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Proof step 5 from proof ls (previous step)

0 < i, 0 < j, j0 < i, i1 < (i1 * i + j0) / j, i ≤ j, 0 ≤ j0, j0 < j, 0 ≤ i1, 0 < (i1 * i + j0) / j ⊦

User applied rule insert elim lemma

And obtained the following new goals

Rule argument

spec: int-div inst: name: div-elim seq: ⊦ 0 < j ∧ 0 < i → ( i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j) subst: [i, j, i0, j0] → [i1 * i + j0, j, i0, j1] flags: current, []

Additional information

Used simplifier rules: div-ls-same: ⊦ 0 < j → (i < i / j ↔ i < 0 ∧ 1 < j);%div-zero-ngr: ⊦ i < 0 ∧ 0 < j → ¬ 0 < i / j;%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%forward-lels: ⊦ i ≤ j ∧ j < j0 → i < j0; (forward) from specification int-abs%forward-le-04: ⊦ j ≤ i ∧ i ≤ 0 → j ≤ 0; (forward) from specification int-abs%forward-lele: ⊦ j ≤ j0 ∧ j0 ≤ j1 → j ≤ j1; (forward) from specification int-abs%ls-add: ⊦ 0 ≤ j1 ∧ j1 < j → (i0 * j + j1 < j ↔ i0 ≤ 0); from specification int-mult%anti: ⊦ i ≤ j → (j ≤ i ↔ i = j); from specification int-abs%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%zero-gels-03: ⊦ i < 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%lep-01: ⊦ i + j ≤ i ↔ j ≤ 0; from specification int-abs%zero-le: ⊦ 0 < i → (i * j ≤ 0 ↔ j ≤ 0); from specification int-mult Used terms: i1 * i + j0, j, i0, j1