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Proof step 8 from proof range-put (previous step)

n < # zbrar, n < nused, nused ≤ # zbrar, n ≠ 0 → zbrar[n - 1].key < key ⊦ zbrar[n].key < key, (key0 ∈ key-range(zbrar, n, nused, ri[key, adr]) ↔ key0 = key ∨ key0 ∈ key-range(zbrar, n, nused, ri)) ∧ (key0 ∈ key-range(zbrar, n, nused, ri[key, adr]) → ri[key, adr][key0] = key-range(zbrar, n, nused, ri)[key, adr][key0])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ n ≠ 0 ∧ n + 1 < nused ∧ nused ≤ # zbrar → key-range(zbrar, n, nused, ri) = (ri |> zbrar[n - 1].key) |≤ zbrar[n].key
Proof step 10:
n < nused, n < # zbrar, nused ≤ # zbrar, n ≠ 0 → zbrar[n - 1].key < key ⊦ zbrar[n].key < key, (key0 ∈ key-range(zbrar, n, nused, ri[key, adr]) ↔ key0 = key ∨ key0 ∈ key-range(zbrar, n, nused, ri)) ∧ (key0 ∈ key-range(zbrar, n, nused, ri[key, adr]) → ri[key, adr][key0] = key-range(zbrar, n, nused, ri)[key, adr][key0]), n ≠ 0 ∧ n + 1 < nused ∧ nused ≤ # zbrar
Proof step 25:
n < nused, n < # zbrar, n + 1 < nused, nused ≤ # zbrar, n ≠ 0 → zbrar[n - 1].key < key ⊦ n = 0, zbrar[n].key < key, (key0 ∈ (ri[key, adr] |> zbrar[n - 1].key) |≤ zbrar[n].key ↔ key0 = key ∨ key0 ∈ key-range(zbrar, n, nused, ri)) ∧ (key0 ∈ (ri[key, adr] |> zbrar[n - 1].key) |≤ zbrar[n].key → ri[key, adr][key0] = key-range(zbrar, n, nused, ri)[key, adr][key0])

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, adr], nused, n, zbrar] flags: <all paths>

Additional information

Used terms: ri[key, adr], nused, n, zbrar