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Proof step 15 from proof keys-inodekeys (previous step)

∀ ns1. # ns1 < #(ns0 ++ n) → true → ns1.keys.inodes? ⊦ ns0.keys.inodes?, (ns0.keys ++ inodekey(n)).inodes?, n ∈ ns0

User applied rule apply induction

And closed the goal!

Rule argument

subst: [ns1] → [ns0]

Additional information

Used simplifier rules: ⊦ ¬ n ∈ ns → #(ns ++ n) = # ns + 1; (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ ¬ n < n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: ns0