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Proof step 30 from proof equal-minus-one (previous step)

1 / i ^ ~ j = ~1, j < 0 ⊦ i = 0, i = ~1, j = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 31:
0 < j, j ≤ i ⊦ i / j = (i - j) / j + 1
Proof step 32:
¬(0 < i ^ ~ j ∧ i ^ ~ j ≤ 1), 1 / i ^ ~ j = ~1, j < 0 ⊦ i = 0, i = ~1, j = 0

Rule argument

spec: int-div inst: name: div-rec seq: 0 < j, j ≤ i ⊦ i / j = (i - j) / j + 1 subst: [j, i] → [i ^ ~ j, 1] flags: rewrite

Additional information

Used simplifier rules: mone-02: ⊦ i + 1 = ~1 ↔ i + 1 + 1 = 0; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%numadd-01: ⊦ (* j) = i0 + (* j0) ↔ j - j0 = i0; from specification int-abs%bet: ⊦ 0 < i → (i ≤ 1 ↔ i = 1); from specification int-abs%sub-eq-01: ⊦ i = j → i - j = 0; from specification int-abs%zero: j ≠ 0 ⊦ 0 / j = 0; from specification int-div Used terms: i ^ ~ j, 1