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Proof step 7 from proof modmod-add2-negpos-neg (previous step)

i + i1 * j < j, i + i1 * j < 0, i1 * j + i + j0 < j, j0 < j, 0 < j, i + 1 < j, i < j, i < 0, (i1 * j + i + j0) + 1 < j, 0 ≤ j0, 0 ≤ i1, i1 * j + i + j0 < 0, (i + i1 * j) + 1 < j ⊦ (i + j0) % j = (i1 * j + i + j0) % j

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-div inst: name: mod-add1-posneg-neg seq: ⊦ i % j = 0 ∧ i + i0 < 0 ∧ 0 ≤ i ∧ i0 < 0 ∧ 0 < j → (i + i0) % j = i0 % j subst: [j, i0, i] → [j, i + j0, i1 * j] flags: rewrite, current

Additional information

Used simplifier rules: mod-mult-04: j ≠ 0 ⊦ (i * j) % j = 0;%mod-mult-pos-02: ⊦ 0 < j ∧ 0 ≤ i ∧ 0 ≤ i0 → (i * j + i0) % j = i0 % j;%zero-left: ⊦ 0 * i = 0; 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%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%zero-ge: ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); from specification int-mult%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs Used terms: j, i + j0, i1 * j