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Proof step 3 from proof modeq-pos-04 (previous step)

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

User applied rule insert lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: modeq-pos-03 seq: 0 ≤ i0 * j + j0, 0 < j, j0 < 0, j0 % j ≠ 0 ⊦ (i0 * j + j0) % j = j0 % j + j subst: [i0, j, j0] → [i0, j, ~ j0]

Additional information

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