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Proof step 8 from proof modmod-add2-posneg-neg-01 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ i % j = 0 ∧ i + i0 < 0 ∧ i < 0 ∧ 0 ≤ i0 ∧ i0 % j ≠ 0 ∧ 0 < j → (i + i0) % j = i0 % j - j

Rule argument

spec: <toplevel> name: mod-add1-negpos-neg seq: ⊦ i % j = 0 ∧ i + i0 < 0 ∧ i < 0 ∧ 0 ≤ i0 ∧ i0 % j ≠ 0 ∧ 0 < j → (i + i0) % j = i0 % j - j subst: [j, i0, i] → [j, i + j0, i1 * j] flags: <all paths>

Additional information

Used simplifier rules: mod-mult-04: j ≠ 0 ⊦ (i * j) % j = 0;% ⊦ i + (j - i) = j; (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ i * 0 = 0; (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ i + 0 = i; (s)% ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)% ⊦ 0 < i → (i * j < 0 ↔ j < 0); (s)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s) Used terms: j, i + j0, i1 * j