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Proof step 42 from proof div-def-01 (previous step)

rational(i, j) / rational(abs(i0), j0) = rational(i * j0, j * abs(i0)), 0 < j, 0 < j0, i < 0 ⊦ i = 0, rational(i, j) / rational(i0, j0) = rational(i * j0, j * i0), i0 = 0, rational(abs(i0), j0) = 0r

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 43:
⊦ i < 0 → abs(i) = ~ i
Proof step 44:
i0 < 0, rational(i, j) / rational(~ i0, j0) = rational(i * j0, j * ~ i0), 0 < j, 0 < j0, i < 0 ⊦ i = 0, rational(i, j) / rational(i0, j0) = rational(i * j0, j * i0), i0 = 0, rational(~ i0, j0) = 0r

Rule argument

seq: ⊦ i < 0 → abs(i) = ~ i subst: [i] → [i0] rotate: F spec: int-abs inst: name: abs-08

Additional information

Used simplifier rules: ⊦ i * j = j * i;% ⊦ ¬ j < 0 → abs(j) = j;% ⊦ ¬ i < j → (i ≠ j ↔ j < i); Used terms: i0