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Proof step 11 from proof ex-modeq-half (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 12:
odd(j), 0 < j, j / 2 < abs(i0 |%| j), i0 |%| j < j ⊦ ∃ i. ≡mod(i, i0, j) ∧ abs(i) ≤ j / 2, (i0 |%| j - j) |%| j = i0 |%| j ∧ j ≤ i0 |%| j + j / 2

Rule argument

Additional information

Used simplifier rules: sub-le: ⊦ i - j ≤ i0 ↔ i ≤ j + i0; from specification int-abs%minus-sub: ⊦ ~(i - j) = j - i; from specification int-abs%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%ax-04: ⊦ i + 0 = i; from specification int-basic2%sub-ls: ⊦ i - j < i0 ↔ i < j + i0; from specification int-abs