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Proof step 4 from proof div-rec-01 (previous step)

~ i / ~ j = (~ i - ~ j) / ~ j + 1, j < 0, i < 0, i ≤ j ⊦ i / j = (i - j) / j + 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
1 = i / j + (j - i) / j, i < 0, j < 0, i ≤ j ⊦ i / j = (i - j) / j + 1

Rule argument

Additional information

Used simplifier rules: neg-01: j ≠ 0 ⊦ ~ i / j = ~(i / j);%neg: j ≠ 0 ⊦ i / ~ j = ~(i / j);% ⊦ i - j = i0 ↔ i = i0 + j; (s)% ⊦ i + ~ j = i - j; (s)% ⊦ ~ i - ~ j = j - i; (s)% ⊦ ~ ~ i = i; (s)% ⊦ i < 0 ∧ 0 ≤ j → i ≤ j; (s)%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2% ⊦ i = i + j ↔ j = 0; (s)% ⊦ i - 0 = i; (s)% ⊦ ~ i = 0 ↔ i = 0; (s)