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Proof step 3 from proof orphans-cons-gc-one (previous step)

orphans-cons(fs, ri, ro), ri-cons(ri, fs), key ∈ ri ⊦ orphans-cons(fs[adr, fs[ri[key]]], ri[key, adr], ro), adr ∈ fs

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ orphans-cons(fs, ri, ro) ↔ (∀ ino. inodekey(ino) ∈ ri → (fs[ri[inodekey(ino)]].nlink = 0 ∧ ino ≠ ROOT_INO ↔ inodekey(ino) ∈ ro))
Proof step 5:
orphans-cons(fs, ri, ro), ri-cons(ri, fs), key ∈ ri ⊦ adr ∈ fs, ∀ ino. inodekey(ino) ∈ ri[key, adr] → (fs[adr, fs[ri[key]]][ri[key, adr][inodekey(ino)]].nlink = 0 ∧ ino ≠ ROOT_INO ↔ inodekey(ino) ∈ ro)

Rule argument

spec: <toplevel> name: orphans-cons seq: ⊦ orphans-cons(fs, ri, ro) ↔ (∀ ino. inodekey(ino) ∈ ri → ( fs[ri[inodekey(ino)]].nlink = 0 ∧ ino ≠ ROOT_INO ↔ inodekey(ino) ∈ ro)) subst: [ro, ri, fs] → [ro, ri[key, adr], fs[adr, fs[ri[key]]]] flags: <all paths>

Additional information

Used terms: ro, ri[key, adr], fs[adr, fs[ri[key]]]