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Proof step 14 from proof modmod-mult-03 (previous step)

0 < j0, 0 < j, i < 0 ⊦ (i % (j * j0)) % j = i % j

User applied rule insert lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: modmod-mult-04 seq: ⊦ 0 < j ∧ 0 ≤ i ∧ 0 < j0 → (i % (j * j0)) % j = i % j subst: [j, i, j0] → [j, ~ i, j0]

Additional information

Used simplifier rules: mod-neg-01: j ≠ 0 ⊦ ~ i % j = ~(i % j);% ⊦ i ≤ ~ j ↔ i + j ≤ 0; (s)% ⊦ i + 0 = i; (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ 0 < i → (i * j = 0 ↔ j = 0); (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ i * 0 = 0; (s)% ⊦ ~ i = ~ j ↔ i = j; (s)% ⊦ i ≤ j ↔ ¬ j < i; (ss)% ⊦ i < j → ¬ j < i; (s) Used terms: j, ~ i, j0