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Proof step 14 from proof modmod-add-04 (previous step)

j1 < j, j0 < j, 0 < j, 0 ≤ j0, 0 ≤ i1, 0 ≤ j1, 0 ≤ i2 ⊦ (j0 + j1) % j = ((j0 + j1) + j * (i2 + i1)) % j

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: inst: name: mod-add-pos seq: ⊦ i0 % j = 0 ∧ 0 ≤ i ∧ 0 ≤ i0 ∧ 0 < j → (i + i0) % j = i % j subst: [j, i0, i] → [j, j * (i2 + i1), j0 + j1] flags: rewrite, current

Additional information

Used simplifier rules: mod-mult-04: j ≠ 0 ⊦ (i * j) % j = 0;%Mult-zero: ⊦ i * 0 = 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: ⊦ i + 0 = 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%zero-ge: ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); from specification int-mult%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%zero-gs: ⊦ 0 ≤ i ∧ 0 ≤ j → ¬ i + j < 0; from specification int-abs Used terms: j, j * (i2 + i1), j0 + j1