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Proof step 15 from proof ex-modeq-half (previous step)

i0 % j < 0, i0 |%| j < j, j / 2 < abs(i0 |%| j), 0 < j, odd(j) ⊦ (i0 |%| j - j) |%| j = i0 |%| j ∧ j ≤ i0 |%| j + j / 2, ∃ i. ≡mod(i, i0, j) ∧ abs(i) ≤ j / 2

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 % j < 0, i0 % j < 0, i0 % j + j < j, j / 2 < abs(i0 % j + j), 0 < j, odd(j) ⊦ (i0 % j + j - j) |%| j = i0 % j + j ∧ j ≤ (i0 % j + j) + j / 2, ∃ i. ≡mod(i, i0, j) ∧ abs(i) ≤ j / 2

Rule argument

spec: 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