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Proof step 39 from proof congmod-def-mod-02 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 40:
⊦ i % j < 0 → i |%| j = i % j + j
Proof step 41:
i0 % j < 0, i0 % j < 0, 0 ≤ i % abs(j), i0 < 0, i < 0, 0 ≤ i % j ⊦ j = 0, i % abs(j) = i0 % j + abs(j) ↔ i % j = i0 % j + j

Rule argument

spec: int-mod inst: name: mod-neg-def seq: ⊦ i % j < 0 → i |%| j = i % j + j subst: [j, i] → [j, i0] flags: rewrite, current

Additional information

Used terms: j, i0