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Proof step 15 from proof congmod-def-mod-01 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 16:
⊦ i % j < 0 → i |%| j = i % j + j
Proof step 17:
¬ i0 % abs(j) < 0, 0 ≤ i, 0 ≤ i0, 0 ≤ i % abs(j), 0 ≤ i % j ⊦ i % abs(j) = i0 |%| abs(j) ↔ i % j = i0 |%| j, j = 0
Proof step 33:
i0 % abs(j) < 0, 0 ≤ i, 0 ≤ i0, 0 ≤ i % abs(j), 0 ≤ i % j ⊦ i % abs(j) = i0 % abs(j) + abs(j) ↔ i % j = i0 |%| j, j = 0

Rule argument

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

Additional information

Used terms: abs(j), i0