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Proof step 13 from proof mod-zero-one (previous step)

i0 % j + i % j < 2 * j, 0 ≤ i, 0 ≤ i0, 0 < j, 1 < (i % j + i0 % j) / j, 0 < (i % j + i0 % j) / j ⊦ (i % j + i0 % j) / j < (2 * j) / j, (i % j + i0 % j) / j = 1

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-div inst: name: num-ls seq: ⊦ 0 < j ∧ 0 ≤ i → (i / (* j) < j0 ↔ i < j * j0) subst: [j0, j, i] → [(2 * j) / j, j, i % j + i0 % j] flags: rewrite, current

Additional information

Used simplifier rules: div-mult-03: j ≠ 0 ⊦ (i * j) / j = i;%div-zero-ngr: ⊦ i < 0 ∧ 0 < j → ¬ 0 < i / j;%div-mult-03: j ≠ 0 ⊦ (i * j) / j = i;%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%add-c: ⊦ i + j = j + i; from specification int-basic2%lels-03: ⊦ i < j → (¬ ¬ j ≤ i ↔ false); from specification int-abs%two-ls: ⊦ 0 < j → (j < 2 ↔ j = 1); from specification int-basic1%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs Used terms: (2 * j) / j, j, i % j + i0 % j