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Proof step 26 from proof max-exists (previous step)

ino ∈ dirs, ∀ dirs0, files0. # dirs0 < # dirs → (∃ ino. ino ≠ 0 ∧ (∀ n. n ∈ dirs0 ∨ n ∈ files0 → n < ino)), ∀ files0, dirs. # files0 < # ∅ → (∀ dirs0, files. # dirs0 < # dirs → true → (∃ ino. ino ≠ 0 ∧ (∀ n. n ∈ dirs0 ∨ n ∈ files → n < ino))) → (∃ ino. ino ≠ 0 ∧ (∀ n. n ∈ dirs ∨ n ∈ files0 → n < ino)) ⊦ ∃ ino. ino ≠ 0 ∧ (∀ n. n ∈ dirs → n < ino)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 27:
⊦ ino ≠ 0 → (ino ∈ dirs ↔ (∃ dirs0, ad. dirs = (dirs0 ++ ino)[ino, ad] ∧ ¬ ino ∈ dirs0))
Proof step 28:
∀ dirs0, files0. # dirs0 < # dirs → (∃ ino. ino ≠ 0 ∧ (∀ n. n ∈ dirs0 ∨ n ∈ files0 → n < ino)), ∃ dirs0, ad. dirs = (dirs0 ++ ino)[ino, ad] ∧ ¬ ino ∈ dirs0, ∀ files0, dirs. # files0 < # ∅ → (∀ dirs0, files. # dirs0 < # dirs → true → (∃ ino. ino ≠ 0 ∧ (∀ n. n ∈ dirs0 ∨ n ∈ files → n < ino))) → (∃ ino. ino ≠ 0 ∧ (∀ n. n ∈ dirs ∨ n ∈ files0 → n < ino)) ⊦ ino = 0, ∃ ino. ino ≠ 0 ∧ (∀ n. n ∈ dirs → n < ino)

Rule argument

spec: heap, inst: dir-heap name: exin seq: ⊦ ino ≠ 0 → ( ino ∈ dirs ↔ (∃ dirs0, ad. dirs = (dirs0 ++ ino)[ino, ad] ∧ ¬ ino ∈ dirs0)) subst: [dirs, ino] → [dirs, ino] flags: <all paths>

Additional information

Used simplifier rules: ⊦ ¬ 0 ∈ dirs; (s) Used terms: dirs, ino