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Proof step 8 from proof gcd-negone-eq (previous step)

~1 = j2 / gcd(i2, j2), 0 ≤ gcd(i2, j2) ⊦ j2 + gcd(i2, j2) = 0, 1 = ~ j2 / gcd(i2, j2)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
j ≠ 0 ⊦ i < 0 → (* i) / j = ~(~ i / j)

Rule argument

spec: int-div name: num-neg seq: j ≠ 0 ⊦ i < 0 → (* i) / j = ~(~ i / j) subst: [j, i] → [gcd(i2, j2), j2] flags: <all paths>

Additional information

Used simplifier rules: gcd-zero-zero: ⊦ gcd(i, j) = 0 ↔ i = 0 ∧ j = 0;%gcd-zero: ⊦ gcd(0, 0) = 0;%j ≠ 0 ⊦ ~ i / j = ~(i / j); (s)% ⊦ ~ ~ i = i; (s)% ⊦ ~ i = j ↔ i + j = 0; (s)% ⊦ (* j) = i0 + (* j0) ↔ j - j0 = i0; (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs Used terms: gcd(i2, j2), j2