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

n < nused, n < # zbrar, nused ≤ # zbrar ⊦ key-range(zbrar, n, nused, ri |> key) = key-range(zbrar, n, nused, ri), n = 0, zbrar[n - 1].key < 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

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 simplifier rules: range-last: ⊦ n ≠ 0 ∧ n < # zbrar → key-range(zbrar, n, n + 1, ri) = ri |> zbrar[n - 1].key;%range-inner: ⊦ n ≠ 0 ∧ n + 1 < nused ∧ nused ≤ # zbrar → key-range(zbrar, n, nused, ri) = (ri |> zbrar[n - 1].key) |≤ zbrar[n].key;% ⊦ ¬ key1 < key0 → ri |> key0 |> key1 = ri |> key1; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m + 1 ↔ n = m + 1); (s)%eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic Used terms: ri |> key, nused, n, zbrar