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Proof step 4 from proof div-elim2 (previous step)

0 < j ∧ 0 < ~ i → (~ i0 = ~ i / j ∧ ~ j0 = ~ i % j ↔ ~ i = ~ i0 * j + ~ j0 ∧ 0 ≤ ~ j0 ∧ ~ j0 < j), i < j, i < 0, 0 < j, i + 1 < j ⊦ i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ j0 ≤ 0 ∧ abs(j0) < j

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: neg-01: j ≠ 0 ⊦ ~ i / j = ~(i / j);%mod-neg-01: j ≠ 0 ⊦ ~ i % j = ~(i % j);%neg-13: ⊦ ~ i < j ↔ 0 < j + i; from specification int-abs%negle-07: ⊦ i ≤ ~ j ↔ i + j ≤ 0; from specification int-abs%neg-11: ⊦ ~ i + ~ j = ~(i + j); from specification int-abs%minus-left: ⊦ ~ i * j = ~(i * j); from specification int-mult%neg-12: ⊦ ~ i = ~ j ↔ i = j; from specification int-abs%numls-03: ⊦ i + (* j) < (* j0) ↔ i < j0 - j; from specification int-abs%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1%neg-05: ⊦ ~ i < 0 ↔ 0 < i; from specification int-abs%abs-08: ⊦ i < 0 → abs(i) = ~ i; from specification int-abs%zero-gels-03: ⊦ i < 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%absls: ⊦ ¬ abs(i) < 0; from specification int-abs%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-zero-rev: ⊦ 0 + i = i; from specification int-basic2%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%zero-left: ⊦ 0 * i = 0; from specification int-mult%negeq: ⊦ ~ i = j ↔ i + j = 0; from specification int-abs%zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%neg: ⊦ 0 < ~ i ↔ i < 0; from specification int-abs