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Proof step 42 from proof ex-modeq-half (previous step)

(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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 43:
⊦ abs(i) < j → i % j = i
Proof step 44:
¬ abs(i0 % j - j) < j, (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
Proof step 47:
abs(i0 % j - j) < j, i0 % 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

Rule argument

spec: int-div inst: name: mod-zero-05 seq: ⊦ abs(i) < j → i % j = i subst: [j, i] → [j, i0 % j - j] flags: rewrite

Additional information

Used terms: j, i0 % j - j