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Proof step 9 from proof mod-mult-neg-02 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: mod-mult-neg seq: ⊦ 0 < j ∧ i ≤ 0 ∧ i0 ≤ 0 ∧ abs(i0) < j → (j * i + i0) % j = i0 subst: [i0, i, j] → [j0, i1 + i, j] flags: <all paths>

Additional information

Used simplifier rules: ⊦ i ≤ j ↔ ¬ j < i; (ss)%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs% ⊦ i < i + j ↔ 0 < j; (s)% ⊦ i ≤ j → (j < i ↔ false); (as)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: j0, i1 + i, j