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Proof step 8 from proof in-below (previous step)

∀ ns0, n, m. # ns0 < # ns → true → (n ∈ below(ns0, m) ↔ n ∈ ns0 ∧ n < m) ⊦ # ns = 0, n ∈ below(ns, m) ↔ n ∈ ns ∧ n < m, ∀ n. n ∈ ns ↔ n ∈ ∅

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
n0 ∈ ns, ∀ ns0, n, m. # ns0 < # ns → true → (n ∈ below(ns0, m) ↔ n ∈ ns0 ∧ n < m) ⊦ # ns = 0, n ∈ below(ns, m) ↔ n ∈ ns ∧ n < m

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ n ∈ ∅; (s)