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Proof step 17 from proof dups-eqsize-ls (previous step)

key ∈ ks, ∀ key. key ∈ ks → key.dentry?, ∀ key. key.dentry? → (key ∈ ks ↔ mklink(key.ino, key.name) ∈ ls), ∀ ls0, ks. # ls0 < # ls → (∀ key. key ∈ ks → key.dentry?) ∧ (∀ key. key.dentry? → (key ∈ ks ↔ mklink(key.ino, key.name) ∈ ls0)) → # ks = # ls0 ⊦ # ks = 0, # ks = # ls

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 18:
⊦ key ∈ ks ↔ (∃ ks0. ks = ks0 ++ key ∧ ¬ key ∈ ks0)
Proof step 19:
∀ key. key.dentry? → (key ∈ ks ↔ mklink(key.ino, key.name) ∈ ls), ∀ key. key ∈ ks → key.dentry?, ∃ ks0. ks = ks0 ++ key ∧ ¬ key ∈ ks0, ∀ ls0, ks. # ls0 < # ls → (∀ key. key ∈ ks → key.dentry?) ∧ (∀ key. key.dentry? → (key ∈ ks ↔ mklink(key.ino, key.name) ∈ ls0)) → # ks = # ls0 ⊦ # ks = # ls, # ks = 0

Rule argument

spec: set-basic, inst: node-index name: indef seq: ⊦ key ∈ ks ↔ (∃ ks0. ks = ks0 ++ key ∧ ¬ key ∈ ks0) subst: [ks, key] → [ks, key] flags: <all paths>

Additional information

Used terms: ks, key