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Proof step 38 from proof replay-one-update (previous step)

fs[ri[dentrykey(ino, str)]] = dentrynode(dentrykey(ino, str), m), ro.inodes?, orphans-cons(fs, ri, ro), ri-closed(ri, fs, ro), ri[dentrykey(ino, str)] ∈ fs, dentrykey(ino, str) ∈ ri, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]) ⊦ (m ≠ 0 ⊃ (ri ⊖ ro)[dentrykey(ino, str), adr];(ri ⊖ ro) -- dentrykey(ino, str)) = ri[dentrykey(ino, str), adr] ⊖ ro

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 39:
⊦ ri-closed(ri, fs, ro) ↔ (∀ key. key ∈ ri → ri[key] ∈ fs ∧ node-closed(fs[ri[key]], ri, fs, ro))
Proof step 40:
fs[ri[dentrykey(ino, str)]] = dentrynode(dentrykey(ino, str), m), ro.inodes?, orphans-cons(fs, ri, ro), dentrykey(ino, str) ∈ ri, ri[dentrykey(ino, str)] ∈ fs, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]), ∀ key. key ∈ ri → ri[key] ∈ fs ∧ node-closed(fs[ri[key]], ri, fs, ro) ⊦ (m ≠ 0 ⊃ (ri ⊖ ro)[dentrykey(ino, str), adr];(ri ⊖ ro) -- dentrykey(ino, str)) = ri[dentrykey(ino, str), adr] ⊖ ro

Rule argument

spec: aubifs-inv name: ri-closed seq: ⊦ ri-closed(ri, fs, ro) ↔ (∀ key. key ∈ ri → ri[key] ∈ fs ∧ node-closed(fs[ri[key]], ri, fs, ro)) subst: [ro, fs, ri] → [ro, fs, ri] flags: <all paths>

Additional information

Used terms: ro, fs, ri