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Proof step 8 from proof scan-entry (previous step)

zbrar[m + n0 + 1] = mkzentry(key, adr), nused = nused0 +1, n0 + 1 < nused, len0 < nused, n0 < len0, n0 < nused, sorted(zbrar, m, nused), m + nused ≤ # zbrar, ∀ zbrar, m, n, key, adr, nused. zbrar[m + n] = mkzentry(key, adr) ∧ n < len0 ∧ n < nused ∧ sorted(zbrar, m, nused) ∧ m + nused ≤ # zbrar ∧ len0 ≤ nused → scan(key, zbrar, m, len0) = m + n ⊦ scan(key, zbrar, m + 1, len0) = m + n0 + 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
zbrar[m + n0 + 1] = mkzentry(key, adr), n0 < nused0, n0 < len0, m + nused0 < # zbrar, sorted(zbrar, m, nused0 + 1), len0 ≤ nused0, ∀ zbrar, m, n, key, adr, nused. zbrar[m + n] = mkzentry(key, adr) ∧ n < len0 ∧ n < nused ∧ sorted(zbrar, m, nused) ∧ m + nused ≤ # zbrar ∧ len0 ≤ nused → scan(key, zbrar, m, len0) = m + n ⊦ scan(key, zbrar, m + 1, len0) = m + n0 + 1

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s)%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → ¬ n < m; (s)% ⊦ m + n < m + n0 ↔ n < n0; (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m + 1 ≤ n ↔ m < n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m +1 = m + 1; (s)