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Proof step 7 from proof modmod-add2-posneg-pos (previous step)

i1 * j + j0 < j, 0 ≤ i1 * j + i + j0, abs(j0) < j, 0 ≤ i, 0 < j, (i1 * j + j0) + 1 < j, j0 ≤ 0, j0 < j, i1 ≤ 0, i1 < j, i1 * j + j0 < 0, 0 < i, j0 < i, i1 < i, (i1 * j + j0) + 1 < i ⊦ (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-negpos-pos seq: ⊦ i % j = 0 ∧ 0 ≤ i + i0 ∧ i < 0 ∧ 0 ≤ i0 ∧ 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;%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-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%zero-ls: ⊦ 0 < i → (i * j < 0 ↔ j < 0); from specification int-mult%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%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-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%succ-01: ⊦ i +1 = i + 1; from specification int-basic2%ls: ⊦ i + j < j ↔ i < 0; from specification int-basic2%numls-03: ⊦ i + (* j) < (* j0) ↔ i < j0 - j; from specification int-abs%ls-01: ⊦ i + j < i + i0 ↔ j < i0; from specification int-basic2%ls-02: ⊦ i < i + j ↔ 0 < j; from specification int-basic2%zero-gr: ⊦ 0 < i → (0 < i * j ↔ 0 < j); from specification int-mult%lenls: ⊦ i ≤ j → ¬ j < i; from specification int-abs%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%anti: ⊦ i ≤ j → (j ≤ i ↔ i = j); 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%lf-02: ⊦ i < j ∧ j = i0 → i < i0; (forward) from specification int-basic1 Used terms: j, i + j0, i1 * j