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Proof step 39 from proof congmod-inverse (previous step)

gcd(i, j) = 1, ~1 = (i * i0) % abs(j), 0 < i0, i < 0, i < i0 ⊦ j = 0, j | i * i0 - 1, j | i0 * i + 1, ∃ i0. ≡mod(i * i0, 1, j)

User applied rule insert elim lemma

And obtained the following new goals

Proof step 40:
i * i0 = i1 * abs(j) + j0 ∧ j0 ≤ 0 ∧ abs(j0) < abs(j) ∧ i1 ≤ 0 ∧ 0 < abs(j) ∧ i * i0 < 0, gcd(i, j) = 1, ~1 = j0, 0 < i0, i < 0, i < i0 ⊦ j = 0, j | i * i0 - 1, j | i0 * i + 1, ∃ i0. ≡mod(i * i0, 1, j)

Rule argument

spec: int-div inst: name: div-elim-negpos seq: ⊦ 0 < j ∧ i < 0 → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j ∧ i0 ≤ 0) subst: [i, j, i0, j0] → [i * i0, abs(j), i1, j0] flags:

Additional information

Used simplifier rules: abs-06: ⊦ 0 < abs(i) ↔ i ≠ 0; from specification int-abs%zero-ls: ⊦ 0 < i → (i * j < 0 ↔ j < 0); from specification int-mult%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%⊦ 0 < j ∧ i < 0 → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j ∧ i0 ≤ 0); (elim) Used terms: i * i0, abs(j), i1, j0