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Proof step 4 from proof scan-branch (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
zbrar[m + n] = mkzbranch(key, adr, r), n < nused, len0 < nused, sorted(zbrar, m, nused), m + nused ≤ # zbrar, n ≤ len0, ∀ zbrar, m, n, key, adr, r, nused. zbrar[m + n] = mkzbranch(key, adr, r) ∧ n < len0 ∧ n < nused ∧ sorted(zbrar, m, nused) ∧ m + nused ≤ # zbrar ∧ len0 ≤ nused → scan(key, zbrar, m, len0) = m + n ⊦ (0 = n ⊃ m;scan(key, zbrar, m + 1, len0)) = m + n

Rule argument

Additional information

Used simplifier rules: scan-rec: ⊦ scan(key, zbrar, m, n + 1) = (key ≤ zbrar[m].key ⊃ m;scan(key, zbrar, m + 1, n));% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ m + 1 ≤ n ↔ m < n; (s)%⊦ m1 < m ∧ zbrar[n + m1] = mkzbranch(key, adr, r) ∧ sorted(zbrar, n, m) → (zbrar[n].key < key ↔ 0 ≠ m1); (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ key0 ≤ key1 ↔ ¬ key1 < key0; (s)% ⊦ m +1 = m + 1; (s)