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Proof step 7 from proof modmod-add-case (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 8:
⊦ i + i0 < 0 ∧ i < 0 ∧ 0 ≤ i0 ∧ 0 < j → (i + i0 % j) % j = (i + i0) % j
Proof step 9:
¬(i + i0 < 0 ∧ i < 0 ∧ 0 ≤ i0 ∧ 0 < j), i < j, i < 0, 0 < j, i + 1 < j ⊦ (i + i0) % j = (i + i0 % j) % j + j, (i + i0 % j) % j = (i + i0) % j + j, (i + i0 % j) % j = (i + i0) % j

Rule argument

spec: int-div inst: name: modmod-add2-negpos-neg seq: ⊦ i + i0 < 0 ∧ i < 0 ∧ 0 ≤ i0 ∧ 0 < j → (i + i0 % j) % j = (i + i0) % j subst: [j, i0, i] → [j, i0, i] flags: rewrite, current

Additional information

Used terms: j, i0, i