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Proof step 6 from proof ri-range-ext (previous step)

nused < # zbrar, sorted(zbrar, 0, nused), scan(key, zbrar, 0, nused) ≤ nused → key-range(zbrar, scan(key, zbrar, 0, nused), nused + 1, ri0) = key-range(zbrar, scan(key, zbrar, 0, nused), nused + 1, ri1), ∀ n. n ≤ nused → key-range(zbrar, n, nused + 1, ri0) = key-range(zbrar, n, nused + 1, ri1) ⊦ (key ∈ ri0 ↔ key ∈ ri1) ∧ (key ∈ ri0 → ri0[key] = ri1[key])

User applied rule apply lemma

And obtained the following new goals

Proof step 7:
nused ≤ # zbrar, sorted(zbrar, 0, nused), n = scan(key, zbrar, 0, nused) ⊦ n ≤ nused ∧ (n < nused → key ≤ zbrar[n].key) ∧ (∀ n0. n0 < n → zbrar[n0].key < key) ∧ (∀ n1. n < n1 ∧ n1 < nused → key < zbrar[n1].key)
Proof step 8:
nused < # zbrar, sorted(zbrar, 0, nused), scan(key, zbrar, 0, nused) ≤ nused, scan(key, zbrar, 0, nused) ≤ nused → key-range(zbrar, scan(key, zbrar, 0, nused), nused + 1, ri0) = key-range(zbrar, scan(key, zbrar, 0, nused), nused + 1, ri1), scan(key, zbrar, 0, nused) < nused → key ≤ zbrar[scan(key, zbrar, 0, nused)].key, ∀ n. n ≤ nused → key-range(zbrar, n, nused + 1, ri0) = key-range(zbrar, n, nused + 1, ri1), ∀ n0. n0 < scan(key, zbrar, 0, nused) → zbrar[n0].key < key, ∀ n1. scan(key, zbrar, 0, nused) < n1 ∧ n1 < nused → key < zbrar[n1].key ⊦ (key ∈ ri0 ↔ key ∈ ri1) ∧ (key ∈ ri0 → ri0[key] = ri1[key])

Rule argument

spec: znode-ops name: scan-strong seq: nused ≤ # zbrar, sorted(zbrar, 0, nused), n = scan(key, zbrar, 0, nused) ⊦ n ≤ nused ∧ (n < nused → key ≤ zbrar[n].key) ∧ (∀ n0. n0 < n → zbrar[n0].key < key) ∧ (∀ n1. n < n1 ∧ n1 < nused → key < zbrar[n1].key) subst: [nused, zbrar, n, key] → [nused, zbrar, scan(key, zbrar, 0, nused), key]

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s) Used terms: nused, zbrar, scan(key, zbrar, 0, nused), key