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Proof step 2 from proof D-ext (previous step)

∀ ino. (ino ∈ dirs ↔ inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory) ∧ ( ino ∈ dirs → dirs[ino] = 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))) ⊦ dirs = abs-dirs(fs, ri)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 3:
⊦ dirs = dirs0 ↔ (∀ ino. (ino ∈ dirs ↔ ino ∈ dirs0) ∧ (ino ∈ dirs → dirs[ino] = dirs0[ino]))
Proof step 4:
∀ ino. (ino ∈ dirs ↔ inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory) ∧ ( ino ∈ dirs → dirs[ino] = 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))) ⊦ ∀ ino. (ino ∈ dirs ↔ ino ∈ abs-dirs(fs, ri)) ∧ (ino ∈ dirs → dirs[ino] = abs-dirs(fs, ri)[ino])

Rule argument

spec: heap, inst: dir-heap name: ext seq: ⊦ dirs = dirs0 ↔ (∀ ino. (ino ∈ dirs ↔ ino ∈ dirs0) ∧ (ino ∈ dirs → dirs[ino] = dirs0[ino])) subst: [dirs0, dirs] → [abs-dirs(fs, ri), dirs] flags: <all paths>

Additional information

Used terms: abs-dirs(fs, ri), dirs