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Proof step 16 from proof ri-range-ext (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 17:
⊦ n ≠ 0 ∧ n + 1 < nused ∧ nused ≤ # zbrar → key-range(zbrar, n, nused, ri) = (ri |> zbrar[n - 1].key) |≤ zbrar[n].key
Proof step 18:
nused < # zbrar, scan(key, zbrar, 0, nused) < # zbrar, sorted(zbrar, 0, nused), scan(key, zbrar, 0, nused) ≤ nused, scan(key, zbrar, 0, nused) < nused → ¬ zbrar[scan(key, zbrar, 0, nused)].key < key, ∀ n1. scan(key, zbrar, 0, nused) < n1 ∧ n1 < nused → key < zbrar[n1].key, ∀ n0. n0 < scan(key, zbrar, 0, nused) → zbrar[n0].key < key, ∀ n. n ≤ nused → key-range(zbrar, n, nused + 1, ri0) = key-range(zbrar, n, nused + 1, ri1) ⊦ key ∈ key-range(zbrar, scan(key, zbrar, 0, nused), nused + 1, ri1), key ∈ key-range(zbrar, scan(key, zbrar, 0, nused), nused + 1, ri0), (key ∈ ri0 ↔ key ∈ ri1) ∧ (key ∈ ri0 → ri0[key] = ri1[key]), scan(key, zbrar, 0, nused) ≠ 0 ∧ scan(key, zbrar, 0, nused) + 1 < nused + 1 ∧ nused + 1 ≤ # zbrar
Proof step 29:
nused < # zbrar, scan(key, zbrar, 0, nused) < # zbrar, scan(key, zbrar, 0, nused) + 1 < nused + 1, sorted(zbrar, 0, nused), scan(key, zbrar, 0, nused) ≤ nused, nused + 1 ≤ # zbrar, scan(key, zbrar, 0, nused) < nused → ¬ zbrar[scan(key, zbrar, 0, nused)].key < key, ∀ n1. scan(key, zbrar, 0, nused) < n1 ∧ n1 < nused → key < zbrar[n1].key, ∀ n0. n0 < scan(key, zbrar, 0, nused) → zbrar[n0].key < key, ∀ n. n ≤ nused → key-range(zbrar, n, nused + 1, ri0) = key-range(zbrar, n, nused + 1, ri1) ⊦ scan(key, zbrar, 0, nused) = 0, key ∈ key-range(zbrar, scan(key, zbrar, 0, nused), nused + 1, ri1), key ∈ (ri0 |> zbrar[scan(key, zbrar, 0, nused) - 1].key) |≤ zbrar[scan(key, zbrar, 0, nused)].key, (key ∈ ri0 ↔ key ∈ ri1) ∧ (key ∈ ri0 → ri0[key] = ri1[key])

Rule argument

spec: zbranch-array-ops name: range-inner seq: ⊦ n ≠ 0 ∧ n + 1 < nused ∧ nused ≤ # zbrar → key-range(zbrar, n, nused, ri) = (ri |> zbrar[n - 1].key) |≤ zbrar[n].key subst: [ri, nused, n, zbrar] → [ri0, nused + 1, scan(key, zbrar, 0, nused), zbrar] flags: <all paths>

Additional information

Used terms: ri0, nused + 1, scan(key, zbrar, 0, nused), zbrar