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Proof step 31 from proof ri-range-ext (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 32:
⊦ n ≠ 0 ∧ n + 1 < nused ∧ nused ≤ # zbrar → key-range(zbrar, n, nused, ri) = (ri |> zbrar[n - 1].key) |≤ zbrar[n].key
Proof step 33:
scan(key, zbrar, 0, nused) = m + 1, scan(key, zbrar, 0, nused) < nused, nused < # zbrar, scan(key, zbrar, 0, nused) < # zbrar, scan(key, zbrar, 0, nused) + 1 < # zbrar, scan(key, zbrar, 0, nused) + 1 < nused + 1, sorted(zbrar, 0, nused), nused + 1 ≤ # zbrar, ∀ 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, zbrar[scan(key, zbrar, 0, nused)].key < key, key ∈ (ri1 |> zbrar[scan(key, zbrar, 0, nused) - 1].key) |≤ zbrar[scan(key, zbrar, 0, nused)].key, key ∈ ri0 ∧ zbrar[m].key < 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] → [ri1, nused + 1, scan(key, zbrar, 0, nused), zbrar] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m + n < m + n0 ↔ n < n0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m + 1 ≤ n ↔ m < n; (s) Used terms: ri1, nused + 1, scan(key, zbrar, 0, nused), zbrar