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

⊦ # ns = 0, n = 0, ns = ∅, ns.min ∈ below(ns, ns.min + n)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ n ∈ below(ns, m) ↔ n ∈ ns ∧ n < m

Rule argument

spec: <toplevel> name: in-below seq: ⊦ n ∈ below(ns, m) ↔ n ∈ ns ∧ n < m subst: [m, ns, n] → [ns.min + n, ns, ns.min] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m < m + n ↔ n ≠ 0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ ns.min ∈ ns ↔ ns ≠ ∅; (ss) Used terms: ns.min + n, ns, ns.min