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Proof step 16 from proof replay-cons-add-orphan (previous step)

log-cons(log, fs), replay-safe(log, fs, fi ⊖ fo), key ∈ ri, key.data? → key.ino ≠ ino ∧ ¬ inodekey(key.ino) ∈ ro, key.data? → ¬ inodekey(key.ino) ∈ ro, key.data? → key.ino ≠ ino ⊦ (ri ⊖ (ro ++ inodekey(ino)))[key] = ((ri ⊖ inodekey(ino)) ⊖ ro)[key], key = inodekey(ino), key ∈ ro

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 17:
⊦ key ∈ ri ∧ ¬ key ∈ ks ∧ ¬ key.data? → (ri ⊖ ks)[key] = ri[key]
Proof step 18:
log-cons(log, fs), replay-safe(log, fs, fi ⊖ fo), key ∈ ri, key.data? → key.ino ≠ ino, key.data? → ¬ inodekey(key.ino) ∈ ro, key.data? → key.ino ≠ ino ∧ ¬ inodekey(key.ino) ∈ ro ⊦ (ri ⊖ (ro ++ inodekey(ino)))[key] = ((ri ⊖ inodekey(ino)) ⊖ ro)[key], key = inodekey(ino), key ∈ ro, key ∈ ri ⊖ inodekey(ino) ∧ ¬ key ∈ ro ∧ ¬ key.data?
Proof step 32:
log-cons(log, fs), replay-safe(log, fs, fi ⊖ fo), key ∈ ri, key ∈ ri ⊖ inodekey(ino), key.data? → key.ino ≠ ino, key.data? → ¬ inodekey(key.ino) ∈ ro, key.data? → key.ino ≠ ino ∧ ¬ inodekey(key.ino) ∈ ro ⊦ (ri ⊖ (ro ++ inodekey(ino)))[key] = (ri ⊖ inodekey(ino))[key], key = inodekey(ino), key.data?, key ∈ ro

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 ⊖ inodekey(ino)] flags: <all paths>

Additional information

Used terms: key, ro, ri ⊖ inodekey(ino)