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Proof step 20 from proof replay-orphans-inode (previous step)

key ∈ ri, key.data? → ¬ inodekey(key.ino) ∈ ro ⊦ (ri ⊖ ro)[key] = (ri[inodekey(ino), adr] ⊖ ro)[key], key = inodekey(ino), inodekey(ino) ∈ ro, key ∈ ro

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 21:
⊦ key ∈ ri ∧ ¬ key ∈ ks ∧ ¬ key.data? → (ri ⊖ ks)[key] = ri[key]
Proof step 22:
key ∈ ri, key.data? → ¬ inodekey(key.ino) ∈ ro ⊦ (ri ⊖ ro)[key] = (ri[inodekey(ino), adr] ⊖ ro)[key], key = inodekey(ino), key ∈ ro, inodekey(ino) ∈ ro, key ∈ ri ∧ ¬ key ∈ ro ∧ ¬ key.data?
Proof step 31:
key ∈ ri, key.data? → ¬ inodekey(key.ino) ∈ ro ⊦ ri[key] = (ri[inodekey(ino), adr] ⊖ ro)[key], key = inodekey(ino), key.data?, inodekey(ino) ∈ ro, 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] flags: <all paths>

Additional information

Used terms: key, ro, ri