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Proof step 3 from proof keys-mod (previous step)

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

User applied rule structural induction

And obtained the following new goals

Proof step 4:
n < nused0 +1, sorted(zbrar, m, nused0 +1), m + nused0 +1 ≤ # zbrar, ∀ n, zbrar, m, adr. n < nused0 ∧ sorted(zbrar, m, nused0) ∧ m + nused0 ≤ # zbrar → keys(zbrar[m + n, mkzentry(zbrar[m + n].key, adr)], m, nused0) = keys(zbrar, m, nused0) ++ zbrar[m + n].key ⊦ keys(zbrar[m + n, mkzentry(zbrar[m + n].key, adr)], m, nused0 +1) = keys(zbrar, m, nused0 +1) ++ zbrar[m + n].key

Rule argument

indvar: nused

Additional information

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