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Proof step 13 from proof mod-rec-neg (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ i % j < 0 → i |%| j = i % j + j
Proof step 15:
¬ ((i0 + 1) * j + j0) % j < 0, (i0 + 1) * j + j0 < 0, j0 < j, j0 < 0, i0 < j + j0, i0 < j, 0 < j, 0 < j + j0, i0 < 0 ⊦ j0 + j = ((i0 + 1) * j + j0) |%| j
Proof step 21:
((i0 + 1) * j + j0) % j < 0, (i0 + 1) * j + j0 < 0, j0 < j, j0 < 0, i0 < j + j0, i0 < j, 0 < j, 0 < j + j0, i0 < 0 ⊦ j0 + j = ((i0 + 1) * j + j0) % j + j

Rule argument

spec: inst: name: mod-neg-def seq: ⊦ i % j < 0 → i |%| j = i % j + j subst: [j, i] → [j, (i0 + 1) * j + j0] flags: rewrite, current

Additional information

Used terms: j, (i0 + 1) * j + j0