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Proof step 9 from proof mod-neg (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ i % j < 0 → i |%| j = i % j + j
Proof step 11:
¬ ~(i0 * j + j0) % j < 0, 0 < j + j0, i0 * j + j0 < j, i0 * j + j0 < 0, j0 < j, j0 < 0, i0 < j + j0, i0 < j, i0 ≤ 0, 0 < j ⊦ j0 + ~(i0 * j + j0) |%| j = 0

Rule argument

spec: int-mod inst: name: mod-neg-def seq: ⊦ i % j < 0 → i |%| j = i % j + j subst: [j, i] → [j, ~(i0 * j + j0)] flags: rewrite, current

Additional information

Used simplifier rules: 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%ax-04: ⊦ 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%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1%mod-neg-01: j ≠ 0 ⊦ ~ i % j = ~(i % j); from specification int-div%neg-05: ⊦ ~ i < 0 ↔ 0 < i; from specification int-abs%mod-grz: ⊦ 0 < j ∧ i < 0 → ¬ 0 < i % j; from specification int-div Used terms: j, ~(i0 * j + j0)