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Proof step 2 from proof div-elim1 (previous step)

0 ≤ i, j < 0 ⊦ i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 + j < 0

User applied rule insert lemma

And obtained the following new goals

Rule argument

spec: int-div inst: name: div-elim-pos seq: ⊦ 0 < j ∧ 0 ≤ i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j) subst: [j, i, i0, j0] → [~ j, i, ~ i0, j0] flags: current, []

Additional information

Used simplifier rules: neg: j ≠ 0 ⊦ i / ~ j = ~(i / j);%mod-neg: j ≠ 0 ⊦ i % ~ j = i % j;%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%zero-left: ⊦ 0 * i = 0; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%add-zero-rev: ⊦ 0 + i = i; 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%lenls: ⊦ i ≤ j → ¬ j < i; from specification int-abs%neg-12: ⊦ ~ i = ~ j ↔ i = j; from specification int-abs%minus-left: ⊦ ~ i * j = ~(i * j); from specification int-mult%neg-02: ⊦ ~ ~ i = i; from specification int-abs%neg-07: ⊦ i < ~ j ↔ i + j < 0; from specification int-abs Used terms: ~ j, i, ~ i0, j0