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

scan(key, zbrar, m, len0 +1) < m + len0 +1, zbrar[scan(key, zbrar, m, len0 +1)].key < key, ∀ zbrar, key, m. ¬(zbrar[scan(key, zbrar, m, len0)].key < key ∧ scan(key, zbrar, m, len0) < m + len0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
zbrar[(¬ zbrar[m].key < key ⊃ m;scan(key, zbrar, m + 1, len0))].key < key, (¬ zbrar[m].key < key ⊃ m;scan(key, zbrar, m + 1, len0)) ≤ m + len0, ∀ zbrar0, key0, m0. ¬ zbrar0[scan(key0, zbrar0, m0, len0)].key < key0 ∨ ¬ scan(key0, zbrar0, m0, len0) < m0 + len0 ⊦

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)%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)