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Proof step 29 from proof ri-range-ext (previous step)

nused < # zbrar, scan(key, zbrar, 0, nused) < # zbrar, scan(key, zbrar, 0, nused) + 1 < nused + 1, sorted(zbrar, 0, nused), scan(key, zbrar, 0, nused) ≤ nused, nused + 1 ≤ # zbrar, scan(key, zbrar, 0, nused) < nused → ¬ zbrar[scan(key, zbrar, 0, nused)].key < key, ∀ n1. scan(key, zbrar, 0, nused) < n1 ∧ n1 < nused → key < zbrar[n1].key, ∀ n0. n0 < scan(key, zbrar, 0, nused) → zbrar[n0].key < key, ∀ n. n ≤ nused → key-range(zbrar, n, nused + 1, ri0) = key-range(zbrar, n, nused + 1, ri1) ⊦ scan(key, zbrar, 0, nused) = 0, key ∈ key-range(zbrar, scan(key, zbrar, 0, nused), nused + 1, ri1), key ∈ (ri0 |> zbrar[scan(key, zbrar, 0, nused) - 1].key) |≤ zbrar[scan(key, zbrar, 0, nused)].key, (key ∈ ri0 ↔ key ∈ ri1) ∧ (key ∈ ri0 → ri0[key] = ri1[key])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 30:
scan(key, zbrar, 0, nused) < nused, nused < # zbrar, scan(key, zbrar, 0, nused) < # zbrar, scan(key, zbrar, 0, nused) + 1 < # zbrar, sorted(zbrar, 0, nused), ∀ n1. scan(key, zbrar, 0, nused) < n1 ∧ n1 < nused → key < zbrar[n1].key, ∀ n0. n0 < scan(key, zbrar, 0, nused) → zbrar[n0].key < key, ∀ n. n ≤ nused → key-range(zbrar, n, nused + 1, ri0) = key-range(zbrar, n, nused + 1, ri1) ⊦ scan(key, zbrar, 0, nused) = 0, zbrar[scan(key, zbrar, 0, nused)].key < key, key ∈ key-range(zbrar, scan(key, zbrar, 0, nused), nused + 1, ri1), key ∈ ri0 ∧ zbrar[scan(key, zbrar, 0, nused) - 1].key < key, (key ∈ ri0 ↔ key ∈ ri1) ∧ (key ∈ ri0 → ri0[key] = ri1[key])

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s)% ⊦ m < n → n ≠ 0; (s)%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ key0 ∈ ri |> key1 ↔ key0 ∈ ri ∧ key1 < key0; (s)% ⊦ key0 ∈ ri |≤ key1 ↔ key0 ∈ ri ∧ ¬ key1 < key0; (s)% ⊦ m < n → m ≤ n; (s)% ⊦ m + n < m + n0 ↔ n < n0; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≤ n ↔ m < n; (s)