1 ↔ i = 0); from specification int-abs%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%zero-02: ⊦ 0 < i ∧ 0 < j → 0 < i + j; from specification int-basic2%sub-le-01: ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; from specification int-abs%zero-gr: ⊦ 0 < i → (0 < i * j ↔ 0 < j); from specification int-mult%lf-02: ⊦ i < j ∧ j = i0 → i < i0; (forward) from specification int-basic1%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%forward-pos: ⊦ i < j ∧ 0 < i → 0 < j ∧ 1 < j; (forward) from specification int-basic1%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs Used terms: j, ~1, ~ i