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Proof step 5 from proof abs-dirs-unused-adrs (previous step)

range ri ∩ as = ∅, ri[inodekey(ino)] ∈ as ⊦ # as = 0, as = ∅, (inodekey(ino) ∈ ri → (fs[ri[inodekey(ino)]].directory ↔ (∅)[ri[inodekey(ino)]].directory)) ∧ ( inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory → mkdir (fs[ri[inodekey(ino)]].size, fs[ri[inodekey(ino)]].meta, fs[ri[inodekey(ino)]].nlink, fs[ri[inodekey(ino)]].nsubdirs, abs-children(ino, fs, ri)) = mkdir ((∅)[ri[inodekey(ino)]].size, (∅)[ri[inodekey(ino)]].meta, (∅)[ri[inodekey(ino)]].nlink, (∅)[ri[inodekey(ino)]].nsubdirs, abs-children(ino, fs \ as, ri)))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ as = as0 ↔ (∀ adr. adr ∈ as ↔ adr ∈ as0)
Proof step 7:
ri[inodekey(ino)] ∈ as, ∀ adr. adr ∈ range ri ∩ as ↔ adr ∈ ∅ ⊦ # as = 0, as = ∅, (inodekey(ino) ∈ ri → (fs[ri[inodekey(ino)]].directory ↔ (∅)[ri[inodekey(ino)]].directory)) ∧ ( inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory → mkdir (fs[ri[inodekey(ino)]].size, fs[ri[inodekey(ino)]].meta, fs[ri[inodekey(ino)]].nlink, fs[ri[inodekey(ino)]].nsubdirs, abs-children(ino, fs, ri)) = mkdir ((∅)[ri[inodekey(ino)]].size, (∅)[ri[inodekey(ino)]].meta, (∅)[ri[inodekey(ino)]].nlink, (∅)[ri[inodekey(ino)]].nsubdirs, abs-children(ino, fs \ as, ri)))

Rule argument

spec: set-basic, inst: node-store name: Extension seq: ⊦ as = as0 ↔ (∀ adr. adr ∈ as ↔ adr ∈ as0) subst: [as0, as] → [∅, range ri ∩ as] flags: <all paths>

Additional information

Used terms: ∅, range ri ∩ as