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Proof step 5 from proof mod-abs-neg-neg (previous step)

i < 0, j < 0, i % j < 0 ⊦ i % j = 0, j + i |%| ~ j = i % j

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

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

Additional information

Used simplifier rules: zero-gels-01: ⊦ i < 0 ∧ 0 ≤ j → i < j; from specification int-abs%mod-neg: j ≠ 0 ⊦ i % ~ j = i % j; from specification int-div%neg-03: ⊦ ~ i + i = 0; from specification int-abs%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%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%mod-ngez-01: ⊦ i < 0 ∧ j < 0 → (0 ≤ i % j ↔ i % j = 0); from specification int-div Used terms: ~ j, i