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Proof step 7 from proof keys-drop-upper (previous step)

m + nused0 < n, m + nused0 < # zbrar, m < # zbrar[n, zbr], n ≤ # zbrar, ∀ m, n, zbrar, zbr. m + nused0 ≤ n ∧ n ≤ # zbrar ∧ m + nused0 ≤ # zbrar → keys(zbrar[n, zbr], m, nused0) = keys(zbrar, m, nused0) ⊦ keys(zbrar[n, zbr], m + 1, nused0) ++ zbrar[n, zbr][m].key = keys(zbrar, m, nused0 + 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 8:
m + nused0 < # zbrar, m + nused0 < n, m < # zbrar, n ≤ # zbrar, ∀ m, n, zbrar, zbr. m + nused0 ≤ n ∧ n ≤ # zbrar ∧ m + nused0 ≤ # zbrar → keys(zbrar[n, zbr], m, nused0) = keys(zbrar, m, nused0) ⊦ keys(zbrar[n, zbr], m + 1, nused0) ++ zbrar[m].key = keys(zbrar, m + 1, nused0) ++ zbrar[m].key

Rule argument

Additional information

Used simplifier rules: keys-rec: ⊦ m < # zbrar → keys(zbrar, m, nused + 1) = keys(zbrar, m + 1, nused) ++ zbrar[m].key;% ⊦ # zbrar[n, zbr] = # zbrar; (s)% ⊦ n + n0 < m → zbrar[m, zbr][n] = zbrar[n]; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic