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Proof step 3 from proof range-le (previous step)

n + 1 < nused, n + 1 < # zbrar, nused ≤ # zbrar ⊦ key-range(zbrar, n, nused, ri |≤ key) = key-range(zbrar, n, nused, ri), key < zbrar[n].key

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ n ≠ 0 ∧ n + 1 < nused ∧ nused ≤ # zbrar → key-range(zbrar, n, nused, ri) = (ri |> zbrar[n - 1].key) |≤ zbrar[n].key
Proof step 5:
n + 1 < # zbrar, n + 1 < nused, nused ≤ # zbrar ⊦ key-range(zbrar, n, nused, ri |≤ key) = key-range(zbrar, n, nused, ri), key < zbrar[n].key, n ≠ 0 ∧ n + 1 < nused ∧ nused ≤ # zbrar
Proof step 7:
n + 1 < # zbrar, n + 1 < nused, nused ≤ # zbrar ⊦ n = 0, ((ri |≤ key) |> zbrar[n - 1].key) |≤ zbrar[n].key = key-range(zbrar, n, nused, ri), key < zbrar[n].key

Rule argument

spec: <toplevel> 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] → [ri |≤ key, nused, n, zbrar] flags: <all paths>

Additional information

Used terms: ri |≤ key, nused, n, zbrar