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Proof step 6 from proof scan-branch (previous step)

zbrar[m + n] = mkzbranch(key, adr, r), n = n0 +1, len0 < nused, n < nused, sorted(zbrar, m, nused), n ≤ len0, 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 ⊦ (0 = n ⊃ m;scan(key, zbrar, m + 1, len0)) = m + n

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 7:
zbrar[m + n0 + 1] = mkzbranch(key, adr, r), n0 < nused, n0 < len0, len0 < nused, n0 + 1 < nused, sorted(zbrar, m, 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 + 1, len0) = m + n0 + 1

Rule argument

Additional information

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