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Proof step 12 from proof range-put (previous step)

n < nused, n < # zbrar, nused ≤ # zbrar, n ≠ 0 → zbrar[n - 1].key < key ⊦ n = 0, zbrar[n].key < key, n ≠ 0 ∧ n + 1 < nused, (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])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
zbrar[n - 1].key < key, n < # zbrar ⊦ n = 0, zbrar[n].key < key, (key0 = key ∨ key0 ∈ ri ∧ zbrar[n - 1].key < key0 → key0 ∈ (ri |> zbrar[n - 1].key)[key, adr]) ∧ (key0 ∈ (ri |> zbrar[n - 1].key)[key, adr] → ri[key, adr][key0] = (ri |> zbrar[n - 1].key)[key, adr][key0])

Rule argument

Additional information

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