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Proof step 5 from proof keys-mod (previous step)

m + nused0 < # zbrar, sorted(zbrar, m, nused0 + 1), n ≤ nused0, ∀ 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

User applied rule constructor cut

And obtained the following new goals

Proof step 6:
n = 0, m + nused0 < # zbrar, sorted(zbrar, m, nused0 + 1), n ≤ nused0, ∀ 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
Proof step 12:
n = n0 +1, m + nused0 < # zbrar, sorted(zbrar, m, nused0 + 1), n ≤ nused0, ∀ 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

var: n

Additional information

Used simplifier rules: nat freely generated by 0, +1;;