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Proof step 25 from proof range-put (previous step)

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])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 26:
n + 1 < nused, n < # zbrar, n < nused, zbrar[n - 1].key < key, n + 1 < # zbrar, nused ≤ # zbrar ⊦ n = 0, zbrar[n].key < key, (key0 = key ∨ key0 ∈ ri ∧ zbrar[n - 1].key < key0 ∧ ¬ zbrar[n].key < key0 → key0 ∈ ((ri |> zbrar[n - 1].key) |≤ zbrar[n].key)[key, adr]) ∧ ( key0 ∈ ((ri |> zbrar[n - 1].key) |≤ zbrar[n].key)[key, adr] → ri[key, adr][key0] = ((ri |> zbrar[n - 1].key) |≤ zbrar[n].key)[key, adr][key0])

Rule argument

Additional information

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