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Proof step 22 from proof mod-rec-neg (previous step)

(i0 + 1) * j + j0 < 0, ((i0 + 1) * j + j0) % j < j + j0, ((i0 + 1) * j + j0) % j < 0, i0 < 0, 0 < j + j0, 0 < j, i0 < j, i0 < j + j0, j0 < 0, j0 < j ⊦ j0 = ((i0 + 1) * j + j0) % j

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 23:
⊦ 0 < j ∧ i ≤ 0 ∧ i0 ≤ 0 → (i * j + i0) % j = i0 % j
Proof step 24:
0 < j ∧ i0 + 1 ≤ 0 ∧ j0 ≤ 0, (i0 + 1) * j + j0 < 0, j0 % j < j + j0, j0 % j < 0, i0 < 0, 0 < j + j0, 0 < j, i0 < j, i0 < j + j0, j0 < 0, j0 < j ⊦ j0 = j0 % j

Rule argument

spec: int-div inst: name: mod-mult-neg-03 seq: ⊦ 0 < j ∧ i ≤ 0 ∧ i0 ≤ 0 → (i * j + i0) % j = i0 % j subst: [i0, j, i] → [j0, j, i0 + 1] flags: rewrite

Additional information

Used simplifier rules: numle-03: ⊦ i + (* j) ≤ (* j0) ↔ i ≤ j0 - j; from specification int-abs%mone-05: ⊦ i ≤ ~1 ↔ i < 0; from specification int-abs%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%zero-gels-03: ⊦ i < 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs Used terms: j0, j, i0 + 1