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Proof step 45 from proof ex-modeq-half (previous step)

(i0 % j - j) % j < 0, j ≤ abs(i0 % j - j), 0 ≤ i0 % j, j / 2 < i0 % j, 0 < j, odd(j) ⊦ (i0 % j - j) % j + j = i0 % j ∧ j ≤ i0 % j + j / 2, ∃ i. ≡mod(i, i0, j) ∧ abs(i) ≤ j / 2

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-abs inst: name: Abs-neg seq: ⊦ j < 0 → abs(j) = 0 - j subst: [j] → [i0 % j - j] flags: rewrite

Additional information

Used simplifier rules: sub-ls: ⊦ i - j < i0 ↔ i < j + i0; from specification int-abs%ax-04: ⊦ i + 0 = i; 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%mod-less-01: ⊦ 0 < j → i % j < j; from specification int-div%neg-06: ⊦ 0 - i = ~ i; from specification int-abs%minus-sub: ⊦ ~(i - j) = j - i; from specification int-abs%sub-le-01: ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; from specification int-abs%lep-01: ⊦ i + j ≤ i ↔ j ≤ 0; from specification int-abs%anti: ⊦ i ≤ j → (j ≤ i ↔ i = j); from specification int-abs%transitivity: ⊦ i < i0 ∧ i0 < i1 → i < i1; (forward) from specification oelem[intlista]%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista]%div2-neg: ⊦ i / 2 < 0 → i < 0; (forward) from specification int-div%forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1%Abs-pos: ⊦ ¬ j < 0 → abs(j) = j; from specification int-abs%add-mult-two: ⊦ i + i = 2 * i; from specification int-mult Used terms: i0 % j - j