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Proof step 15 from proof ri-range-ext (previous step)

key ∈ key-range(zbrar, scan(key, zbrar, 0, nused), nused + 1, ri0) ↔ key ∈ key-range(zbrar, scan(key, zbrar, 0, nused), nused + 1, ri1), key-range(zbrar, scan(key, zbrar, 0, nused), nused + 1, ri0)[key] = key-range(zbrar, scan(key, zbrar, 0, nused), nused + 1, ri1)[key], nused < # zbrar, scan(key, zbrar, 0, nused) < # zbrar, sorted(zbrar, 0, nused), scan(key, zbrar, 0, nused) ≤ nused, 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) ⊦ (key ∈ ri0 ↔ key ∈ ri1) ∧ (key ∈ ri0 → ri0[key] = ri1[key]), key ∈ key-range(zbrar, scan(key, zbrar, 0, nused), nused + 1, ri0) ∧ scan(key, zbrar, 0, nused) < nused + 1 ∧ nused + 1 ≤ # zbrar

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 16:
scan(key, zbrar, 0, nused) < # zbrar, nused < # zbrar, sorted(zbrar, 0, nused), scan(key, zbrar, 0, nused) ≤ nused, scan(key, zbrar, 0, nused) < nused → ¬ zbrar[scan(key, zbrar, 0, nused)].key < key, ∀ n. n ≤ nused → key-range(zbrar, n, nused + 1, ri0) = key-range(zbrar, n, nused + 1, ri1), ∀ n0. n0 < scan(key, zbrar, 0, nused) → zbrar[n0].key < key, ∀ n1. scan(key, zbrar, 0, nused) < n1 ∧ n1 < nused → key < zbrar[n1].key ⊦ key ∈ key-range(zbrar, scan(key, zbrar, 0, nused), nused + 1, ri0), key ∈ key-range(zbrar, scan(key, zbrar, 0, nused), nused + 1, ri1), (key ∈ ri0 ↔ key ∈ ri1) ∧ (key ∈ ri0 → ri0[key] = ri1[key])

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ key ∈ ri → ri[key] = (∅)[key]; (s)% ⊦ m + 1 ≤ n ↔ m < n; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)