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Proof step 3 from proof scan-entry (previous step)

zbrar[m + n] = mkzentry(key, adr), n < len, n < nused, sorted(zbrar, m, nused), m + nused ≤ # zbrar, len ≤ nused ⊦ scan(key, zbrar, m, len) = m + n

User applied rule structural induction

And obtained the following new goals

Proof step 4:
zbrar[m + n] = mkzentry(key, adr), n < nused, n < len0 +1, sorted(zbrar, m, nused), len0 +1 ≤ 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, len0 +1) = m + n

Rule argument

indvar: len

Additional information

Used simplifier rules: scan-base: ⊦ scan(key, zbrar, m, 0) = m;%nat freely generated by 0, +1;;% ⊦ m = m + n ↔ n = 0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ 0 ≤ m; (s)% ⊦ ¬ m < 0; (s)