[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 31 from proof dups-eqsize-entries (previous step)

key0.dentry?, key ∈ ks0 → key.dentry? ∧ key.ino = key0.ino, ∀ key. key.dentry? ∧ key.ino = key0.ino → (key = dentrykey(key.ino, str) ∨ key ∈ ks0 ↔ key.name ∈ entries0[str, ino]), ∀ key. key ∈ ks0 → key.dentry? ∧ key.ino = key0.ino, ∀ ks1, ino, entries. # ks1 < #(ks0 ++ dentrykey(key0.ino, str)) → (∀ key. key ∈ ks1 → key.dentry? ∧ key.ino = ino) ∧ (∀ key. key.dentry? ∧ key.ino = ino → (key ∈ ks1 ↔ key.name ∈ entries)) → # ks1 = # entries ⊦ # ks0 = # entries0, key0 ∈ ks0 ↔ key0.name ∈ entries0, str ∈ entries0, dentrykey(key0.ino, str) ∈ ks0

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 32:
key0 = dentrykey(m, str0), key0.dentry?, key ∈ ks0 → key.dentry? ∧ key.ino = m, ∀ key. key ∈ ks0 → key.dentry? ∧ key.ino = m, ∀ key. key.dentry? ∧ key.ino = m → (key = dentrykey(key.ino, str) ∨ key ∈ ks0 ↔ key.name ∈ entries0[str, ino]), ∀ ks1, ino, entries. # ks1 < #(ks0 ++ dentrykey(m, str)) → (∀ key. key ∈ ks1 → key.dentry? ∧ key.ino = ino) ∧ (∀ key. key.dentry? ∧ key.ino = ino → (key ∈ ks1 ↔ key.name ∈ entries)) → # ks1 = # entries ⊦ # ks0 = # entries0, key0 ∈ ks0 ↔ str0 ∈ entries0, dentrykey(m, str) ∈ ks0, str ∈ entries0

Rule argument

spec: key name: elim-03 seq: ⊦ key.dentry? → (m = key.ino ∧ str = key.name ↔ key = dentrykey(m, str)) subst: [key, m, str] → [key0, m, str0]

Additional information

Used simplifier rules: ⊦ key.dentry? → (m = key.ino ∧ str = key.name ↔ key = dentrykey(m, str)); (elim) Used terms: key0, m, str0