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Proof step 23 from proof modmod-add-case (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 24:
⊦ i + i0 % j < 0 ∧ 0 ≤ i ∧ i0 < 0 ∧ 0 < j → (i + i0 % j) % j = (i + i0) % j
Proof step 25:
i0 < 0, 0 < j, i + i0 < 0, 0 ≤ i ⊦ (i + i0) % j = (i + i0 % j) % j + j, (i + i0 % j) % j = (i + i0) % j, (i + i0 % j) % j = (i + i0) % j + j, i + i0 % j < 0 ∧ 0 ≤ i ∧ i0 < 0 ∧ 0 < j

Rule argument

spec: <toplevel> name: modmod-add2-posneg-neg seq: ⊦ i + i0 % j < 0 ∧ 0 ≤ i ∧ i0 < 0 ∧ 0 < j → (i + i0 % j) % j = (i + i0) % j subst: [j, i0, i] → [j, i0, i] flags: <all paths>

Additional information

Used terms: j, i0, i