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Proof step 29 from proof div-mult-01 (previous step)

j < 0, 0 < i0 ⊦ i < 0, j = 0, (i * i0) / (j * i0) = i / j

User applied rule insert lemma

And obtained the following new goals

Rule argument

spec: int-div inst: name: div-mult-02 seq: 0 < j, 0 < i0, 0 ≤ i ⊦ (i * i0) / (j * i0) = i / j subst: [j, i0, i] → [~ j, i0, i] flags: current, []

Additional information

Used simplifier rules: neg: j ≠ 0 ⊦ i / ~ j = ~(i / j);%Div-right: ⊦ j ≠ 0 → i / ~ j = ~(i / j);%minus-left: ⊦ ~ i * j = ~(i * j); from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%zero-neq-01: ⊦ 0 < i → (i * j = 0 ↔ j = 0); from specification int-mult%neg-12: ⊦ ~ i = ~ j ↔ i = j; from specification int-abs%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%neg: ⊦ 0 < ~ i ↔ i < 0; from specification int-abs%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs Used terms: ~ j, i0, i