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Proof step 5 from proof ri-cons-del-orphans (previous step)

key-cons(key, fs[ri[key]]), key ∈ ri, ri[key] ∈ fs, key.data? → ¬ inodekey(key.ino) ∈ ro ⊦ key ∈ ro, (ri ⊖ ro)[key] ∈ fs ∧ key-cons(key, fs[(ri ⊖ ro)[key]])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ key ∈ ri ∧ ¬ key ∈ ks ∧ ¬ key.data? → (ri ⊖ ks)[key] = ri[key]
Proof step 7:
key-cons(key, fs[ri[key]]), ri[key] ∈ fs, key ∈ ri, key.data? → ¬ inodekey(key.ino) ∈ ro ⊦ key ∈ ro, (ri ⊖ ro)[key] ∈ fs ∧ key-cons(key, fs[(ri ⊖ ro)[key]]), key ∈ ri ∧ ¬ key ∈ ro ∧ ¬ key.data?

Rule argument

spec: aubifs-base name: at-dels-ro-nodata seq: ⊦ key ∈ ri ∧ ¬ key ∈ ks ∧ ¬ key.data? → (ri ⊖ ks)[key] = ri[key] subst: [key, ks, ri] → [key, ro, ri] flags: <all paths>

Additional information

Used terms: key, ro, ri