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Proof step 11 from proof modmod-ma (previous step)

i % j * i0 % j < j, i % j * i0 % j < 0, 0 < j, 0 ≤ j0, 0 ≤ i0, 0 ≤ i, i % j * i0 % j + 1 < j ⊦ (i % j * i0 % j + j0) % j = ((i % j * i0 % j) % j + j0) % j

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-mult inst: name: zero-ls seq: ⊦ 0 < i → (i * j < 0 ↔ j < 0) subst: [j, i] → [i0 % j, i % j] flags: rewrite

Additional information

Used simplifier rules: mod-zero-02: ⊦ 0 ≤ i ∧ 0 < j → (i % j ≤ 0 ↔ i % j = 0);%mod-zero-01: ⊦ 0 < j ∧ 0 ≤ i → ¬ i % j < 0;%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%lf-01: ⊦ i < j ∧ j < i0 → i +1 < i0; (forward) from specification int-basic1%trans: ⊦ i < j ∧ j < j0 → i < j0; (forward) from specification int-basic1%lf-02: ⊦ i < j ∧ j = i0 → i < i0; (forward) from specification int-basic1%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%zero-left: ⊦ 0 * i = 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-rev: ⊦ 0 + i = 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 Used terms: i0 % j, i % j