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Proof step 8 from proof repalyone-monotonic (previous step)

key ∈ replay-one(adr, fs[adr], ri0), key ∈ ri0 → key ∈ ri1, ∀ key. key ∈ ri0 → key ∈ ri1 ⊦ key ∈ replay-one(adr, fs[adr], ri1)

User applied rule structural induction

And obtained the following new goals

Proof step 9:
inodenode(key0, md, boolvar, ino, ino0, ino1) = fs[adr], key ∈ replay-one(adr, fs[adr], ri0), key ∈ ri0 → key ∈ ri1, ∀ key. key ∈ ri0 → key ∈ ri1 ⊦ key ∈ replay-one(adr, fs[adr], ri1)
Proof step 13:
dentrynode(key0, ino) = fs[adr], key ∈ replay-one(adr, fs[adr], ri0), key ∈ ri0 → key ∈ ri1, ∀ key. key ∈ ri0 → key ∈ ri1 ⊦ key ∈ replay-one(adr, fs[adr], ri1)
Proof step 15:
datanode(key0, buf) = fs[adr], key ∈ replay-one(adr, fs[adr], ri0), key ∈ ri0 → key ∈ ri1, ∀ key. key ∈ ri0 → key ∈ ri1 ⊦ key ∈ replay-one(adr, fs[adr], ri1)

Rule argument

term: fs[adr]

Additional information

Used simplifier rules: replay-trunc: ⊦ replay-one(adr, truncnode(key, n), ri) = ri \ datakeys(key, n, ri);%node freely generated by inodenode, dentrynode, datanode, truncnode;;% ⊦ key ∈ ri \ ks ↔ key ∈ ri ∧ ¬ key ∈ ks; (ws)%⊦ key ∈ datakeys(key0, n, ri) ↔ key ∈ datakeys(key0, ri) ∧ n ≤ key.part * VFS_PAGE_SIZE; (s)%⊦ key ∈ datakeys(key0, ri) ↔ key ∈ ri ∧ key.data? ∧ key0.inode? ∧ key.ino = key0.ino; (s)