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Proof step 33 from proof ex-modeq-half (previous step)

(i0 % j - j) % j = 0, 0 ≤ i0 % j, j / 2 < 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 34:
j ≠ 0 ⊦ i % j = 0 ↔ (∃ j0. i = j0 * j)
Proof step 35:
j ≠ 0, ∃ j0. i0 % j - j = j0 * j, 0 ≤ i0 % j, j / 2 < 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

Rule argument

spec: int-div inst: name: modex seq: j ≠ 0 ⊦ i % j = 0 ↔ (∃ j0. i = j0 * j) subst: [j, i] → [j, i0 % j - j] flags: rewrite

Additional information

Used simplifier rules: sub-zero-01: ⊦ i - 0 = i; from specification int-abs Used terms: j, i0 % j - j