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Proof step 3 from proof replay-orphans-dentry (previous step)

⊦ (ri ⊖ ro)[dentrykey(ino, str), adr] = ri[dentrykey(ino, str), adr] ⊖ ro, dentrykey(ino, str) ∈ ro

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ ri = ri0 ↔ (∀ key. (key ∈ ri ↔ key ∈ ri0) ∧ (key ∈ ri → ri[key] = ri0[key]))
Proof step 5:
⊦ dentrykey(ino, str) ∈ ro, ∀ key. (key ∈ (ri ⊖ ro)[dentrykey(ino, str), adr] ↔ key ∈ ri[dentrykey(ino, str), adr] ⊖ ro) ∧ (key ∈ (ri ⊖ ro)[dentrykey(ino, str), adr] → (ri ⊖ ro)[dentrykey(ino, str), adr][key] = (ri[dentrykey(ino, str), adr] ⊖ ro)[key])

Rule argument

spec: store, inst: node-index name: k-Extension seq: ⊦ ri = ri0 ↔ (∀ key. (key ∈ ri ↔ key ∈ ri0) ∧ (key ∈ ri → ri[key] = ri0[key])) subst: [ri0, ri] → [ri[dentrykey(ino, str), adr] ⊖ ro, (ri ⊖ ro)[dentrykey(ino, str), adr]] flags: <all paths>

Additional information

Used terms: ri[dentrykey(ino, str), adr] ⊖ ro, (ri ⊖ ro)[dentrykey(ino, str), adr]