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Proof step 20 from proof add-div-mult (previous step)

0 < j, j0 < j, j1 < j, 0 ≤ j * i2 + j0 + j1, 0 ≤ j0, 0 ≤ i1, 0 ≤ j1, 0 ≤ i2, (j0 + j1) / j = 1 ∨ (j0 + j1) / j = 0 ⊦ i2 + (j0 + j1) / j = (j * i2 + j0 + j1) / j

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: div-add1-pospos seq: ⊦ i % j = 0 ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 < j → (i + i0) / j = i / j + i0 / j subst: [j, i0, i] → [j, j0 + j1, j * i2] flags: <all paths>

Additional information

Used simplifier rules: div-mult-pos-01: ⊦ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 < j → (i + i0 * j) / j = i / j + i0;%div-mult-03: j ≠ 0 ⊦ (i * j) / j = i;% ⊦ i * 0 = 0; (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ i < 0 ∧ j < 0 → ¬ 0 ≤ i + j; (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ i + j = i + i0 ↔ j = i0; (s)% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); (s)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)% ⊦ i ≤ j ↔ ¬ j < i; (ss)% ⊦ 0 ≤ i ∧ 0 ≤ j → ¬ i + j < 0; (s) Used terms: j, j0 + j1, j * i2