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Proof step 27 from proof ex-modeq-half (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 28:
⊦ i % j < 0 → i |%| j = i % j + j
Proof step 29:
¬ (i0 % j - j) % j < 0, odd(j), 0 < j, j / 2 < i0 % j, 0 ≤ i0 % j ⊦ ∃ i. ≡mod(i, i0, j) ∧ abs(i) ≤ j / 2, (i0 % j - j) |%| j = i0 % j ∧ j ≤ i0 % j + j / 2
Proof step 42:
(i0 % j - j) % j < 0, odd(j), 0 < j, j / 2 < i0 % j, 0 ≤ i0 % j ⊦ ∃ i. ≡mod(i, i0, j) ∧ abs(i) ≤ j / 2, (i0 % j - j) % j + j = i0 % j ∧ j ≤ i0 % j + j / 2

Rule argument

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

Additional information

Used terms: j, i0 % j - j