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Proof step 26 from proof ri-range-ext (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 27:
scan(key, zbrar, 0, nused) = 0, nused < # zbrar, sorted(zbrar, 0, nused), nused ≠ 0 → ¬ zbrar[0].key < key, ∀ n. n ≤ nused → key-range(zbrar, n, nused + 1, ri0) = key-range(zbrar, n, nused + 1, ri1), ∀ n. n ≠ 0 ∧ n < nused → key < zbrar[n].key ⊦ key ∈ key-range(zbrar, 0, nused + 1, ri1), key ∈ key-range(zbrar, 0, nused + 1, ri0), (key ∈ ri0 ↔ key ∈ ri1) ∧ (key ∈ ri0 → ri0[key] = ri1[key])

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ m < 0; (s)% ⊦ 0 ≤ m; (s)% ⊦ m < n → n ≠ 0; (s)% ⊦ 0 < m ↔ m ≠ 0; (s)%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%scan-min-eq: ⊦ m0 = scan(key, zbrar, m1, len) → m1 ≤ m0; (forward) from specification znode-ops